University of Virginia Topology Seminar 2007-08

I will outline the problem of determining which 3-manifolds embed in the 4-sphere. This will have essentially 3 parts:
(i) Known obstructions to a 3-manifold embedding in the 4-sphere such as torsion linking forms, Alexander polynomials, the Rochlin invariant, the Oszvath-Szabo d-invariant, Casson-Gordon invariants.
(ii) Procedures for constructing embeddings.
(iii) Computations: Recent independent work of Matveev, Petronio and Burton has produced lists of all the 3-manifolds (up to homeomorphism) that can be triangulated with 11 or less tetrahedra.
Computer implementation of known obstructions gives gives us a list of all 3-manifolds which pass the tests (1) but for which we still, as of yet, do not know if they embed in S^4. I will describe some characteristic manifolds from this list.

(Joint with Ruth Charney and Karen Vogtmann) We determine the virtual dimension of the group $P\Sigma(n,k)\subset Out(F_n)$ generated by automorphisms which send the first $k$ generators to conjugates of themselves (each x_i is going to a conjugate $wx_iw^{-1}$). Thm: $vcd(P\Sigma(n,k))=2n-k-2$.

(Joint with Ruth Charney and Karen Vogtmann) We determine the virtual dimension of the group $P\Sigma(n,k)\subset Out(F_n)$ generated by automorphisms which send the first $k$ generators to conjugates of themselves (each x_i is going to a conjugate $wx_iw^{-1}$). Thm: $vcd(P\Sigma(n,k))=2n-k-2$.

I will talk about two recent approaches to the study of smooth knot concordance, one coming from Donaldson theory, the other from Heegaard Floer homology. Then I will discuss how to combine the two to give a novel test for a knot to be slice, and apply the test to determine the concordance order of three-stranded pretzel knots.

The chain complexes in Novikov and Floer homology theories carry natural real-valued filtrations, allowing one to define the spectral number of any homology class as the infimum of the filtration levels of all chains representing that class; in the case of Hamiltonian Floer homology these numbers have had interesting applications to symplectic topology due to Schwarz, Oh, and others. I'll sketch a proof that the infimum in the definition of the spectral number is always attained. This fact allows some of the applications of the spectral numbers that have been carried out for special classes (e.g., rational or semipositive) of closed symplectic manifolds to be generalized to all closed symplectic manifolds.