IMAGES OF CERTMN MAMFOLDS UNDER MAPPINGS 0F DEGREE ONE Thesis for the Degree of Ph. D. MECHEGAN STATE UNIVERSITY LAWRENCE EDWARD SPENCE 1970 “JV-“Q Michigan Scam UniVmitY - This is to certify that the thesis entitled IMAGES OF CERTAIN MANIFOLDS UNDER MAPPINGS OF DEGREE ONE presented by Lawrence Edward Spence has been accepted towards fulfillment of the requirements for Ph.D. degree in Mathematics A AW. LIBRARY Major professor 0-169 rm 4 m. y amomc av HMS & SflNS' HELME‘LJEE . germ, '1‘ M3 . Q . . _---A- an exam—cm W _ f-V'm" ABSTRACT IMAGES OF CERTAIN MANIFOLDS UNDER MAPPINGS OF DEGREE ONE By Lawrence Edward Spence This thesis considers two distinct problems. In the second chapter a classification of the images of certain products of Spheres under mappings of degree one is obtained. The principal results are the following theorems. Theorem 2.1: Let f: SIn X Sn.m a M be a mapping of degree one into a closed, connected, orientable n-manifold M. Then either M has the homotopy type of Sn, or f is a homotopy equivalence. Theorem 2.3: Let M be a closed, connected, orientable 3-manifold and f: S1 X S1 X S1 a M a mapping of degree one. Then either f is a homotopy equivalence, or M has the homotopy 2 type of S1 X S or 33. l l 1 2 In the third chapter S X S X S1 and S x S are characterized in the class ml of closed, connected 3-manifolds M having the property that each connected, finite-sheeted cover- ing space over M is homeomorphic to M. Both S x S1 X S1 and S1 X 82 are members of this class with non-zero fundamental groups; whether there are other such 3-manifolds remains unanswered. 1 1 But S1 X S X S and S1 X 32 can be shown to be the only Lawrence Edward Spence members of 5m satisfying certain additional conditions. Theorem 3.1: Let M be a member of the class ml having a non-zero, nilpotent fundamental group. Then M is homeomorphic to SIXSIXS1 orto SIXSZ. Theorem 3.6: Let M be a member of the class Fm such that each double covering of M is proper. (A double covering p: M a M is said to be prOper.if the non-trivial covering trans- formation over p is homotopic to In.) If H1(M) is infinite, 1 then M is homeomorphic t0v S1 X S X S1 or to S1 X 82. IMAGES OF CERTAIN MANIFOLDS UNDER.MAPPINGS OF DEGREE ONE By Lawrence Edward Spence A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1970 \e (,2. 1‘ .'-'V I'- y ‘ (r; ,a ...J --. _,. L‘J/j- - 7 / ACKNOWLEDGMENTS The author wishes to express his gratitude to Professor Kyung W. Kwun for suggesting the problems considered in this thesis and for providing many helpful suggestions during the research. He also wishes to acknowledge the financial support of NSF grant GP 19462 during the latter part of this research. ii TABLE OF CONTENTS Page ACKNOWLEDGMENTS ..... ............ ...... ........... ii INTRODUCTION ..... .... ........ ............ ..... ... 1 CHAPTER I. THE DEGREE OF A MAP .............................. 3 II. IMAGES OF CERTAIN MANIFOLDS UNDER MAPPINGS 0F DEGREE ONE 000......00.00..........OOOOOOOOOOO0.0. 11 1 1 1 1 2 III. A CHARACTERIZATION OF S X S X S AND S X S 18 BIBLIOGRAH'IY so ooooooooooo 00.000000000000000000000 27 iii INTRODUCTION This thesis considers two distinct problems. In chapter II a classification by homotopy type is obtained for those closed, connected, orientable n-manifolds ‘M which admit a degree one - l mapping Sm X Sn m!» M. In the third chapter 81 X 81 X S and S1 X 82 are characterized in the class 5]! of closed, connected 3-manifolds M having the property that each finite-sheeted covering space over M is homeomorphic to M. There is a well-known result which states: For n 2 5, any closed, orientable n-manifold M admitting a degree one map Sn a M is homeomorphic to Sn. The principal theorem of the second chapter generalizes this result to the case in which the domain consists of a product of two Spheres; of necessity, the classification is by homotopy type rather than homeomorphism. Two additional results are obtained by taking as domain certain other products of Spheres. In [5] Kyung W. Kwun asks which closed, connected, orientable 3-mainfolds admit double coverings (or proper double coverings) of themselves. More recently, Jeffrey L. Tollefson has proved [12, Theorem 2] that a closed, connected, orientable 3-manifold properly covers itself k times for every prime k if and only if it is the product of a 2-manifold and SI. The class ‘Dl described above consists of those closed, connected 3-manifolds which admit no finite-sheeted coverings other than 1 .' Ill ll.llll.ll I‘ll! Illl'l' II‘IIIIIJIIEII" I III" I I}! i‘ Illli coverings of themselves. This class is examined in chapter III, 2 and S1 X S1 X S1 and S1 X S are shown to be the only mani- folds in this class which satisfy certain additional conditions. CHAPTER I THE DEGREE OF A MAP In this chapter two definitions of the degree of a proper mapping between n-manifolds will be given. These definitions will then be used to obtain two theorems (Theorem 1.1 and Theorem 1.4) which will be applied frequently in the next chapter. The nth sheaf cohomology group of X with compact supports will be denoted by 1120!; 8), where 8 is the constant sheaf on X with stalk Z, the infinite cyclic group. For a connected, orientable n-dimensional manifold M, HZCM; 8) - Z. Each such manifold will be assumed to have a preferred free generator pH 6 H201; 8). The (algebraic) degree of a mapping is defined for proper maps between connected, orientable n-manifolds. (A mapping is proper if the inverse image of each compact subset of the range is a compact subset of the domain.) If f: (M, 5M) a (N, 5N) is such a mapping, then the degree of f is the integer denoted by deg(f) which satisfies the equality f*(pN) - deg(f)pM. This purely algebraic definition of degree has no geometric interpretation. In order to recognize the geometric significance of the degree of a map, it is necessary to intro- duce an alternate definition of degree. .The geometric degree C(f) of a proper mapping f: (M, 3M) ... (N, 3N) between 3 n-manifolds is defined to be infinite unless there exists an n-disk D in the interior of N such that f-IGD) is the union of a finite number of disjoint n-disks each mapped homeomorphically onto D under f. When such disks do exist, G(f) is defined to be the minimal number of components in the inverse image of each such disk. The two definitions of degree are related by the in- equality |deg(f)| s G(f), which is obvious if so?) is infinite. If G(f) is a positive integer k, then there is an n-disk D in the interior of N such that f-1(D) - D1 U D2 U°--U Dk‘ is the union of k disks each mapped homeomorphically onto *D under f. Let equal 1 or -1 ‘1 according to whether f: Di a D is orientation preserving or orientation reversing. Then deg(f) -=z:g1 ei [3, Lemma 2.1b], so that ‘deg(f)| s G(f) in this case also. The existence of a mapping f: (M, 5M) a (N, 3N) of degree one between two connected, orientable n-manifolds has important implications for the algebraic invariants of the manifolds, as the following fundamental result shows. Theorem 1.1: If f: (M, 3M) ~’(N, 5N) is a proper mapping of degree one between connected, orientable n-manifolds, then (a.) the induced mapping of fundamental groups f#:.111(M) ... n1 (N) is epimorphic; (b.) the induced mapping of homology groups f*: H*(M, 3M) e H*(N, 3N) is a split epimorphism. Proof of (a.): Let p: N a N denote the covering space of N corresponding to the subgroup f#(n10M)) of n1(N). Then f can be lifted to a map f: Mw* N which is necessarily proper (because f is proper). Now 1 - deg(f) - deg = deg

deg<%>. and therefore deg(p) - 1:1. Thus N I‘N [3, §2], and hence f# (11100) " "1(N) - Proof of (b.): Since the Borel-Moore homology groups of n-manifolds coincide with the corresponding singular homology groups [1, V. 12.6], there is a commutative diagram q * q new) (a, L Hem) f in which the vertical maps are the Poincare Duality isomorphisms induced by the cap product [1, V. 9.4 and V. 10.2]. That f* is a split epimorphism follows from the cap product rule f*(a'n f*(5)) - f*(a) n 5, which implies the commutativity of the diagram. The homotopy results used in chapter II often require that the manifolds under consideration be simply connected. So when f: (M, 3M) a (N, 3N) is a mapping of degree one between manifolds which are not simply connected, it will be necessary to pass to the universal covering spaces of M. and N. Thus it is important to know conditions under which f will induce-a mapping of degree one on the universal covering spaces of M and N. One such situation is described in Theorem 1.4, the proof of which requires two lemmas. Lemma 1.2: Let f: X «'Y be a continuous function which induces an epimorphism f#: n1(X) a n1(Y) of fundamental groups. If p: T «’Y is a fibration with unique path lifting such that Y is path-connected, then P, the fibered product of f and p, is also path-connected. Proof: In the diagram P A)? '13 JP $Y X let f and 3 denote the maps induced by f and p, respectively. Then 5 is a fibration with unique path lifting [10, 2.8.6]. In order to show that P is path-connected, it suffices to prove that there is a path between any two points of p-1(x) for an arbitrary x E X; so let (x,yi) 6 p-1(x) (i I 0,1). Since T is path-connected, there is a path w: I «‘T from y to yl. Now p(yi) 3 f(x) for i - 0,1, so that 0 [pm] 6 n1(Y,f(x)). Because f# is epimorphic, there is a loop A: I n X based at x such that f) :.pm rel {0,1}. Let X: I a P be a lifting of A such that X(0) - (x,yo). ’Now WEEK] 3 [PEN] g [£51.] =1 [fA] '3 [PU-l] ‘- P#[w]- But since p#: "1(§) w n1(Y) is a monomorphism [10, 2.3.4], EX 3 w rel {0,1}. In particular, fi(1) - m(1) - yl. So i(1) ' (x9y1)3 and X is the required path. Lemma 1.3: In the commutative diagram f P >‘Y 00! 00 X 4¢‘W let f and g be continuous maps of Hausdorff spaces, P be the fibered product of f and g, and i and g be the maps induced by f and g, reSpectively. If f is a prOper map, then f is also a proper map. Proof: Recall that P = {(x,y) 6 X X Y: f(x) I g(y)] and that f and g are defined by f(x,y) - y and §(x,y) - x. If X is a compact subset of Y, then f-1(X) is a closed sub- set of P n (f-1(gK) X K). Because W is a Hausdorff Space, P is a closed subset of X X Y [2, V11.l.5]. Moreover, f-1(gK) X K is compact since f is proper and g is con- tinuous. Thus f-1(K) is a closed subset of a compact set in the Hausdorff Space X X Y and hence is compact. Theorem 1.4: Let M and N be compact, connected, orientable n-manifolds, and let f: M.~'N be a mapping of degree one which induces a monomorphism f#: NICM) “'WICN) of fundamental groups. If q: N ~>N is the universal covering Space of N .and P is the fibered product of f and q, then: (a.) The induced covering projection p: P a M is the universal covering space of M; (b.) If G(f) I 1, then any map f: P!» N induced by f has geometric degree one; (c.) If G(f) I 1, there is a proper map f: P.» N of degree one such that qf = fp. Proof of (a.): There is a commutative diagram P f ) ’U m i5€----¢z: ..O M or? in which both vertical maps are covering projections [10, 2.8.6]. Since f#p#: n1(P) a n1(N) factors through n1(N) = O, f#p# is the zero homomorphism. But both p# and f# are monomorphisms; so n1(P) I 0. Because P is path-connected by Lemma 1.2, P is a simply connected covering space of M, and hence, the universal covering Space of M [10, 2.5.7]. Proof of (b.): If G(f) I 1, there exists an n-disk D1 in the interior of N such that f-1(D1) is homeomorphic to D1 under f. Choose an n-disk DZCD1 so that D2 lies in some open subset of N which is evenly covered by q and so that f-1(D2) lies in some open subset of ML which is evenly covered by p. Let D be any component of q-1(D2); then q maps D homeomorphically onto D2. Since I is a proper map (Lemma 1.3), f-1(D) is compact. l ~-1 - - ~-1 Now f (D)<: p f 1(D2), and therefore f (D) is the union of a finite number of disjoint disks, each of which is mapped homeomorphically onto D by f. If (xi,yi) E f-1(D) (i I 1,2) lie in the same fiber, then y1 I f(x1,y1) I f(x2,y2) I y2 and hence f(xl) I p(y1) I p(y2) I f(xz). But since x1,x2 E f-ICDZ), it follows that x1 I x2. 80 each fiber of f contains a single point, and therefore G(f) I 1. Proof of (c.): There is a map g: M.« N homotopic to f and having geometric degree one [3, Theorem 4.1]. If Pg denotes the fibered product of g and q, then the fibration p': Pg I M induced by q is fiber homotopy equivalent to p: P I’M [10, 2.8.14], and the fiber homotopy equivalences between p and p' are easily seen to be homeomorphisms. Hence P and P8 may both be identified with the universal covering space M of M by part (a.). Denote the covering projeCtion M1» M by n, and let H: M,X I I'N be a homotopy from g to f. Let g: M.» N denote any mapping induced by g; then G(g) I 1 by part (b.). The homotOpy lifting property guarantees the existence of a map H: M X I I»N such that H(x,o) I §(x) and qH(x,t) I H(n X lI)(x,t) for all x E M, t E I. The desired map f: M e‘N is defined by f(x) I H(x,1). In order to prove that f is a proper map of degree one, it suffices to Show that H is a proper map (for the degrees of properly homotopic maps are equal). Since .nlz M X I a M, the projection onto the first factor, is a homotopy equivalence, the homotopy commutative ‘I’..l , E I x I I ...... lhl'lllllllll 10 diagram M X I j;N M shows that H satisfies the hypotheses of part (a.). Thus ~ M X I is the fibered product of H and q, and H is proper by Lemma 1.3. CHAPTER II IMAGES OF CERTAIN MANIFOLDS UNDER MAPPINGS OF DEGREE ONE In this chapter those manifolds which admit a mapping of degree one from certain products of Spheres will be classified. The principal result (Theorem 2.1) gives a classification of such manifolds by homotopy type for the case in which the domain is a cartesian product of two Spheres. This theorem generalizes the fact that for n 2 S the only n-manifold M. which admits a mapping Sn I M of degree one is Sn itself. (This result follows immediately from Theorem 1.1 and the n-dimensional Poincaré Conjecture.) A similar result is obtained in Theorem 2.3 when the domain is S1 X S1 X 51, and Theorem 2.2 proves that there are no mappings of degree one from the n-dimensional torus Tn (the product of n copies of 51) into an n-manifold with funda- mental group equal to 151 Z. i=1 Theorem 2.1: Let M be a closed, connected, orientable n-manifold and f: Sm X Sn-m I'M (1 s m s n-m) a mapping of degree one. Then either M has the homotopy type of Sn, or f is a homotopy equivalence. Proof: Since n1(M) is abelian (Theorem 1.1 (a.)), 11104) = ulna) is a direct summand of 111(3m x s“'“‘) = “1(3‘“ x 3““) by Theorem 1.1 (b.). Thus, because the infinite cyclic group is indecomposable, n1(M) is a free abelian group. 11 E.' (II! IIIIEI‘I‘. ‘11 II 12 If n I 2, so that m I n-l I l, the conclusion follows easily from the classification theorem for closed, connected 2-manifolds [6, 1.5.1]. In fact, M must be homeomorphic to either S2 or S1 x 81. Therefore it will be assumed that n 2 3. Suppose first that m I 1. Since 111(81 X Sn-l) I 2, either n1(M) I 0 or n1(M) I Z. Consider first the case that n1(M) I 0. In this case H1(M) I 0, and Hn-1(M) I H1(M) I 0 by Poincaré Duality and the universal-coefficient theorem for cohomology [10, 5.5.3]. Thus, by Theorem 1.1 (b.) , Hk(M) is trivial except for k I 0 or k I n, in which case it is infinite cyclic. The absolute Hurewicz isomorphism theorem [10, 7.5.5] then implies that the Hurewicz homomorphism m: "n(M) I Hn(M) is an isomorphism. Let “M E Hn(M) and Vn E Hn(Sn) be the preferred generators, and select a map g: Sn I M representing the class ¢-1(pM). The definition of m shows that ”M I m[g] I 8*(vn); hence g is a mapping of degree one. So g*: H*(Sn) I H*(M) is epimorphic by Theorem 1.1. Since every epimorphic endomorphism of the infinite cyclic group is an isomorphism, g* is actually an isomorphism. It follows that g is a weak homotopy equivalence [10, 7.6.25] and hence, a homotopy equivalence [10, 7.6.24]. Now assume that n1(M) I Z. As above, Theorem 1.1 implies that f#: 111(S1 X Sn-1) I n1(M) is an isomorphism. Thus, in order to prove that f is a homotopy equivalence, it suffices 1 - to show that f#: nk(S X Sn 1) a nk(M) is an isomorphism for 13 k 2 2. It follows from Theorem 1.4 that there is a commutative diagram P'hl R x Sn"1 % in which q: M I‘M is the universal covering space of M. and f is a prOper mapping of degree one. If Hn_1(M) I 2, then Hn_1(M) I 0 (Theorem 1.1). So the absolute Hurewicz isomorphism theorem implies that M is contractible and thus implies that M is a space of type (2,1) [10, 7.2.11]. But then M is homotopically equivalent to S1 [15, 2.10.4], contradicting ~ ~ “-1 ~ that Hn(M) - 2. Therefore Hn_1(M) - z, and f*: H*(R x s ) ~H*(M) ~ -1 ~ is an isomorphism, As before, it follows that f#: nk(R X Sn ) I "k(M) is an isomorphism for k.2 2, and so f#: nk(S1 X Sn-l) I NRC”) is an isomorphism for k 2 2 [10, 7.2.11]. This completes the proof of the case that m I l. m n-m For m 2 2 S X S is simply connected, and therefore M is simply connected. Since the only non-trivial homology m n-m . . . groups of S X 8 occur in dimenSions 0, m, n-m, and n, Hk(M) I 0 except possibly for k I 0, k I m, k I n-m, and k I n. Moreover, Poincaré Duality implies that HO(M) I HnCM) I Z a d that H (M) - H (M) Because H (8m x 3““) = 2 if n m n-m ° m m I n-m, either Hm(M) I O or Hm(M) I 2 if m I n-m. If Hm(M) I 0, then the Hurewicz homomorphism ¢:‘"n(M)'T Hnflfl) is again an isomorphism. As before, any representative of the 14 class w-1(”M) is a mapping of degree one from Sn to M, and such a map is necessarily a homotopy equivalence. When Hm(M) - z, 13*: H*(Sm x 3““) -. H*(M) is an iso- morphism. Hence f: Sm X Sn-m I M is a homotopy equivalence. Suppose now that m I n-m. Since Hm(SIn X Sm) I Z @ Z, it follows that Hm(M) I 0, HmCM) I Z, or HmCM) I Z 62. When Hm(M) I O or Hm(M) I Z @>Z, the preceding arguments prove that M has the homotopy type of Sn or that f is a homotopy equivalence, reSpectively. So it remains to show only that Hm(M) I Z is impossible. Assume that Hm(M) I Z, and choose a generator a E Hm(M) I Hm(M). Poincaré Duality gives I e((aU a) n “‘M) " 2(an (an “M” ‘Z. A similar result is true for n-manifolds, as the following theorem shows. Theorem 2.2: Let M be a closed, connected,orientable n-manifold with n1(M) I Eé: Z. Then there exists no mapping f: Tn I M of degree one. Proof: Assume that f: Tn I’M is a mapping of degree one. Since the Kernel of f#: n1(Tn) I n1(M) is Z, the cover- ing space of Tn corresponding to the Kernel of f# is homeo- morphic to Rn-1 X 81. Let p: Rn-1 X 31 I Tn denote this cover- ing Space, and let q: M«I M. be the universal covering space of M. An argument similar to that used in the proof of Theorem: 1.4 (c.) gives a commutative diagram 16 Rn‘l x s1 f _;174 P [<1 f T“ 3M in which I is a mapping of degree one. Thus Theorem 1.1 implies that f*: H.k(Rn-1 X SI) I H*(M) is an epimorphism. So M is homologically trivial and hence contractible. But then [10, n-l 7.2.11] implies that M is a Space of type (6 2,1) - an i=1 impossibility. This chapter concludes with the previously mentioned analogue of Theorem 2.1. Theorem 2.3: Let M. be a closed, connected, orientable 3-manifold and f: S1 X S1 X S1 I M a mapping of degree one. Then one of the following is true: (a.) M. has the homotopy type of 83; (b.) M has the homotopy type of S1 x 82; (c.) f is a homotopy equivalence. Proof: It follows from Theorem 1.1 that H1(M) I 0, H104) I Z, H1(M) I Z @Z, or H1(M) I Z 92 @Z; thus, in view of Theorem 2.2, either 111(M) I 0, 111(M) I Z, or 111(M) I Z 6 Z 6 Z. Moreover, H1(M) I H2(M) by Poincaré Duality and the universal coefficient theorem for cohomology. If n1(M) I 0, then the Hurewicz homomorphism T“"3(M) I H3(M) is an isomorphism. As before, any representative of the class m-1(uM), where “M is the preferred generator of 17 H3(M), is a homotopy equivalence between M and 83. When “1(M) I Z, then M is homotopically equivalent to a prime manifold, for M has a decomposition as a connected sum P1 # P2 #'°°# Pk of prime manifolds [7, Theorem 1], and 111(P1 # P2 #---# Pk) I n1(P1) * n1(P2) *--°* "1(Pk). Hence M is homotopically equivalent to S1 X 82 or to an irreducible 3-manifold [7, Lemma 1]. If the latter were true, then the universal covering space of M would be contractible. But then M would be a Space of type (2,1) and would be homotopically equivalent to SI. This contradiction shows that M is homotopically equivalent to S1 X S2. Finally, if nICM) I Z ® Z 6 Z, then f induces a mapping ~ 3 ... of degree one f: R. I M on the universal covering spaces of S X S X S and M. Thus M is contractible, and M. is a space of type (Z (+3 2 ® 2,1). Hence there are isomorphisms 1 X 81) I«nkan) for all k, and therefore f is f#: nk(S1 x S a homotopy equivalence. Here, as in Theorem 2.1, the three possibilities in the conclusion of the theorem actually occur, for two copies of S1 can be collapsed to give 82, and the resulting 81 X 82 can be collapsed as before to give S CHAPTER III 1 1 1 l 2 A CHARACTERIZATION OF S X S x S AND S X S Kyung W. Kwun.[5] and Jeffrey L. Tollefson [12, 13] have investigated conditions under which a 3-manifold admits a finite-sheeted covering of itself. In this chapter a study will be made of those closed, connected 3-manifolds M. with the property that every connected finite-sheeted covering space over M is homeomorphic to M. The ciass of such 3-manifolds will be denoted by {02. Notice first that any closed, simply connected 3-manifold trivially belongs to ‘ML for a simply connected manifold has no non-trivial, connected, finite-sheeted coverings. In fact, the fundamental group of a manifold in ED! must be either zero or infinite, else the universal covering space is homeomorphic to the manifold. Moreover, since every manifold has an orientable double covering, each manifold in ED! is orientabler The only 3-manifold which is not prime and admits a k-fold covering (k 2 2) of itself is P3 # P3, the connected sum of two copies of projective 3-Space [13, Theorem 1]. Hence any manifold in £0: with non-zero fundamental group is a prime manifold (since S1 x 82 double covers P3 # P3) and thus is homeomorphic to S1 X 82 or is irreducible. Furthermore, if M Efm’ is irreducible and H1(M) is infinite, then [5, Theorem 2] shows that M fibers over S1 since the hypothesis that 18 II El.- III III! I'll: ‘I I III. .i 19 H1(M) contain no element of order 2 is required only for the proof of [5, Proposition 5.1], where the assumption that HICM) is infinite is clearly sufficient. Both 81 X S1 X S1 and S1 X 82 are members of 5]! having non-zero fundamental groups. Whether other Such 3-manifolds exist remains unanswered. In this chapter two theorems will be 1 l 1 1 2 obtained which characterize S X S X S and S X S in the class ER. The first of these appears below. Theorem 3.1: Let M be a manifold in the class :13 with non-zero, nilpotent fundamental group. Then M is homeo- l morphic either to S1 X S X S1 or to S1 X 32. Proof: If M is not homeomorphic to S1 X 52, then M is irreducible. Since nICM) must be infinite, the universal covering space of M is non-compact and hence, contractible. Thus nk(M) I 0 for k 2 2, and 11104) has no elements of finite order. It follows from [11, Theorem N] that 11104) I Z, nICM) I Z Q Z 6 Z, or 111(M) is a split extension of Z e Z by Z in which the action of Z is defined by the matrix where m is a non-zero integer. Now n1(M) I Z, since otherwise M would be a space of type (2,1), and 111(M) I Z 6 Z 6 Z implies that M is homeomorphic to S1 X S1 X S1 [8, Theorem 1]. So it suffices to show that no 3-manifold in SI]! has as its fundamental group a split extension of the type described 20 above. In view of [8, Theorem 1] there is a unique irreducible 3-manifold N with fundamental group isomorphic to the Split extension 1IZ®ZI111(N)IZ-ol in which the action of Z is defined by the matrix where m is a non-zero integer. From [11] it can be seen that 1 1 this manifold can be obtained from S X S X I by identifying (e2n1a, 821115, 0) with (ezma, shim”), 1). Let 1?: be the 3-manifold obtained from S1 X 81 X I by identifying (eZTTia, ez‘fla, 0) with (e2"i°’, e2"i(3+2““), 1). A double covering p: N I N is defined by p(e2flia, e21118 (e4nia’ ezflia, t). (That p is well-defined follows from the equality p(eZ1-ria, 82ni(3+2ma), 1) = (elm-rial, eZ-rri(a+2ma)’ 1) (9.2111(201), e211i(3+m(2a)) 1) ~ (821110.01), e2nia, 0) 3 3 o 2 ' p