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Register a new userHomology cobordism and triangulations
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Authors
Ciprian Manolescu
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The study of triangulations on manifolds is closely related to understanding the three-dimensional homology cobordism group. We review here what is known about this group, with an emphasis on the local equivalence methods coming from Pin(2)- equivariant Seiberg-Witten Floer spectra and involutive Heegaard Floer homology.
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In this paper we define and investigate Z/2-homology cobordism invariants of Z/2-homology 3-spheres which turn out to be related to classical invariants of knots. As an application we show that many lens spaces have infinite order in the Z/2-homology cobordism group and we prove a lower bound for the slice genus of a knot on which integral surgery yields a given Z/2-homology sphere. We also give some new examples of 3-manifolds which cannot be obtained by integral surgery on a knot.
For each integral homology sphere $Y$, a function $Gamma_Y$ on the set of integers is constructed. It is established that $Gamma_Y$ depends only on the homology cobordism of $Y$ and it recovers the Fr{o}yshov invariant. A relation between $Gamma_Y$ and Fintushel-Sterns $R$-invariant is stated. It is shown that the value of $Gamma_Y$ at each integer is related to the critical values of the Chern-Simons functional. Some topological applications of $Gamma_Y$ are given. In particular, it is shown that if $Gamma_Y$ is trivial, then there is no simply connected homology cobordism from $Y$ to itself.
We give simple homological conditions for a rational homology 3-sphere Y to have infinite order in the rational homology cobordism group, and for a collection of rational homology spheres to be linearly independent. These translate immediately to statements about knot concordance when Y is the branched double cover of a knot, recovering some results of Livingston and Naik. The statements depend only on the homology groups of the 3-manifolds, but are proven through an analysis of correction terms and their behavior under connected sums.
We show that if a prime homology sphere has the same Floer homology as the standard three-sphere, it does not contain any incompressible tori.
We review the construction and context of a stable homotopy refinement of Khovanov homology.
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