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Register a new userVanishing of all equivariant obstructions and the mapping degree
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Suppose that $n eq p^k$ and $n eq 2p^k$ for all $k$ and all primes $p$. We prove that for any Hausdorff compactum $X$ with a free action of the symmetric group $mathfrak S_n$ there exists an $mathfrak S_n$-equivariant map $X to {mathbb R}^n$ whose image avoids the diagonal ${(x,xdots,x)in {mathbb R}^n|xin {mathbb R}}$. Previously, the special cases of this statement for certain $X$ were usually proved using the equivartiant obstruction theory. Such calculations are difficult and may become infeasible past the first (primary) obstruction. We take a different approach which allows us to prove the vanishing of all obstructions simultaneously. The essential step in the proof is classifying the possible degrees of $mathfrak S_n$-equivariant maps from the boundary $partialDelta^{n-1}$ of $(n-1)$-simplex to itself. Existence of equivariant maps between spaces is important for many questions arising from discrete mathematics and geometry, such as Knesers conjecture, the Square Peg conjecture, the Splitting Necklace problem, and the Topological Tverberg conjecture, etc. We demonstrate the utility of our result applying it to one such question, a specific instance of envy-free division problem.
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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.
Given a null-homologous knot $K$ in a rational homology 3-sphere $M$, and the standard infinite cyclic covering $tilde{X}$ of $(M,K)$, we define an invariant of triples of curves in $tilde{X}$, by means of equivariant triple intersections of surfaces. We prove that this invariant provides a map $phi$ on $Al^{otimes 3}$, where $Al$ is the Alexander module of $(M,K)$, and that the isomorphism class of $phi$ is an invariant of the pair $(M,K)$. For a fixed Blanchfield module $(Al,bl)$, we consider pairs $(M,K)$ whose Blanchfield modules are isomorphic to $(Al,bl)$, equipped with a marking, {em i.e.} a fixed isomorphism from $(Al,bl)$ to the Blanchfield module of $(M,K)$. In this setting, we compute the variation of $phi$ under null borromean surgeries, and we describe the set of all maps $phi$. Finally, we prove that the map $phi$ is a finite type invariant of degree 1 of marked pairs $(M,K)$ with respect to null Lagrangian-preserving surgeries, and we determine the space of all degree 1 invariants of marked pairs $(M,K)$ with rational values.
These are the lecture notes for my course at the 2011 Park City Mathematics Graduate Summer School. The first two lectures covered the basics of the Torelli group and the Johnson homomorphism, and the third and fourth lectures discussed the second cohomology group of the level p congruence subgroup of the mapping class group, following my papers The second rational homology group of the moduli space of curves with level structures and The Picard group of the moduli space of curves with level structures.
We prove that any mapping torus of a closed 3-manifold has zero simplicial volume. When the fiber is a prime 3-manifold, classification results can be applied to show vanishing of the simplicial volume, however the case of reducible fibers is by far more subtle. We thus analyse the possible self-homeomorphisms of reducible 3-manifolds, and use this analysis to produce an explicit representative of the fundamental class of the corresponding mapping tori. To this end, we introduce a new technique for understanding self-homeomorphisms of connected sums in arbitrary dimensions on the level of classifying spaces and for computing the simplicial volume. In particular, we extend our computations to mapping tori of certain connected sums in higher dimensions. Our main result completes the picture for the vanishing of the simplicial volume of fiber bundles in dimension four. Moreover, we deduce that dimension four together with the trivial case of dimension two are the only dimensions where all mapping tori have vanishing simplicial volume. As a group theoretic consequence, we derive an alternative proof of the fact that the fundamental group $G$ of a mapping torus of a 3-manifold $M$ is Gromov hyperbolic if and only if $M$ is virtually a connected sum $# S^2times S^1$ and $G$ does not contain $mathbb{Z}^2$.
In the setting of finite type invariants for null-homologous knots in rational homology 3-spheres with respect to null Lagrangian-preserving surgeries, there are two candidates to be universal invariants, defined respectively by Kricker and Lescop. In a previous paper, the second author defined maps between spaces of Jacobi diagrams. Injectivity for these maps would imply that Kricker and Lescop invariants are indeed universal invariants; this would prove in particular that these two invariants are equivalent. In the present paper, we investigate the injectivity status of these maps for degree 2 invariants, in the case of knots whose Blanchfield modules are direct sums of isomorphic Blanchfield modules of Q-dimension two. We prove that they are always injective except in one case, for which we determine explicitly the kernel.
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