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CIHGAN STATE

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This is to certify that the

dissertation entitled

Real Toric Manifolds

presented by

Radhouane Sellami

has been accepted towards fulfillment
of the requirements for

Ph.D. degree in Mathematics

 

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W39- 1

_..——- .27.,

REAL TORIC MAN IFOLDS

By

Radhouane S ellamz'

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Mathematics

1993

ABSTRACT

REAL TORIC MAN IFOLDS
By

Radhouane Sellamz'

In this thesis, we study toric manifolds as a particular case of the monomial
manifolds, and give an identification of the two structures under weak conditions.
Toric manifolds of dimension r have a (Z;)’ action, while their complexifications
have a T' action and the two actions on the real and the corresponding complex toric
manifolds have the same orbit space. For r = 2 or 3, the manifolds with T' action are
well studied, and we use the known results about them to classify the dimensional 2
toric manifolds and give a characterization up to surgery and connected sum with RP3
of the 3-dimensional case. Also we give a Heegard characterization of the orientable
toric 3 manifolds and get a restriction on the manifolds which can support a toric

structure.

To my grandparents, my parents, my wife, and my son youssef.

iii

ACKNOWLED GEMEN TS

I would like to express my infinite gratitude to my dissertation adviser Dr. Selman
Akbulut for all his help, advice and patience with me throughout my work.
My sincere thanks to Dr. Ronald Fintushel for his help, encouragement and support
to complete this dessertation.
I would also like to thank my dissertation committee members Dr. J .Mc Carthy, Dr.
J .Wolfson, Dr. N .Ivanov for their time.
I would like to thank the authorities at the Ministry of Defense and the Ministry
of Education of the goverment'of Tunisia for providing me with the opportunity to
complete my graduate studies in the US.
Finally I would like to thank my friend Ammar Gharbi for his help with some com-
puter skills.

iv

TABLE OF CONTENTS

mmhh

LIST OF FIGURES vi
1 Introduction . 1
2 Real Toric Manifolds
2.1 Construction ...............................
2.2 Nonsingularity ...............................
2.3 Toric Manifolds ..............................
2.4 Compactness ............................... 13
2.5 Equivariant isomorphisms ...................... 15
3 Monomial Structures and Uniformization 19
3.1 Monomial Structures ........................... 19
3.2 Uniformization .............................. 23
4 Study of The 2 and 3 Dimensional Cases 27
4.1 Complex Toric manifolds ......................... 27
4.2 Classification of Compact Toric Manifolds of Dimension two ..... 30
4.3 Cross Sections ..................... A .......... 34
4.4 Classification ............................... 37
4.5 Dimension 3 Compact Toric Manifolds ................. 41
4.6 Cross Sections ............................... 50
4.7 Orientation ................................ 52
4.8 Heegaard Diagrams for Orientable Toric 3-Manifolds ......... 56
4.9 Surgery .................................. 58
4.10 First Homology Groups for Orientable Toric Manifolds ........ 66
BIBLIOGRAPHY 63

4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
4.13
4.14
4.15
4.16
4.17
4.18
4.19
4.20
4.21
4.22
4.23
4.24
4.25
4.26
4.27
4.28
4.29

LIST OF FIGURES

Representation of the orbit space .................... 31
Orbit space of the orthogonal action of T2 on D“ ........... 32
X; ..................................... 33
Duality picture .............................. 34
Orbit space of the action of (Z2)2 on D2 ................ 34
The disk .................................. 35
M ‘ and cutting it into cones ....................... 36
Cutting an edge .............. ' ................ 37
The graphs and their dual fans with t = 3, 4 .............. 38
Two different cases for the graph with t 2 5 .............. 39
Example of reducing G to the triangle and the dual action on fans . . 40
Obtaining Mg from G when t = 3,4 ................... 42
Reducing M to MlthP2 ......................... 43
(Ds)‘ .................................... 45
Example .................................. 48
G ...................................... 49
(D3) ..................... 50
Cutting G into cones ........................... 51
G ...................................... 52
H ...................................... 53
Cutting H ................................. 53
An edge .................................. 55
RP3 .................................... 59
A,GandPG ................................ 60
H1 ............................ I ......... 61
Surgery .................................. 62
H1 for the surgery ............................ 63
Reducing G ................................ 64
Blowing up ................................ 65

vi

CHAPTER 1

Introduction

Algebraic spaces are topological spaces modelled on algebraic sets with the glueing
functions being birational isomorphisms. If in addition, the charts are nonsingular
then the space is a smooth manifold.

Toric manifolds are a particular case of nonsingular algebraic spaces,where the glueing
maps are monomials [9, 8, 5].

In [1] S.Akbulut discussed the real algebraic structures on smooth manifolds where a
real algebraic space (X, {¢0}) is a topological space X with a collection of imbeddings
4),, : Sc. - S; —> X such that S; C 5., are real algebraic sets, the images of 43,, cover

X and
i Each Tap = 5", U (if (X — image 435) is a real algebraic set
ii Each 455143,, : (Sc. — Tag) —> (53 — T3,.) is a birational isomorphism.

And in particular he considered the rational algebraic structures where a rational al-
gebraic space is a real algebraic space with the extra condition that S, = R”, and he
asked the questionzls every smooth compact manifold diffeomorphic to a nonsingular
rational space?

In this paper, we look at a particular case of rational structures, namely the case

where the transition maps are monomials. It turns out that the toric manifolds [9]

l

admit this monomial structure i.e the real compact toric manifolds of dimension n
are obtained by glueing copies of R“ using monomials as transition maps (so they
are a particular case of rational manifolds). Conversely, here we give a necessary and
sufficient condition for a monomial manifold to be toric. Also we show that the toric
structure is not developable. Then we specialize in the toric structures on compact
2 and 3 dimensional manifolds. The coefficients of the monomials are determined
by a collection of integral nonsingular cones of dimension n, called a fan. Starting
with the same fan, we construct a complex toric manifold by glueing copies of C"
together with the transition maps being the same monomials as in the real case, this
gives a natural complexification of the real toric manifolds. These complex manifolds
admit a T" smooth action on them, this action induces an action of (22)" on the
corresponding real toric manifolds and the orbit spaces of the two actions are equal,
while the isotropy groups in the real case are (22)“0 ( isotropy groups in the complex
case ).

For the cases we are interested in, the orbit spaces are dual to the fans, hence by
starting from a fan we obtain directly the orbit space, without the need to identify
the manifold.

If n = 2, the orbit spaces are D2 with weights on the boundary 5'1 [10]. The 2-
dimensional toric manifolds are obtained by glueing‘four copies of the orbit spaces
along the boundary, and we are able to identify all compact real toric manifolds of
dimension 2.’

If n = 3 the orbit spaces are D3 with weighted graphs G on the boundary 52 [7], the
corresponding toric manifolds are obtained by glueing eight copies of the orbit spaces
following the informations given by the weights on S2 [7]. The graphs corresponding
to orientable manifolds are colored by only 4 colors, and to identify the orientable
toric manifolds (and hence to partially answer the question of S.Akbulut in the case

of toric structures), we glue the eight copies of the orbit space along the cells sur-

rounding a vertex in the graph, so that we obtain a 3 ball with a graph PG on its
boundary. The cells on PG are identified two by two. We bore out small cylinders
around the edges of the graph G as done in [12] to obtain a Heegaard representation
of the manifold.

We show that all orientable compact toric 3 manifolds are obtained from RP3 by
a sequence of blowing up points (the fixed points of the (Z2)3 action), which corre-
sponds to connected summing with RP3, and 1/ 2 surgeries along some special circles
corresponding to edges in the graph. Choices of these circles are importants since all
3-manifolds can be obtained from [11.5“ x S2 by 1/2 surgeries [2]. We use this rep-
resentation to draw some conclusions about the homology groups of these manifolds
and show that some 3—manifolds such as lens spaces L(2s + 1, q) can not admit toric

structures.

CHAPTER 2

Real Toric Manifolds

2.1 Construction

In this section we will recall how toric manifolds are constructed from rational cones
[9]. Let N = Z', M = Hom(N,Z), NR = N®R r: R', M3 = M ® R and let
<, > denote the duality pairing of M and N as well as its extension to MR and NR.
N and M are groups in the obvious way. We assume througout the paper that all
splittings NR = V 65 W (where V and W are rational subspaces) have the further
property N = (N n V) $ (N n W) and likewise splittings in MR.

Definition 2.1 I. A subset o of N3 is a convex polyhedral cone, and denoted in
short by crpc if there exists a finite set of vectors {n1, . . . , n,} of N3 such that
0’ == Rgoni + - - - + R20".-

2. A convex polyhedral cone in N3 is called rational if its generating vectors

{n1,...,n,} are in N.

3. Such a o' is called strongly convex rational polyhedral cone and denoted in short

by scrpc ifo n (-o) = {O}.

4. For a crpc a we define (lime to be the dimension of the vector subspace of NR

generated by a

Definition 2.2 Let a be an scrpc in N3. Then

av {yEMR|<y,x>ZOVx€a)

{y 6 MR I< y,n,- > Z 0Vi=1,...,s} is called the dual ofo

01' = {yEMR|<y,x>=OVx€o'}

into is the usual interior of a regarded as a subset of the real vector space Ra'.

Remark :

By theorem 19.1 [11] there exist m1,--~,m, in MR such that o" = Rzom1 + +
Rzom: and since a is rational i.e the generators {n1, . . . ,n.} are in N, it is easily
seen that the m,- can be chosen to be in M, hence we get that o" is a crpc but not

necessarily strongly convex.

Proposition 2.1 Let a be an scrpc in NR then
I. (0")V = 0'.
2. (0 F) o’)" = o" + 0".
3. xeintoé <x,y>>0 VyEoV\o‘L ©0"fl{x}* =04.

Proof: Theorem A.l, lemma A.4 in [9]. 0

Definition 2.3 Leta be an scrpc in N3, define S, = Mno" = {y 6 M |< y,x >2
0 Vx 6 0'}

Proposition 2.2 Let a' be an scrpc, then

N

. So is a subsemigroup of M.

(9

. S, is finitely generated as a semigroup.

5°

5', generates M as a group.

Proof: (Prop 1.1 in [9]). c1

3 Definition 2.4 Let a be an scrpc in N3. Define
U, = {11:33, -—e R I u(m + m') = u(m)u(m’) and u(0) = 1}

Remarks :
Let S, = ZZ°m1 + + Zzom, for some (m1,...,mp) C M, then every u in U, is

completely determined by (u(m1), . . . , u(m,)) i.e

U,—vR’

u u—-> (u(m1), . . . , u(m,))

defines a coordinate system on U,.

Proposition 2.3 If we identify U, with its image in R", then

U, = {(x1,. . . ,xp) 6 R’ I xi" ...x:’ = xf‘ . ..xg’for allomfli 6 220 wicha,m,- = Zfi,m,}

so U, is an algebraic subset of R".

Proof: (Prop 1.2 of [9]). U

2.2 Nonsingularity

Proposition 2.4 U, is nonsingular if a is generated by a Z subbasis of N.
Definition 2.5 We call a cone generated by a Z subbasis of N a nonsingular cone.

Proof :

(4:) Let 0 = R2071: + + Rzon, where (m);l is a Z subbasis of N, we complete

this subbasis to {n1,...,n,} a Z basis of N and let {m1,...,m,} be the dual Z
basis in M, then a" = Rzom1 + + Rzom, + Rm,“ + - -- + Rm,, and for u in
U,, u(m,-) 7‘- 0 for p + 1 S i S r, since m,- and -m,- are in o" for such 2'. Therefore

U',=Rx---xRxR‘x-~ x113

 

F J,
(=>) Let Nil be the smallest vector subspace of N3 generated by 0', then NR = NfieV

where V E at. We will first show that we can assume Nil = N3. Otherwise, let
M]; = Hom(Ni1,R), we can view Mil as a subspace of Mg by letting m(V) =
0 Vm 6 Mil: hence Mil 21’ MR/oi, so MR 3 Mi 69 a", and M E M’ EB (M fl 0*).
Since 0' C Nil and if 0” denote its dual in Mil: then a” is the image of 0" under
the above identifiCation. Let S; = M’ {'1 0” hence S, = M n 0" E (M n at) x 3;.
Let (m1, . . . ,m,,m§, . . . ,m;) be a family of generators for S, where (m1, . . . , m,) are

chosen to form a basis for the vector space a", and (m’1,.. . , min) are generators for

5",. Since there is no relation between (mg), and (mg-)1. then U = U x U; where

U = {u : M n at ——; R' | u(m + m’) = u(m)u(m') and u(0) = l}

g R‘XouxR:

‘1

 

Therefore U, is nonsingular ifi' U}, is nonsingular. So we assume that or generates NR
i.e dima = r, hence 0" is strongly convex because (5V) (1 (-o") = a"L = {0}, hence
0 6 U,. Let

U, = {(x1,. . . ,xp) 6 R’ | x‘," ...x"’ = x?‘ ...x£’for alla;,fi.- 6 Z20W1th20imi = Efiimi}

P

LCt {mi}? be a minimal set of generators of 5, so that there is no i such that m,- =
23-1 ajmj and a, E 220. We prove that p = r.
J¢i

Let auml + - - ~ + agpmp = flamI + - - - + flgpm, with the condition that if my > 0 then

fig,- = 0 and if 13,-, > 0 then 01,-,- = 0. We remark that U, contains Um} E R" as an

open subset, hence dim U, is r. If U, is nonsingular then it is nonsingular at 0, thus

there exists a finite number of polynomials {f.- = x?“ .. .253" - x?“ . . . xfi") in [(U,)
a - - a

such that rank (55;- [0) = p — r, and ifp 96 r then rank (55;- [0) > 0.

Assume without loss of generality that 3% 75 O with f1 (1:1,. . .,x,,) = x?“ . . a?" —

1‘13” . . . mg", hence

_ a —1 a 311-1 B

so

 

 

0

If on = 0 then 3% lo: -fl110”"'109” . . .051? which is difi'erent of zero only if flu = 1
and fi,,_ = 0 Vj = 2.. . p i.e m1 = 021722 + ' - - + apm, which contradict the hypothesis.
Therefore an 2 1 then by assumption flu =0, but this is just the same replacing an-
by 311, hence there is no relation between (mg), i.e (m,) form a basis for M. Hence
(11,-) form a basis for N. D

From now on we assume that all our cones are nonsingular.

2.3 Toric Manifolds

Definition 2.6 Let a be a scrpc in N3. A subset r of a' is called a face of a (

denoted r < o) if there exists mo in (7" such that r = a n {may

Proposition 2.5 I. Since a is rational then mo can be chosen to be in 5,.

2. By definition 1' is also an scrpc.

Proof: (Prop 1.3 of [9]).

Proposition 2.6 If r < a are nonsingular cones then there exists {111, . . . ,n,} a Z
basis 0f N such that O’ = R2071} + ° - - + R2012? and T = R2071] + ° ' ' + R2071, with

1333p.

Propositionv2.7 If r < a so that r = 0’ fl {y}‘L for some y in 5', then 7'" =

UV + R20(—y), and Sf = So + ZZO(-y)'

Proof:

r<o¢>r = ofl{y}iforsomey65,

= a n (Rgo(y))v n (Rzo(-y))"

hence 1'" = o" + Rzo(y) + R20(-y) by proposition (2.1) but y 6 0" hence 1'" =

0" + R20(-y). ' Cl
Proposition 2.8 If r = 01 0 a; is a face of both 01 and 02, then S, = 3,, + 5,,.
Proof:

(2) r" = (010%)" =o]’+o;’,henceS' =Mfl(0'1r'102)" =Mfl(o';’+oz)’) D

(Mfloi’)+(Mflog’)=S,, +3,,.

(9;) The proof is by induction on dim 0’; + (1111102. We assume that 0',- ¢ 0,. Then
into; 0 into; = 0. By the separation theorem in [11] there exists a hyperplane
H of N3 such that 01 is contained in one of the closed half spaces limited by H,
and 0'; is contained in the other closed half space, and since into, 0 into; = 0
we can assume that they do not both lie in H. Now let H = {mo}; for some
mo in MR, then since 1' C H and r is a rational cone, mo can be chosen to be in
M, so 01 and a; lie on mutually opposite sides with respect to the hyperplane
{mo}"', so that 01 C {x 6 NR |< x,mo >2 0}, hence mo 6 S,,, and
a; C {x 6 NR l< x,mo >5 0} = {x 6 NR |< x,-mo >2 0} hence (—mo) 6
5,2. 1' = 0'1 002 C o,- n {mo}‘L for i = 1,2.

Let 0'"- = o,- 0 {mo}*, note that since 01 and a; don’t both lie in H then

dime; + dime; < dimol + dimoz. Then 5,; = 5,, + ZZo(-mo) C So, +

10

5,,, 5,5 = Z20(mo) + 5,, c 5,1 + 5,,.Hence 5,; + 5,; c: 5,l + 5,,

Since 1' = a; n 0;, the induction hypothesis implies that S, C 5,; + S,;. C]

Proposition 2.9 Let a be an scrpc in N3, and r < 0' then U, is an open subset of

V
U,.

Proof : Since we are assuming that our cones are nonsingular, we give a proof
for that case only. Let 0' = Rzonl + + Rzon, where n1,...,n, is a Z -basis
for N3 hence o" = Rzoml + - - - + Rzom, + Rm,“ + + Rm,, and let 1' < 0'.
Without loss of generality we can assume 1' = RzonI + + Rzonjors S p, then
we have 1'" = Rzoml +---+Rzom,+Rm.+1 +~~+Rm,, so S, = Zzom1+~~+
Zzom, + Zmp.” + - -- + Zm,, S, = Zzoml + - - - + Zzom. + Zm.+1 + - - - + Zm,, and
U,2Rx...xRxR‘x...xR3Usz...xRxR‘x...xR;. D

f

 

v

p r-p I r-s

Definition 2.7 A fan in N3 is a nonempty set A of scrpc in N3 such that:
i IfoEA and r<o ,theanA.

ii Ifo',o’€A , then oflo’<o and oflo’<o’.

Theorem 2.1 Let A be a fan in-NR. Then we can naturally glue {U,,a E A}

together to obtain a manifold XA = U,EAU,.

Proof : We have to prove that X A is Hausdorff, which is equivalent to proving that

the map

X4 -—-) XAXXA

u r—» (u,u)

is closed.

The only problem is with the identification of U, in U,, with U, in U,2 where

11

r = 01 fl 0;. Now let 5,1 = Zzoml + + 220m, and 5,, = Zzom’1 + + 220mg,
‘ then by proposition 2.8 5, = Z20 + - - - + Zzom, + Zzom’1 + - -- + 220m; (we do not
exclude the cases m, = -—m,- or m2 = -—m;-). We identify U, (resp U,,, respU,,) with
its image in RP“ (resp RP, resp R9). Thus U, and U,1 x U,2 can be regarded as
closed subsets of RP“ and since U, is contained in U,, x U,2 therefore U, is closed
in U,, x U,,. C]

Remark :

1. {0} 6 A, then UM = {u : M —-. R I u(m +m’) = u(m)u(m’) u(O) = l}, and
sinceM = Zm; +~~~+Zm,, we have U{0} ER‘ x x R‘.
We will denote U{0} by T.

2. For every 0' in A we have {0} < a, then T is open in U, for every 0 in A.

Therefore 7 is open in X a-

3. Let t be in T and u be in U,. Define tu by (tu)(m) = t(m)u(m) for m in 5,;
This defines an action of T on U, and by natural glueing on XA. So the real
toric manifolds are a particular case of manifolds with R‘ x . - - x R' action,
and in particular a Z; x - . - x Z; action. We will see that there are manifolds

which have Z; x - - - x Z2 action but they are not toric manifolds.

Example :

Let a; = Rzofli + Rzonz, where {n1,ng} is the canonical basis for R2, and a; =
Rzo(an1+bng)+Rzo(cn1+dng) where ad—bc = 1. Then a)’ = Rzom1+Rzom2, 5,1 =
Zzom1+Z20m2, a; = R20(dm1 —cm2)+Rzo(-bm1+amg). So u 6 U,, is determined

by (u(mI),u(m2)) and u E U,2 is determined by (u(dm1 - cm2),u(—bm1 + 07712))-

12

And we have the following commutative diagram :

T x (U,, n U,,) c U,, —+ (U,, n U,,) c U,,
(t1,t2),(x1,x2) +-—> (t1x1,t2x2)
l l
T x (U,, n U,,) c U,2 —-» U,, n U,,) c U,,

(
(ta-a trbts). (river. xrbxs) (ta‘xitrzr. trbzr‘tsxs)

I

Definition 2.8 Let a be an scrpc in N3. Define.-
orba = {u : M n 01' —-+ R“ group homomorphism}

Remark :
orbo is canonically embedded in U, by letting u(m) = 0 for any m in 5, not in

Mn 0"”.

Proposition 2.10 Let A be a fan in N3.

For every 0 6 A let orba be as above. Then we have:

. Every ’1' orbit in X4 is of this form and in this way.

N

2. [fr < 0' then orbr C U,.

3. Va 6 A, U, = U,<,orbr.

4. A is in one to one correspondence with the set of T orbits in X A.
5. Fora,r€A r<a¢>orboCo_rE.

6. 3:5; = U,<,orbo.

7. orb(0) = T.

13

Proof: (Prop 1.6 [9]). C1
Example : Let a = Rzonl + Rzong, then A = {{O},n = Rzonh’rg = Rzon2,0'}
is a fan. An element u in U, is determined by (u(ml),u(m2)) and orb{0} =
R“ x R‘, orbrl = {u : M n 134' —> R‘, group homomorphism} = {u : Zmz —,

R' group homomorphism} 2‘ 0 x R‘, 01‘ng = R" x 0 and orbo = 0 x O.

2.4 Compactness

Proposition 2.11

. A is finite
X4 is compact 4:
and I A ]= NR where I A I: U,5Ao
Proof : (=>) XA is compact.

Let A’ = {maximal dimensional cones in A} then XA = U,eA:U, and XA is not

covered by any proper subset of A’, hence
XA compact 2 A’ is finite = A is finite.
Let n 6 N and A E R‘, define

7n(’\) = M —-* R'

m ,__, A<n.m>
Then 7,,(A) E T for every )1.

XA is compact => limA-eo 7n(A) E XA => 30' E A such that limg_.o 7,,(A) 6 U,. i.e
limA-.OI\<"’"‘> 6 U, = <n,m> 20 Vm65, =>n60forsomeo€ A => IA |=
NR.

(<=) Let T = {u E T | u(m) = :t1,Vm 6 M} 2 (Z2)" and

l4

1- : R20n1+~- + Rzonn where {n1...n,} is a basis of N,

TV = Rgomi + + Rzomt + Rm,“ + + Rm,. Let u 6 orbr so that u =

(0.. . . ,O,u(m,+1),.. . ,u(mr)) With u(m.) ¢ 0 for i = t + 1, . . . ,r, we then have:
V

t

orbr / T

III

III

{(0,...,0,x¢+1,...,x,) where x,- > O}
R>Ont+1 + ' - - + R>°nr
Rn.“ + - - - + Rn, ( using the function - log)

NR/RT

Since T is a compact group and acts on XA, it is enough to prove that X, / T is

compact.

XA/T

U (oer/ T)

PEA

U (Na/RP)
PEA

U ( U (a + Rn/Rp) (since I A' |= NR)

PEA OEA’
P<0

U (U (0 + Rp)/Rp)

aEA’ pea
p<o.

Uwo

oEA’

It is enough to show that w, = U,.?(o' + Rp)/Rp is compact Va 6 A’ (since A is
p a

finite).

Let 0' = Rzonl + - - - + Rzon, be a nonsingular cone, and p = R2071] + - - - + R2011, be

a face of a. The image of (o + Rp)/Rp under the identification NR/Rp E orbp/ T

' is given by:

(a + Rp)/Rp g

(Rnl “1" ' ' ' ‘l" Rn: 'l' Rzonc+1+ ' ° ' + Rzon,)/Rn1 + ° - ° + Rn,

RZOnH-l + ' ' ° + RZOnr

15

g (0.1]714.” + ' ' ' +(0,1]n,

Ill

{(0,...,0,$¢+1,...,I,) ] 0 < 513,31}

hence

U(o+Rp)/Rp E U {(x1,...,x,)|x,-=Oifi65,andO<x,-Slotherwise}

pEA SC{1 ..... r}
P<0

III
:5
O-d
“'2‘

which is compact. D
So the compact real toric manifolds of dim r correspond to complete nonsingular fans

in R'.

2.5 Equivariant isomorphisms
Let (N, A), (N’, A’) be fans. N : Z',N’ : z"

Definition 2.9 A map of fans (N,A) ——9 (N’,A’) constitutes of a Z linear homo-

morphism (,0 : N —> N’ whose scalar extension ip : NR —-> Nil satisfies the property
Va 6 A 3 a’ 6 A’ such that cp(o) C 0’

Theorem 2.2 Let cp : (N,A) —» (N’, A’) be a map offans. Then cp gives rise to
a smooth map 90.. : X4 —-> X ’A, which is equivariant with respect to the actions of
T and T’ on X4 and X A. respectively. Conversely : If 7 : T ——> 7" is a group
homomorphism and f : XA —-> X3. is a map equivariant with respect to 7, then
there exists a unique Z linear homomorphism cp : N —t N’ which gives rise to a map

offans gs : (N,A) --+ (N’,A’) such that f = 90..

16

Proof : Let (,0 : N —> N’ be a Z homomorphism, we define :

90': M’-—-> M
m’ i——i cp'(m’) where < cp‘(m'),n >=< m’,<,o(n) >

Suppose 90(0) C 0’, then Va 6 o,Vm’ 6 o’v < go‘(m’),nl >=< m’,ap(n) > 2 0
therefore o-(a'v) c o" and o-( 3,.) c 5, Define cp. : XA —. X’ . as follows:

Let u e XA then there exists a E A such that u E U,, let a" E A’ such that
90(0) C a", then define cp.(u) in U},. such that 9p.(u)(m’) = u(cp‘(m’)) for every
m in 5",“ We have to prove that such w. is well defined and equivariant: So let
u 6 U,1 fl U,2 = ,,n,, = U, let 1" E A’ such that cp(r) C 1" then go. is well defined
in U; hence go. is well defined. Now we prove the equivariance of <p.: cp.(tu)(m’) =
t‘U(<.0"(m')) = t(‘P'(m’))u(<P'(m')) = v-(t)(m’)¢-(u)(M’) = (SO-(t)<e.(U))(m’), hence
so. is equivariant.

Conversely: Let 7 : T —-+ T’ be a group homomorphism and define

cp' : M’ g Hom(’1", R‘) —» M E Hom(T,R‘)

m’ *-* ¢'(m’)(t) = m’(7"(t))
then we get

(,0: N—> N’

n r—-» 90(n) where < cp(n),m’ >=< n,<,o‘(m’) >

Let f : T —-i T’ be an equivariant map with respect to 7, we want to prove that
there exists a unique Z linear morphism so such that Va 6 A,30" 6 A’ such that

cp(o') C 0’. Let a' E A, then 30’ e A’ such that f (orbo) C orbo’ by equivariance

17

of f . Let r < 0 and let 1" E A’ such that f(orbr) C orbr’. Since 1' < 0 then
orba C orbr => f(orbo) C f(orbr) C f(orbr) C orbr’. Since f(orb0) C (orb0’)
we get orbo’ C orbr’ which is equivalent to r’ < 0’. But U, = Ur<00rbT, hence

f(Ua) C U,<,f(orbr) C Ur’<0’orb7’ = U;.. Now let n E 0 then lim1_.o 7,,(A) e U,,

therefore
[f 0 7n( )Tn]( =([7 ((‘nA ))’)l(m )(because 7,0) 6 Tandf = 70117)
= m'[7(7n(l))l
= «e'(m’)[7n(*)]
= A<¢(n).M’>
= [7w(n)(*)l(m’)-

i.e f o 7,, = 7,0,), but f is continuous hence lim,\_.o 7,(n)(z\) = f (limx...o 7,,(/\)) E U;,
(since limlno 7,,(A) e U, and f(U ,) C U’ ), therefore cp(n) 6 0’, so 90(0) C 0’. D

Remarks :

1. Suppose 0" = Zzoml + + ZZom, and 0’V = zgomi + ' - ' + Zzom; and
cp'(m£) = Zaijmj where aij 6 zzo, let u E U, and u = (u(ml),. . .,u(m,,)) =

“'1

(x1,.. .,x,) then cp.(u)= (xm ... ., :1:1 .xgfl') i.e the equivariant maps

are represented by monomials.

2. By the construction of (p. we see that X A and X A. are equivariantly difieomor-
phic if and only if cp is an isomorphism between N and N’ such that for every
0 6 A there exists 0’ such that 90(0) = 0’. So the classification of closed real
toric manifolds of dimr up to equivariant homeomorphism is equivalent to the

classification of complete nonsingular fans in Z' up to fan isomorphism.

This ends our review of toric manifolds. In the next chapters we will show how they

arise naturally from special types of rational structures, and discuss their classification

in dimensions 2 and 3.

18

CHAPTER 3

Monomial Structures and

Uniformization

3.1 Monomial Structures

Let A be a complete nonsingular fan in R'. Then XA is a compact manifold of
dim r. Let 0 (resp 0’) be maximal cones in A, and let {n1, . . . ,n,} (resp {n;, . . . , n’,
) be the generating basis for 0 (resp 0’). Without loss of generality we assume that
{n}, . . . , 12,} is the canonical basis for R', then n:- = 2;.“ aijnj with a,,- 6 Z; letting
A = (a,,-) we get 722 = A‘n, and since (71,-) and (121) are Z bases, then detA = :tl
(without loss of generality we assume it is +1).

Let 0" = Rzoml + - - - + Rzom, and 0’" = Rzomfl + - - - + Rzom; be the dual cones.
Therefore {< m£,n£ >= 1 and < m$,n;- >= 0 ifi 76 j} 4:

{< m;,A‘n,- =1 and < m;,A‘n,- >= 0 ifi #j} 4:

{< Am§,n,- >= 1 and < Am£,n,- >= Oifi 94 j} 4: {m,- = Amf}, hence letting

'r=0r‘10’wehave:

go: U,=U,OU,ICU,: ——*U,CU,

(u(m’l), . . . ,u(m;)) o—-i(u(m1),...,u(m,))

20

So if we let (x1, . . . ,x,) = (u(m'l), . . . ,u(m’,)) then u(m.) = u(a1,-m’1 + a2,-m"2 + ---+

a,,-m:,) = u(m’l)“1' . . . u(m,)°" = 2:?" . . . 1:2" i.e (,9 is defined by:
(2:1,. . . ,r,) .--—+ (x‘f‘lxgz‘ ...x2'1,...,x‘1‘" . . .x3")

50 we see that toric manifolds have a monomial structure, that is they are covered
by charts with the transition maps being monomials, and by the construction above,
we see that the transition maps are completely defined by the coordinates of the
generating vectors of the different maximal cones of the fan.

Now considering the transition map cp : U,(C U,:) —-+ U,(C U,) we remark that
‘PIUm : Um} E (R‘)’ —-> U{0} ’5 (R')' is a homeomorphism (since detA = 1). The
following theorem shows that the toric manifolds are a special case of the rational

manifolds defined in [ ], with the rational functions being replaced bymonomials.

Theorem 3.1 A compact manifold M is toric if M is covered by a finite number of

charts (U,,cpg) such that
1. (1,3 11'.
.2. (p, 0 «pi-1 are monomials.
3. U,- is dense in M for every i.

Lemma 3.1 A monomial (p is a transformation map for a compact toric manifold if

and only if cp and so“ are homeomorphisms on their respective domain of definition.

Proof :
(=>) We consider cp : U, ——* U, as above. The only possibility to extend (,0 is along

some of the coordinate axes (since cp is defined on(R‘)’ ) but

tp: U, —->U,

(mum-n) ~—~()

21

so go can be extended only to the i”‘ axes where a,-,- 2 0 Vj, without loss of generality
,assumei= l. SupposeRxR’ x--- xR" ¢U, i.erR‘ x xR' C U, and that
up is defined on 0 x R‘ x x R', which means that a11,...,a1, 2 0 then we have

either one of the three cases:
(i) All a1.- = 0 then det A = 0 contradiction.

(ii) A unique an, 75 0 => an, == 1 and n’1 = n.-,, therefore Rzon; < 1' hence x1 6 R

i.erR'x---XR‘CU,.

(iii) There exists at least two j1,j2 such that a1,,,a1,-, > 0. Therefore 7:; is in the
interior of a face of 0 namely the face generated by the n;- where 011' > 0, but
this contradicts the definition of a fan. Therefore go is defined only on U, and

by the same method to" is defined on U,.

(<=) Let

co : A —-+ B
(x1,...,x,) o—v (x‘f’l ...xfi",...,x§"...xf,")

be a homeomorphism from A to B and such that go can not be defined on a set bigger
than A, and cp“ can not be defined on a set larger than B. so is a homeomorphism
from R" onto itself, hence det(a,-j) = :i:1, we assume it is 1. We will prove that the
cones 0 = Rzonl + - .. + Rzon, and 0’ = Rzon’1 + - - - + Rzon; where n: = E,- agn,
intersect on a face.

Suppose that cp is defined on R x R‘ x - - - x R‘, so a1,- 2 0 Vj. We have the following

cases:
1. All a1, = 0, then det(a,-,-) = 0, which is not acceptable.

2. There exists a unique an, # 0, hence ab, = l i.e n’l = n,,.

22

3. There exist at least two j1,j2 such that “12': , “1:“: > 0. Without loss of generality

we can assume that co is of the following form:

_ a a1 a a 62 +1 a a a
¢(x1,oce,xr)—(1111000xrr,eee,x11‘oooxrr‘,z2’ ...zrr.+l,...,I22r-..Ir')

r

Hence cp(0,x2,...,x,) = (0,...,0,x;”“ ...x:"+’,...,x;" ...xfi") i.e cp(0 x
R' x x R') C (0 x x 0 x R‘ x x R') injectively which is impos-
sible.

Now if cp is defined on R’ x R"" then n], = m, for 1 5 k S s with i], 71E 2', if k 75 I,
so 0 fl 0’ = Rzon’1 + - -- + Rzon’, = Rzong, + . - - + Rzong, which is a face of both 0
and 0’ because of the nonsingularity of the cones.

So we proved that no face of 0’ is in the interior of a face of 0 unless it is equal to it.
And we prove the same result for the faces of 0 using cp'l D

Example : Let

go: R’XR‘ -—>R'XR'

(may) *—' ($231". fly)

so is a monomial which is a homeomorphism on its domain of definition, but cp is not

a toric transition map because:

90“: R‘XR‘ -—vR’xR‘
($.11) *—-* (rm-15y“)
can be extended to R2 but not as a homeomorphism. Infact (o arises from the cones:

0 = Rzonl + R2011: and 0’ = R2°(2n1 - 12;») + RZ°(-n1 + 7n) and these two cones

do not intersect along a face.

23

Proof of the theorem:

(=>) this is verified by construction.

(c) Having a covering of M verifying the three conditions we need to exhibit a fan
that corresponds to M:
Let (I) = cpl o (o;l : w2(U1 ('1 U2) C R’ —r go1(U1 (’1 U2) C R' if we prove that
902(U1 (1 U2) is the domain of definition of <1> (as a monomial in R') we would
have two r-dimensional cones, and they would be intersecting along a face by
the previous lemma and by repeating the process for all i, j we get a fan that
corresponds to M. So the only thing we have to prove is that tpg(U1 0 U2) is

the domain of definition of (D (the same procedure will apply for 11>").

(Q) this is verified by definition.

(2) let x 6 domain Q C R' = gog(Ug). We want to prove that x E go2(U1 0
U2) i.e cp;1(x) 6 U1 n U2. U1 and U; are open dense in M, therefore
U1 0 U2 is dense in M, therefore there exists a sequence (t,),. C U1 0 U,
such that (t,) converges to (p;l(x). Let y, = 9920,.) and 2,, = go1(t,,) =
(o1(¢p§'1(y,,)) = <I>(y,,). Since lim,,_..3° t, = cp;1(x) then lim,,_.,, y, = 2:
therefore hm..-“ My“) = ¢(x) i.e lint...” 2,. = 0(x), but 2,, and <I>(x) are
in w1(U1) 2 R' therefore lirn,,..,o ta = lim,..,, «of 1(3,.) ,1 cpf1(<l>(x)) hence
<P1'1(‘1’(I)) = cFEW) therefore NI) = (P1 0 5051(3) i-8 2 6 ¢2(U1 0 U2)- 0

3.2 Uniformization
Let 5 be a topological space, G a group of homeomorphisms of S such that every g
in G is determined by its action on any open subset of 5 [6].

Definition 3.1 A topological space M is said to be uniformized by (5, G) if there exist

an atlas (U,,cp,), covering M such that co, : U,, —+ «p,(U,) C 5 is a homeomorphism

24

and gag = co, 0 (p51 are restrictions of elements of G.

Consider the following bundle f with base M and fiber 5

x = x’ inU, ('1 U3
E5 = U,(U, x 5)/.., where (x,s,) ~ (x’,sg) iii

so: = 90633
The bundle E is determined by its characteristic class p : H1(M) -—i G which is
called the holonomy representation of the uniformization. Let K. = kerp, MK = the
corresponding regular covering space of M; let p : MK -—r M then px = p o p. :
<l>1(MK) —v G is trivial, hence p.£ is trivial on MK i.e Em, z MK x 5 . Let

0: M —*E5

3 *-+ it. 9043)]

anddefineogsz—vEp.£,6K=P200K:MK-+E.g—r5

Definition 3.2 6g is a local homeomorphism called the developing map of the uni-

formization.
Theorem 3.2 The toric structure is not uniformizable.

Proof : Assume that the toric structure correspond to a uniformization by (5, G),
where G is isomorphic to a subgroup of the group of monomials so that G is discrete,
and we compute the corresponding holonomy. Let M be a toric manifold, consider
the bundle: as above and let 0 be a curve in M from xo to 2:1 , denote by 5, the fiber
, over 0(t). Let ho : 5 —-v 50 be the identification map, then there exists a bundlemap
h : I x 5 —+ E such that h(0,s) = ho(s) and ph(t,s) = 0(t).

Denote 0” = hooh,’1 : 51 -—* 50 where h.(s) = h(t, 3). Since G is discrete, 0“ depends

only on the homotopy class of 0.

25

If 0 is a closed curve then 0n : 50 —-+ 50 and we can regard 0” as an element of G.
Let 0 C U, be a curve from 1:0 to 2:1 and ho(s) = <1),(xo,s) where (P, : U, x .S' ——»
p"1(U,) is a trivialization chart for the bundle over U,, then h(s, t) = <I>,(0(t), s) =

¢,,,(,)(s) hence 0" = 0a,,(o)<l>;;(1) where s is in 51. If 0 is closed then 0‘1 = 1.
If 0 = 01.02 with 0,- C U.- then 0” = 0’] - 0% = <I>,,,,,(0)03"(,)Qp,,,(o)<1>§:,2(,). Working

with the associated principal bundle we have:

(p1: U1 X G -—*P-1(U1)

(1,9) *—> [1.9]
and

(1,1,: G -—>p'l(x)

g H [3:9]

where [x,g] = [x,gug] if x = U1 (1 U; with gm = cpl o «pg-1.
Let 0 = 01 -02 with 0,- C U.- and 02(1) = 01(0), then 0” 6 G acts on the fiber

0,,(0) = Gaza)

a“[o:(1).g] = so.(amt.m°2m<v°$2m[”3“l’9]
= o1,,,(o)<1>,"},,(1)92.a,(0)(9)
= ‘1’1.c1(0)‘1’1-,ln(1)l°'2(0)i9]
= ¢,,,,(,,o;;1(,,[ai(l).9129]
= @1,a,(0)(9129)
= [01(0).9129]

= [02(lligl

26

i.e 0” = 1.

. Now since M is toric then the charts are dense in M, hence gag 0 gm is defined on an
open set of M and we get gay 0 ya, = g,., ( which is not the case in general since if
U,, 0 U5 0 U, = 9 we cannot define gag 0 gm).

By the same method as above we prove that 0n = l for a general 0 therefore:

p: I11(X,xo) —vG
0 r—-+0"

is trivial which implies that E 2 M x 5 hence the map

(5sz E —v5

I *—* (mun) '—* %($)

is a local homeomorphism since the holonomy is trivial.

If M were a compact manifold then 6 would be a covering map so we see that all
compact toric manifolds are covering spaces of 5 which can not happen since as we
see in the next chapter in the example of dimension 2 the torus and RP2 are toric
varieties and obviously they can not cover the same space. D
Remark : I

We could have concluded the non uniformizability of the toric structures by observing
that if M has an (.S', G) structure then any covering space of M has it also. But we

wanted to show that this is the case because of the holonomy triviality .

CHAPTER 4

Study of The 2 and 3 Dimensional

Cases

4.1 Complex Toric manifolds

We consider the compact complex toric manifolds, they arise in the same way as the
real ones. In this section we will give a short overview of these manifolds, for more
details see [9]. Let A be a nonsingular complete fan in N3, 2:: R” and let 0 be in A.
Define U? = {u : 5, —-> C | u(m+m’) = u(m)u(m’) u(O) = 1} and X3 = U,EAU§.
X E is a simply connected compact manifold of dimension Zr and the transition maps
are monomials with the same coeficients as in the real case. (C‘)" acts smoothly on
X E , therefore we have a smooth action of the r-dimensional torus T’ on XE, since
T' C (C‘)'. The action of (C‘)' is determined by the same coeficients as in the real
case, and we have similar results for the complex case as in proposition 2.10 ( see
Prop 1.6 in [9]).

Proposition 4.1

XE/T' = may

27

28

Proof : Since U5: (resp U, ) is invariant under the action of (C")r ( resp (R')' )
then it is invariant under the action of T' (resp (Z2)” ), and the following diagram

commutes:

Z; x U, —-+ U,

l l

T' x 11$ _. 05
Therefore it is enough to prove that Uac/T’ = U,/(Zg)', and also since XE =
U,.;AIUE’. where A’ = {maximal cones of A}, we need only to consider 0 to be a
maximal cone. So it sufices to show that C'/T' =' R’/(Zg)' under the action de-

fined by the following commutative diagram

(Z;x---xZ2)x(Rx---XR) -—+ RX-HXR
l l
(51x---x51)x(Cx---><C) --v Cx---xC

t1...t, x1...x1 »—> t““...t°"x1,...,t°"...t“"x, with det a,- :1.
1 r 1 r J

Let us denote the equivalence classes in the complex case by [x] and in the real case

by [x]R so we need to show:

Va: 6 C'; 3 ye R" such that [x] = [y]
i and

[W E R' C 0' [yln C [y]

 

So let (x1,...,x,) 6 ' C', then there exists (31,...,s,) 6 T' such that
(31x1,...,s,x,) 6 R’ and since det(a,-,-) = 1, there exists (t1,...,t,) 6 T' such
that (t‘1'1‘...t‘,‘",...,t‘1'" ...t3") = (s1,...,s,). Therefore for every x in 0' there
exists y in R' such that T’(x) = T'(y) i.e [x] = [y]. And since (Z2)' C T' then
Z;(x) C T'(x) for every x e R' i.e [x]R C [x], therefore C'/T' = R'/(Z2)' and

29
hence )(23/7‘r .= XA/(zm :1

Proposition 4.2 The isotropy groups of the action of T” on X E are tori subgroups

ofT'.

Proof : Since every US is invariant under T' it is enough to prove the result in U,,C

where 0 is a maximal cone in A. The action is given by:

(5‘x-~-x51)x(Cx---><C) —-> ‘Cx-nxC

(em, . . . , e"')(x1, . . . , x,) r—r (e’E‘W‘JxI, . . . , e’z“'191x,)

with det(a,,-) = 1. Obviously the points that are fixed by some subgroups of T '
are the ones that have some x,-’s equal to zero, thus if x,-,, = 0 for k = l, . . . , p and
x,, at 0 otherwise, then the isotropy group corresponding to such point is I{,,,,,,,,-,} =
{(e’91,... ,e"') I 2,- agjoj E 0 mod 2n Vi 516 i1, . . .,i,}.

Let A = (a,,-), then since A 6 5L(r, Z) we have

5’——>5'

(eio‘,...,e”') v—> (e‘2°"01,...,e’l:°'101)=(e’¢‘,...,e“’”)

is a change of coordinates of 5', so I{,-,,,",,-,} = {(e‘f‘p ..,e"‘") | cw" = 1 for I: 75
i1, . . . , ip}. Therefore I{,-,,,,,,,-,} is a torus of dimension p. 0
Remark :

Since the action of (Z;)" on XA is just a restriction of the action of T' on XE, the
isotropy groups of the first action are just (Z2)' intersecting the isotropy groups of

, the second action.

30
4.2 Classification of Compact Toric Manifolds of
Dimension two

In [10] P.Orlik and F.Raymond studied the action of the 2-torus on simply connected
closed 4 manifolds, we start this by a brief description of the effective smooth action
of T2 on a closed simply connected 4-manifold and then we give an overview of their

results:

1. If the isotropy group of x is T2 i.e x is a fixed point then the slice at x (which is
the fiber over x of the normal disc bundle of the orbit) is a 4 disc and T2 acts on
it by a rotation in two planes by (m1, n1) and (mg, n2) with mlng -— mgnl = :tl

and the image of x in M " is an isolated boundary point.

2. If the isotropy group of a point x in M is a circle subgroup of T2 denoted
(m,n) = {(893893 E T2 | m0; + n02 E 0 mod 2n and gcd(m,n) = 1}, then
the slice at x is a 3 disc, the isotropy group (m,n) acts on it by rotation, the

image of the orbit in M " is a boundary point.

3. If the isotropy group of x is e, so the orbit of x is a torus, then the slice is a 2

disc and the image of the orbit is an interior point.

Theorem 4.1 (Theorem 1.12 in [10]). If T2 acts efectively and smoothly on a 4
manifold M without boundary, such that there are no nontrivial finite isotropy groups,
and such that the set of fixed points and points of circle isotropy groups is not empty,
then the orbit space is a 2 manifold with boundary, with weights identifying the isotropy

groups.

In section 4.4 of [10] they prove that under the hypothesis of theorem 1.12 the in-
terior points correspond to principal orbits and the boundary points correspond to

orbit with circle isotropy groups or isolated fixed points. In section 5 of [10] they

31

studied the action of T2 on closed simply connected 4-manifolds and they proved:

Theorem 4.2 (Lemma 5.1 in [10]). The action has fixed points and M' is a 2-
disk with interior points corresponding to principal orbits, and the boundary points

correspond to orbits with circle isotropy groups or isolated fixed points.

So if f1, . .., f, denote the fixed points and f: their images in M‘ then the are 5,‘
between f," and ff“ on aM' represents a 2 sphere 5.- and if we denote its stability

group by (a,-,b,-) = {(0,5) 6 T2 I a°‘B”‘ = 1}, we get a representation for M‘ as

. . a' “+1 . . . .
shown in Fig. 4.1. where l i = 1:1. The determinant condition arises because

be bi-l-l

 

Figure 4.1. Representation of the orbit space

the action of T2 on X E is differentiable then by corollary V1.2.4 and the definition of
local smooth actions in [3], the restriction of the toric action to a neighborhood of a

fixed point is equivalent to an orthogonal action of T2 on D‘ i.e to:
9

Tsz4 —-v D4

(t1, tz}(1', y) ’_"l (tint?! 1:,t’1n’t'2’2y)

32

This action where all the m,- and n,- are integers has the orbit space represented in

Fig. 4.2. Now we translate these results to the case of two dimensional toric manifolds:

 

Figure 4.2. Orbit space of the orthogonal action of T2 on D4

We start the study of the two dimensional toric manifolds by fixing the coordinates
in (Z;)2 as follows: We consider an element (t1,t2) in (Z3)2 to be (t(m1),t(m2))
where m1 and m; are the duals of the canonical basis of B“. Let A be a complete

nonsingular 2 fan, let 0 = R20(an1 + bng) + Rzo(cn1 + dng) be a maximal cone in

a c
A with = 1, then U, is isomorphic to R2, and the action of (Z3)2 on U, is
b d .

given by (t1, t2)(1l1, 112) = (t‘l’t;°u1,tf”t;ug) (see example in page 11), the origin is the
unique fixed point in U,, and since U, n U,: for 0 and 0’ in A’ does not contain the
origin of either one of them, then there is a one to one correspondence between the
set of fixed points and A’. The different proper isotropy groups are (d, -c) n (22)2
and (—b, a) fi (Z2)2, where

Z: x 1 if d even and c odd denoted 10
(d, —c)fl(Z2)2 = {(tlatzl 6 (Z?)2 l tit? = 1} = 1 x Z: if d odd and c even denoted 01
{(-1,-l),(1, 1)} if d,c odd denoted 11

33

By the results in [10] presented above and the fact that the orbit space is the same
in the real and complex toric manifolds we get a presentation for X; = X A/ (Z2)2 as
shown in Fig. 4.3. We call such a picture a colored graph ( or graph for short ) and

will be denoted by G, the labels 01, 10,11 are the colors.

Remarks :

Figure 4.3. X 3

1. No two adjacent edges on the graph have the same color, since two adjacent
coloring correspond to the isotropy groups of the action of (Z2)2 on U, for a.

maximal cone and the determinant condition does not allow this to happen.

2. The number of edges is equal to the number of one dimensional cones and the
number of fixed points is equal to the number of maximal cones so we have a
duality picture: if we represent the fan and the orbit space corresponding to it
on the same picture we get Fig. 4.4, where the notation E in the figure represents

the class of almodulo 2.

34

finfbfz ‘2

Figure 4.4. Duality picture

4.3 Cross Sections

Definition 4.1 A cross section for 1r : M —» M" is a continuous map 5 : M * ——i

M such that 1r 0 s is the identity on M ‘.

Lemma 4.1 If (Z2)2 acts on a 2 manifold such that M ‘ 2: D2 with D2 colored as
shown in Fig. 4.5, then there is across section to this action. Furthermore if a cross

section is given on an are A C 5' (where 5" is the horizontal segment in the figure

), then it can be extended to all of D”.

 

Figure 4.5. Orbit space of the action of (Zg)2 on 02

35

Proof : M is obtained from M ‘ by glueing 4 copies of D2 along parts of 5+ (5+ is the

upper half circle of the boundary)two by two in the way shown in Fig. 4.6. Obviously

 

Figure 4.6. The disk

a cross section is just the choice of one quarter of M. And if a cross section is given
in A C 5', then this amounts to just indicating which quarter of M is chosen, and

therefore the cross section is extended to that quarter. D

Theorem 4.3 If (Z2)2 acts on a closed 2-manifold such that M ‘ z D2 with all inte-
rior points of D2 correspond to principal orbits, and points on the boundary correspond
to either fixed points or orbits with 10 , 01 or 11 stability groups, then there is a cross

section.

Proof : Let M "' be as shown in Fig. 4.7(a) with t edges, then we cut D2 into t cones
C,- such that every cone contains a unique fixed point f: as shown in Fig. 4.7(b).
Then by the above lemma there exists a cross section along Cl, and this cross section
is defined along an arc of the southern boundary of Cg, therefore by the same lemma,
it can be extended to 02, and continuing the same procedure to the following cones,

we see that we can extend the cross section to all of 0’. Cl

36

 

(a) (b)

Figure 4.7. M ‘ and cutting it into cones

Theorem 4.4 If (Zg)2 acts on a compact 2-manifold with boundary such that M ’ 2
Dz. If D2 \ 5+ consists of principal orbits, and points on 5+ correspond to either
fixed points or orbits with 10 , 01, 11 stability groups, then there is a cross section to

this action..
Proof : same as the boundaryless case. 0

Definition 4.2 Let (Z2)2 act on M and M’ with M‘ and M” being as in theorem
4.3 then a homeomorphism between M" and M" which carries the weights of M ’ onto

the weights of M" isomorphically is called a weight preserving homeomorphism.

Theorem 4.5 Suppose (Zg)2 acts on two closed 2 manifolds M, N such that M ‘, N‘
satisfy the condition of the theorem above and that there is a weight preserving homeo-

morphism h‘ : M ‘ —+ N“ then there is an equivariant homeomorphism h : M -—+ N.

37

Proof : This follows from theorem 4.1 and theorem 3.3 chapter I in [3]. C1
Remark :

Theorem 4.5 means that if we start with a closed graph and change the colors using
a bijection of {10,01,11} onto itself, then we have thesame manifold. So we can
assume that we have a fixed point f in M where a neighborhood of f‘ in M" is as
shown in Fig. 4.5. We remark that this is just the same assumption done at the end

of chapter 2 for the fans.

4.4 Classification

Definition 4.3 Let G be a graph, then cutting an edge out of G means to replace G

by a new graph G’ as shown in Fig. 4.8.

 

Figure 4.8. Cutting an edge

Now we see which colored graphs correspond to fans:

Proposition 4.3 Lctt be the number of fixed points in the manifold.

If t = 3 and t = 4, the possible graphs and their dual fans are shown in Fig. 4.9.

38

 

 

2
t-3
fl. ,1
“1'32
n2
t-4 10 o-n1 n1
1
n2
I
t“ 11-10 /_‘1
01 -n1.n2
-112

 

Figure 4.9. The graphs and their dual fans with t = 3, 4

39

Proposition 4.4 If t 2 5 then a graph is dual to a nonsingular complete fan if it is

colored by the three colors.

Proof : (=>) Let G be dual to A with t 2 5 then by (Proof of theorem 8.2 in [8])
there exists n,- such that n,- = n,-._1 +n,-+I, and since det(n,-_1, n.) = 1, det(n,~, 12.4.1) = 1
then the parity of n,-1 and n,-+1 are different otherwise the parity of n,- would be 00,
also the parity of n,- is different from the parity of the. two others by the determinant
condition, therefore we have the three colors.

(4:) If G is colored by 3 colors then wlog we have the two cases shown in Fig. 4.10.

So we cut out the edge e in the first case, and we are still left with three colors, and in

 

Figure 4.10. Two different cases for the graph with t 2 5

case 2 we can cut out either edges e or e’, but we make sure that the cut out edge will
still leave us with three colors. We keep doing this operation until we get the triangle
with three colors which was seen for t = 3, then, we start with the fan corresponding
to t = 3 and for each step of the above operation ( beginning from its last step) we
introduce the sum of the two vectors dual to the two edges surrounding the removed

edge until we get our graph back and we get a dual fan for it. D

40

Example : See Fig. 4.11.

Now we see which manifolds correspond to these graphs:

 

 

 

 

 

 

 

 

 

 

Figure 4.11. Example of reducing G to the triangle and the dual action on fans

Let G be a colored graph dual to a fan A, then X A is obtained by glueing 4 copies of
G along the edges. The 4 copies correspond to images of G under the action of the
different elements of (Zg)’. Let us denote 1 = (1,1)G , 2 = (1,-1)G , 3 = (-—1,1)G
and 4 = (-1, —l)G. Therefore, for example, a side of 1 whose color is 10, is identified
with the same side of 2, and a side of 2 whose color is 11, is identified with the same

side of 4. To mark these informations on the graphs, we let

1 6 (resp l (1) denote the color 10'for 1 and 2 (resp 3 and 4)

41

6 l (resp 6 1) denote the color 01 for 1 and 3 (resp 2 and 4)

11 (resp -11) denote the color 11 for 1 and 4 (resp 2 and 3)

And now we can determine the 2 dimensional toric manifolds.

Proposition 4.5 If t = 3 then MG 2 RP’.
If t = 4 then there are two cases, and M0 is either T2 or the Klein bottle as shown

in Fig. 4.12.
Proposition 4.6 If t 2 5 then MG 2 1],-3RP’.

Proof : By preposition 4.4, G is colored by three colors, hence it looks as shown
in Fig. 4.13(a), but G; corresponds to RP2 \ D2, and G1 corresponds to M1 \ D2 for
some manifold M1 as shown in Fig. 4.13(b). Therefore G corresponds to MlllRPz,

and by the proof of proposition 4.4, we have Ma 2 [1,-2RP’. D

4.5 Dimension 3 Compact Toric Manifolds

1n [7] D. Mac Gavrin studied the action of the 3-torus on simply connected closed 6
manifolds, we start this section with a brief description of the efiective smooth action
of T3 on closed simply connected 6-monifolds, and then we give an overview of his

results:

1. If the isotropy group of x is T3 i.e x is a fixed point then the slice
at x is a 6 disc, and T3 acts on it by a rotation in three planes by
T(au,a12,a13),T(a;1,agg,a23),T(a31,a33,a33) with det(a,-,~) = 21 and where
T(ak1,ak2,ak3) = {(e"‘,e“’°,e“’°) | flag-«p,- E 0(2n) for l aé k}.The image

of x in M ‘ is an isolated boundary point.

2. If the isotropy group of a point x in M is a 2-torus then the orbit of x is a

circle, the slice is a 5-disc. The action of the isotrOpy group on the slice is a

42

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3
2-1JJ 3
61 U1 - a
.:l
c ”‘6'“;de
Q
z-i 10 10 61 2 51‘ 151 3
.. 1U 15 2
O 01 1 ‘ I b/\ b
M 1 111
4 3 :i a
3 16 1 3
a 2 t1 a - 3 °
t" 11] 10 1 _ 1 1 - +9
, .. iii if 2
O 01 1 ‘ 1 I b/S /\b
DI. ‘ [513 D1 3-—Z—3
3 11 4-11 3
c Mfielcqissdc

Figure 4.12. Obtaining MG from G when t = 3,4

43

 

(a)

 

 

   

(b)

Figure 4.13. Reducing M to MlllRP2

44

rotation in two planes by T(a, b, c), T(a’, b’, c’). The image of the orbit in M‘ is

a boundary point.

3. If the isotropy group of x is circle then the orbit is a 2-torus, the slice is a 4-disc.
The action of the isotropy group on the slice is a rotation and the image of the

orbit in M ’ is a boundary point.

4. If the isotropy group of x is e then the orbit is T3, the slice is a 3-disc. The

image of the orbit in M‘ is an interior point.

Theorem 4.6 (lemma 4.5 [7])

If T3 acts smoothly and eflectively on a compact connected simply connected 6 man-
ifold M and the only stability groups are torus subgroups of T3, then the orbit space
is simply connected 3 manifold, with the points on the boundary of M ' are orbits of
isotropy type T, T2 or T3 and interior points are principal orbits. The weighted
orbit space M ' can be described by a graph G on the boundary of M ‘, the vertices will
correspond to the fixed points, the points on the edges will be orbits with T 2 stability

groups and the points in the cells correspond to orbits with T1 stability groups.

Theorem 4.7 If the manifold is, closed, then the principal orbits are only in the

interior of M '. If in addition, BM' is connected, then M ‘ 2 0".

Notation : If M is closed and M“ 2 D3, we let Gc denote the orbit space as well

as the graph and Mac denote the manifold.

Proposition 4.7 The orthogonal action of T3 on D6 given by:

T3 X 06 —r 06
(e"”‘,e"”,e’m)(rle'9‘,r2e"’,r3693) , , (rler(91+awi+biw+cm),

,.zer(92+awi +bam+cz (as ) ,

raet(93+03¢i+63vz+cswsl)

45

is a smooth action, it is efl'ective ifl' detl(a,~),(b,-),(c,)] = :1:1. The orbit space is
given in Fig. 4.14 where G,- = {(e‘fl,e‘f’,e""3)]a,~<,o1 + bjCPQ + cigog E O(27r)}, and
T,- = G,- r). G}, where i,j, k are all distinct.

Figure 4.14. (DG)"

Proposition 4.8 Let T3 act on M'3 as in theorem 4.6, then by corollary V1.24 and
definition of local smooth action in [3], the restriction of the action to a neighborhood

of a fixed point is equivalent to the orthogonal action defined above.

Corollary 4.1 In any graph corresponding to such action , there are exactly three

edges emanating from each vertex and exactly three cells that meet at each vertex.

N ow we translate these informations to the case of toric manifolds. Let A be a

complete nonsingular 3 fan, and let 0 = R2062? agng) + 1120(2‘1’ bini) + 1120(2? c,-n,-)

46

be a maximal cone in A with det |a,, b,-, c,-| = 1 then

 

 

112° m: + mg + mg +

 

 

R20 7721 + mg + m3

 

 

U? 2 C3, and the action of T3 on US is given by:
(as. ,.., mm. 2) = (were man, 821%)

Hence, for each maximum cone corresponds a unique fixed point i.e a vertex in G3;

a T 2 isotropy group would be of the form
. v . 3
{(e’“, e'”, e""’)|0 5 go,- _<_ 2n and Zafigo, E 0(21r)}
1
and a T1 isotropy group would be of the form

3 3
K = {(ei‘Pl, efw, 81¢3)|0 S ‘p'. S 21'- and Zaflp" E 0(2W),Zb:-(p‘ E 0(27r)}
I. I

so K is determined by the vectors u = (a’1,a;,a,’,) and v = (b’,, 5,, g) which are

orthogonal to (c1, c2, c3).

47

Proposition‘4.9 K is completely determined by (c1,c2,c3) and will be denoted

T(Cla C2) C3) '

Proof : Let X = (2:1, 2:2, 2:3) be an integral vector orthogonal to (c1, c2, C3), then X =
au+flv and since det |(a2), (bf), (cf)| = 1 then a and H are integers, let Y = (y1, y2, y3)
be another integral vector orthogonal to (c1, c3, c3) and such that the 2—minors of the
matrix [(X), (Y)] are relatively prime ( hence there exists an integral vector Z such
that det |(X), (Y),(Z)| = l ), then because of the determinant conditions, and the

fact that u, 22, X, Y are in the same plane, it is easy to see that
. . ‘ I 3 3
K = {(em, 6"”, e'”’)|0 S «p.- S 21r and 23¢,- E 0(2w),:y,-cp,- E O(21r)}
l 1 __

D

Therefore to each l-dimensional cone of A is associated a T1 isotropy group, to
each 2—dimensional cone is associated a T2 isotropy group as follows: if r =
Rzo(aun1 + 012113 + a13n3) + Rzo(ann1 + 022113 + a33n3) then there exists an in-
tegral vector (031,032,033) such that det(a,-,-) = 1, so let the isotropy group G =
{(t1,t2,t3) e T3 | t‘i'”t§”t§” = 1}. G is easily proved to be uniquely determined by
(an, an, an) and (“name”), and to each 3- dimensional cone is associated a fixed
point, hence we have a duality between A and Ge as follows: .

The l-dimensional cones of A are half lines emanating from 0, each half line is sup-
ported by its generating vector, the 2 dimensional cones are membranes that are
bounded by two 1 dimensional cones so that when we intersect A with S2 we get a
triangulation of 52 whose edges are equal to 5]2 fl (2dim cones) and the vertices are
equal to 5'2 n (1dim cones) . So we can represent A as a weighted triangulation of
S 2 where the weights are adjoined to the vertices, the weights are the coordinates of
the respective generators of the 1 dimensional cones. To represent A on the plane we

project 52 stereographically from a vertex ( usually the considered vertex is adjacent

48

to a maximum number of vertices ).

Example : let

A = { Rzoni + Rzonz + Rzons; RZOnl + Rzonz + R20(—n1 — n2 - n3);
Rgoni + Rzons + R20(—n1 " n2 - n3); Rzonz + Rzons + R20(-n1 - n2 - R3);

the faces of these cones}

Now Go is obtained from A as the dual graph on 32 and the weights of Gc are
determined from the weights of A as shown in Fig. 4.15.

Remark : By construction of Go from A we see that Go is connected and hence

113 I O 11110.1)

 

 

“1

 

Figure 4.15. Example

49

In the real case the Z2 isotropy groups are of the form

K n (Z2)3 = {((-1)k‘,(-1)k’a(—1)k3)|Zaiki E 0(2),}:b2k, E 0(2)}
{((-1)k‘, (-1)k’, (-1)"3)I 2% = 0,235 = 0}. where E = class of a mod ‘2.

but sinceiwe have Xafi-c, = 0 and Ebfi-c; = 0 and det |(a:-), (bi), (cfi)| = 1 then 7; = 5.
Hence T(c1,c2, c3,)fl(Zz)3 = {(1,1,1);((—l)?‘-,(—l)5,(-l)a)} and it will be denoted
by 212223. '

So in the example above we get G (we denote by G the graph in the real case ) as in
Fig. 4.16.

Remark : The isotropy groups around a fixed point verify the determinant condition

 

out
010
11
100
C
Figure 4.16. G

hence for example we can not have {100, 010, 110} as colors around a fixed point .

50
4.6 Cross Sections

Lemma 4.2 If (2:)3 acts on a 3-manifold M such that M" 2 D3 with D3 colored as
shown in Fig. 4 .17 where the interior points and points on 5’ correspond to principal
orbits and det |(h',-'), (b-,-), (EN = :l:l, then the action has a cross section and if a cross

section is given on a disc D C S" then it can be extended to all of D3.

525252 515161

Figure 4.17. (03)"

Proof : M is obtained by glueing eight copies of D3 two by two along the weighted

cells and a cross section is just a choice of one copy among the eight. 0

Theorem 4.8 If (Zz)3 acts on a closed 3—manifold M such that M‘ e: D3 with
all interior points of D3 correspond to principal orbits, and we have a graph on the

boundary as described in the previous section, then the action has a cross section.

Proof : We just cut D3 into cones, with each cone containing one fixed point in its
base as shown in Fig. 4.18. Then by the same argument as in dimension 2 we have a

cross section. D

51

 

cutting G into cones

Figure 4.18. Cutting G into cones

Theorem 4.9 If (23)3 acts on a compact 3 manifold with boundary such that M ' 2
D3. If D3 \ 5'" consists of principal orbits and 5+ has a graph on it, then the action

has a cross section.
Proof : The proof is similar to the boundaryless case. 0

Theorem 4.10 If (Z2)3 acts on two 3 manifolds M,N such that M‘ and N‘ satisfy
the conditions of one of the two theorems above and if there is a weight preserving

homeomorphism between M ‘ and N‘ then M and N are equivariantly homeomorphic.

Proof :The proof is similar to the 2 dimensional case. 0
Note : By the above theorem we assume that there is a fixed point where G looks
as in Fig. 4.19 which means for A that there exists a maximal cone 6 in A whose

generating vectors form the canonical basis for R3.

52

 

Figure 4.19. G

4.7 Orientation

Theorem 4.11 Let G be a graph on 52 as before. Then MG is orientable ifi’
100,010,001 and 111 are the only colors in G.

Corollary 4.2 A 8 dimensional compact toric manifold X4 is orientable ifl‘ every
generator vector of A has the parity 100,010,001 or 111.

Proof of the theorem : Let G be given, then we have a part H of G that has the
representation shown in Fig. 4.20 where T is a subgroup of order 2 of (22)3 and by
the determinant condition we know that T corresponds to one of the following four
colors: 100, 110, 101, 111. But we can represent H as in Fig. 4.21. By theorem 4.10,
both X1 and X2 yield D3 with the obvious action in the case of X2, and in the case

of X1 the action is given by:

(Z2)3 XD1 XD2 -—» D1 x D2

(t1,tg,t3)(:c,y,z) ._+ (t,z,t‘;t2y,t:taz)

 

53

 

 

 

Figure 4.20. H
m ' 01
I
- 001..
x1 1!2

Figure 4.21. Cutting H

10

54

where a and b are 1 or 0 depending on T. Also by the cross section theorem we
have that X1 0 X2 corresponds to 5° x D2 with the induced action from either one
(the two actions are similar on 5° x D2 ), hence My is the union of two copies of

D1 x D2 glued together along 5° x D2 i.e M3 is a D2 bundle over S 1. Such bundles

l 0 l 0
are classified by «0(02) = { , }. Let

01 0—1

f: (S°xD1xD‘)CMx,—+ (1"3'°><D1><D1)CMX1

(2.31. 2) *-* (as. I‘ll. 252)

f is obviously an equivariant homeomorphism, and the action of (Z2)3 on M x, U f M x,
has H as orbit space therefore My = M x; U f M x, by theorem 4.10.

Therefore, M H is orientable e the bundle is trivial 4:)

fr=f(xi-a-): R2—" R2

(y. 2) *--* (75°31, 2’2)

correspond to the identity element in ro(02) e» a = b = 0 or a = b = 1 4: T = 100
or T = 111. I

Now assume that M is orientable and since any edge of G touches exactly four cells
(see Fig. 4.22). Then by the above discussion T4 is either T1 or T1T2T3 (this notation
means that the nontrivial element of T4 is the product of the nontrivial elements of
T1T2T3 ) and since the product of any three of the colors 100, 010 , 001 , 111 is the
fourth we have that G is colored by the 4 colors only.

Conversely let G be colored by the 4 colors only. We have eight copies of G, each one
is the image of G under the reflection by an element of (Z2)3. We glue these eight
copies along the 3 faces surrounding a vertex (call it the central vertex ) so that

we get a weighted 3 ball Po, the glueing is made such that a 100 face ( 010, 001, 111

55

 

Figure 4.22. An edge

resp ) of aG is identified with the same face of 19G iff (-1,1,1)a = 5((1,-l,1)a =
fl, (1, 1, —1)a = ,6,(-1, -l, -1)a = fl resp) then we choose orientation for G (i.e for
the interior of G and the faces ) and reflect these orientations to the other copies
so that we obtain a coherent orientation on Pa. To prove that M9 is orientable, it
is enough to prove that any two identified faces have opposite orientations but as
we remarked above, a face of aGi is identified with the same face of 0G iii 3 is the
product of a by (~1,1,1) ; ( 1,-1,1) . (1,1-1) or (-1,-1,-1) , and all of them are odd
reflections i.e they reverse the orientations. 0
Notation : We denote the eight copies of G by:

l =(l,l,1)G, 2 =(-1,1,1)G , 3=(1,-1,1)G , 4=(1,1-1)G , 5=(—1,- 1,1)G, 6=(-1,1,-1)G,
7=(1,-l,-1)G and 8=(-1,-1,-1)G.

Remark : We notice that in the glueing process of the eight copies of G, we obtain
4 distinct copies of each cell which are attached along their edges, in the same way
as in the 2 dimensional case. Let C be a cell of G with k edges and let n be the

generator of its dual cone in A and n1, . . . ,nk are the generators of the respective

56

dual cones of the adjacent cells to C with det(n,-,n,-+1,n) = 1. Since a Z change of
v basis will change the weights of G bijectively we assume that n is on the z axis. If
n, = (a,, b.-, c,-) then a,-b,-+1 ‘- b,-a,-+1 = 1. And as we have seen in dimension 2, if k 2 5
we have at least 3 different colors on the adjacent cells of C. Therefore in the case of
an orientable toric manifold, a cell with more than four edges has exactly three colors

surrounding it.

4.8 Heegaard Diagrams for Orientable Toric 3-
Manifolds

Let A be a fan corresponding to an orientable compact toric manifold, and G its dual
graph. G is colored by only 4 colors 100, 010, 001 and 111; let n denote the number
of cells of G, it is equal to the number of generating vectors of A, hence the number
of cells of P9 is equal to 8(n -— 3) and MG ( = X4) is obtained from Pa by pairwise
identification of its cells. If we consider the decomposition of MG into the eight copies
of G and bore out the vertices of the decomposition by the procedure of boring out
a small ball surrounding each vertex except the central vertex, likewise we bore out
the edges of the decomposition ( except the edges which have the central vertex as an
end point ) by boring out small full cylinders about them where the cylinders connect
the balls surrounding the end points of the edges, we obtain a handlebody H1. The
subspace which is remaining after one has bored out H1 is a handlebody H2, obviously
it is the one obtained from PC; by identifying small discs inside the cells ( one disc for
each cell ). H1 and H2 have genus h = 4(n—3). M3 is obtained by glueing H1 and H2
along their common boundaries, the glueing is defined by a homeomorphism between
3% and 8H2, and since H1 and H; are handlebodies, such a homeomorphism is

determined by the images of the boundary of the pairwise identified discs in 6H,.

57

But we have seen, in proposition 4.8, that locally (in the neighbohood of a vertex ),
the cells look like coordinate planes, the edges like the coordinate axes and the vertex
like the origin; also the eight copies of a cell C are identified two by two, and the four
unidentified copies form a plane whose intersection with 8H1 is just the image of the
boundary of the disc in C under the identification homeomorphism, hence we don’t
need to represent 0H1 as a full 3 dimensional handlebody, it is enough to draw the
generating circles of H1 with the vertices, and, to mark the planes corresponding to
the cells, we just draw their intersection with the balls around the vertices and mark

the cooresponding weights on them.

Remarks :

1. The skeleton or the generating circles of ’H1 can be obtained from the graph
G by just deleting the central vertex and all the edges that have it as an end
point. We call the circles represented by edges: edge circles, and the circles

that arise from the cell boundaries as cellular circles.

2. A vertex in G which is the endpoint of an edge that has the central vertex as the
other endpoint is the center of only one plane in the Heegaard representation
( H.D for short), this plane corresponds to the cell which does not have the

central vertex on its boundary.

3. A vertex which is in the boundary of the same cell as the central vertex but not

the same edge is the center of two perpendicular planes in the H.D.

4. A vertex that does not share a cell with the central vertex is the center of three

planes.

5. An edge which is in the boundary of a cell 0 carries two of the four unidentified

copies of C on one half of the circle it represents, and the two other copies on

the other half.

58

6. If an edge is in the boundary of a cell that contains the central vertex then the

edge carries only one cell in the H.D.

7. If an edge is in the boundary of two cells that do not contain the central vertex

then the edge carries two cells in the H.D.

Example 1 : Let A and G be as in Fig. 4.15, then Pa, its skeleton and H1 are as
shown in Fig. 4.23..

Example : 2 Let A and G be as shown in Fig. 4.24(a), then P9 is as shown in
Fig. 4.24(b) and H; are as shown in Fig. 4.25.

4.9 Surgery

Suppose that G is as given in Fig. 4.26(a), where the T38 are the 4 different colors,
then as we have seen in the proof of theorem 4.11, the indicated region is D2 x S".
We may do equivariant surgery on the circle by replacing the solid torus X1 around it
with another solid torus X2, to obtain a new manifold with the orbit space shown in
Fig. 4.26(b), (we can do that, because the boundary of the two exchanged parts is the
same ). The corresponding picture of this surgery in the RD is shown in Fig. 4.27.
A meridian curve of X2 is for example the one that is running over the boundary of
the D-square (1D -v 3D -> 4D -> 20). Its image in X1 is running twice along the
longitude and once around the meridian, hence the surgery coefficient is §, i.e the
surgery operations we are doing are % surgeries. Also by construction, the surgery

* is performed only on the edge circles of H; which have a half twist in their normal

bundles; the twist is indicated in H1 by the planes carried by the-edge circle.

Theorem 4.12 If MG is an orientable closed toric manifold, then it is obtained from

RP3 by a series of connected sums of RP3 and the above surgery.

59

v3

 

 

 

 

V2 v1

Q; max!

 

Figure 4.23. RP3

60

n2 - int ini ty

 

 

“a

 

 

'2

‘3 W33

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   

Figure 4.24. A, GandPg

61

 

'<\\'.%\KNWW

2'
3
s

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 
   

 

~:-.'-:.‘,«'~:-. a. '9 ‘
‘~ :-‘\$§§% .«s‘F

 

 

3

 

 

 

 

 

3313

 

 

03 23‘

 

 

 

 

 

 

Figure 4.25. H1

62

 

Figure 4.26. Surgery

Proof : we assume that G is different from the tetrahedron, which corresponds to
RP3. Since there are four colors, then G is as in Fig. 4.28(a). Then we do the surgery
on an edge of the cell G; in the same manner as for the cutout of edges in dimension
2 so that the new cell G{ still has 3 different adjacent colors. We continue doing the
surgeries until we end up with the modified cell 01 having exactly 3 edges as shown
in Fig. 4.28(b). But this corresponds to a connected sum with RP. We remove the
RP3 only if we still have four colors on the graph, otherwise we have to change to
another cell and do the same work on it. After we remove the RP", the graph will be
as shown in Fig. 4.28(c). After doing this, the graph G has one less cell and is still
colored by four colors. We keep repeating the sMe process until we obtain a graph
with only four cells which is the tetrahedron and the associated manifold is RP". 0
Remarks :

63

 

 

 

 

 

Figure 4.27. H1 for the surgery

64

 

graph with graph after graph after
4 colors: surgeries moving

(a) (b) (c)
Figure 4.28. Reducing G

1. In case G has only three colors, the only possibility for G to correspond to a
fan is that G is the cube and hence MG is T3. In that case we connect sum T3
with RP3 obtaining a new graph with a cell having five sides and we proceed

with the moves above.

2. The graphs obtained by the different moves in the process toreduce a given
graph to the tetrahedron may not be dual to fans. It is not known if every

triangulation of the sphere is supported by a nonsingular fan.

Proposition 4.10 Let A be a nonsingular fan, let T = Rzom + + Rzon. be in
A.

We remark that ifr < a then there exists a">< a such that a = r+o" with a’flr = {0}.
Define no = n1 + + n. and r,- = R2071] + . - - + Rzon,-1 + Rzono + Rzon.“ +
+Rzon, forl S i S s. We then let a; = n+0" and A, = (A\{a' 6 Alr <

0}) U {faces ofailo e A,r < 0,1 S i S s}.

65

Then XA, is obtained by blowing up the closed submanifold orbr.

Proof: See Prop. 1.26 in [9]. [:1

Remarks :

1. The corresponding move in dimension 3 for the fan and its dual graph are as

shown in Fig. 4.29.

V‘s

t t

 

Figure 4.29. Blowing up

2. The 1/2 surgery move is a blowing up followed by a blowing down the ap-
propriate circle. We can do such consecutive moves only if T1T4 = T2T3 i.e
T4 = T1T2T3 and to obtaine a toric manifold we need to have n1 + n4 = 112 + n3

so we get a nonsingular fan.

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3. In [4] Danilov showed that any toric manifold is obtained from RP'3 by a se-
quence of blowing up and down along points and circles as with every step of

the sequence corresponding to a toric manifold.

4.10 First Homology Groups for Orientable Toric
Manifolds

A representation of the first homology group of an orientable compact toric 3-manifold
is obtained from its HD as follows: H1 is a 4(n - 3') handlebody, so we have 4(n -
3) generators corresponding to the generating circle of 711 and 4(n - 3) relations
corresponding to the boundaries of the cellular disks (n = number of generating
vectors of the fan = number of cells of G ).

' Every cell in G except the ones carrying the central vertex yield 4 relations, one
for each of the unidentified four copies. The relations are obtained by running over
the boundaries of the cellular disks. The generators that appear in these relations
correspond to the edges and the cellular circle of the cell.

We orient the generating Circle of H; as follows: in the skeleton of H1 , we orient the
edge circle so that the cellular circles are given a coherent orientation i.e the arrows
in the cellular circles have the same direction, so that an edge circle has an opposite
orientation to any cellular circle it contributes to.

Every generator corresponding to an edge appears exactly in two or four relations,
depending on the number of planes it carries, these generators appears with coefficient
1, but each generator corresponding to a cellular circle appear in the four relations
obtained from its cell, and its coefficient is -1. Therefore if we write the coefficients of
the different generators in a matrix where the columns correspond to the relations, and

the rows correspond to the generators, we will have a square 4(n - 3) x 4(n - 3) matrix

67

A with entries being 0 or $1, with every row containing two or four 1’s and the other
coefficients in that row are 0, or it contains four -1’s and the other coefficients are 0.
This matrix is equivalent to a diagonal integral matrix D ( A 2 D 4: 3P, Q invertible
matrices over Z such that D = PAQ) , where the diagonal is {d1,...,d,,0,...,0}
with d.- ¢ 0 Vi and d,-|d,- ifi S j. So the rank of H1 is 4(n — 3) — r and its torsion
coefficients are the d.-’s.

We have A is equivalent to a matrix D whose entries in the first column are either
2’s or 4’s (by adding all the columns of A and replacing it with the first column ),

hence 2| det D i.e if rankH1 = 0 then H1 has an element of torsion 2.

Corollary 4.3 5'3 and L(p,q) where p is odd are not toric manifolds.

BIBLIOGRAPHY

BIBLIOGRAPHY

[1] S. Akbulut. Lectures on real algebraic spaces. Proceedings of Kaist Mathematics
Workshop, pages 1 - 15, 1992.

[2] S. Akbulut and H. King. Rational structures on 3-manifolds. Pacific Journal of
Mathematics, 150(2):201 — 214, 1991.

[3] G. E. Bredon. Introduction to Compact Transformation Groups. Academic Press,
New York, 1972.

[4] V. I. Danilov. The birational geometry of toric 3-folds. Math. USSRsz, 21:269
- 280, 1983.

[5] J. Jurkiewicz. Torus Embeddings, Polyhedra, k'-actions and Homology. Disser-
tationes Mathematicae 236, Rozprawy Mat, 1985.

[6] R. Kulkarni. On the principles of uniformization. J. Difl'erential Geometry,
13:109 - 138, 1978.

[7] D. Mcgavrin. The T3 actions on simply connected 6-manifolds I.
Trans. Amer. Math. Soc, 220:59 - 85, 1976.

[8] T. Oda. Lectures on Torus Embeddings and Applications. Springer Verlag, New
York, 1978.

[9] T. Oda. Convex Bodies and Algebraic Geometry. Springer Verlag, New York,
1987.

[10] P. Orlik and F. Raymond. Actions of the torus on 4-manifolds 1.
Trans. Amer. Math. Soc, 152:531 - 559, 1970.

[11] R. T. Rockafellar. Convex Analysis. Princeton University Press, New Jersey,
1970.

[12] Seifert and Threlfall. A Textbook of Topology. Academic Press, New York, 1980.

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