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Register a new userChern-Simons Functional and the Homology Cobordism Group
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Authors
Aliakbar Daemi
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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.
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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.
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.
Kronheimer and Mrowka asked whether the difference between the four-dimensional clasp number and the slice genus can be arbitrarily large. This question is answered affirmatively by studying a knot invariant derived from equivariant singular instanton theory, and which is closely related to the Chern--Simons functional. This also answers a conjecture of Livingston about slicing numbers. Also studied is the singular instanton Fr{o}yshov invariant of a knot. If defined with integer coefficients, this gives a lower bound for the unoriented slice genus, and is computed for quasi-alternating and torus knots. In contrast, for certain other coefficient rings, the invariant is identified with a multiple of the knot signature. This result is used to address a conjecture by Poudel and Saveliev about traceless $SU(2)$ representations of torus knots. Further, for a concordance between knots with non-zero signature, it is shown that there is a traceless representation of the concordance complement which restricts to non-trivial representations of the knot groups. Finally, some evidence towards an extension of the slice-ribbon conjecture to torus knots is provided.
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.
We develop a theory of equivariant group presentations and relate them to the second homology group of a group. Our main application says that the second homology group of the Torelli subgroup of the mapping class group is finitely generated as an $Sp(2g,mathbb{Z})$-module.
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