This Ph.D. dissertation studies the relationship of an involution acting on a 3-manifold (or a knot K) with the Heegaard Floer homology. There are three main aspects of this project: strong cork detection, studying homology bordism group of diffeomorphisms and explicitly computing the action of symmetry on the Knot Floer complex for symmetric knots. In the second chapter, we study pairs of an integer homology sphere equipped with an involution modulo equivariant homology cobordisms. We show that equivalence classes of the above relation form an abelian group under the group operation as disjoint union. We refer to this group as the homology bordism group of involutions. This group can be thought of as a generalized version of the bordism group of diffeomorphisms, which was first studied by Browder. We define two Floer-theoratic invariants of this group, using the framework of involutive Heegaard Floer homology, recently developed by Hendricks and Manolescu.Corks play an important role in the study of exotic smooth structures on 4-manifolds. As shown by Matveyev and Curtis-Freedman-Hsiang-Stong , any two smooth structures on a simply connected topological 4-manifold are related by the action of cork-twist. Lin-Ruberman-Saveliev studied a more generalised version of a cork, called the strong cork. These are corks for which the cork-twist involution does not extend over any homology 4-ball that the cork may bound. They also constructed the first example of such a strong cork by studying the induced action of a cork-twist on monopole Floer homology. In the third chapter, we show that the invariants developed earlier also detect strong corks. We then go on to establish several new families of corks and prove that various known examples corks are actually strong. Our main computational tool is a monotonicity theorem which constrains the behavior of our invariants under equivariant negative-definite cobordisms, and an explicit method to construct equivariant cobordisms. The contents of second and third Chapter are from a joint work of the author with Irving Dai and Matthew Hedden . In the 4-th chapter we study symmetric knots. We show that each symmetry of a knot induces a map on the knot Floer complex. We further show that these induced maps behave differently according to how the fixed set of the symmetry intersects knot. We then explicitly compute some of those maps.
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