Topology Seminar

This seminar is held on Tuesdays at 3pm in 210 Deady.

October 2, Dev Sinha (UO)
The mod-two cohomology of DX and QX

Abstract: The spaces of maps from an n-sphere to the nth suspension of a space form a directed system, whose limit is called $\Omega^\infty \Sigma^\infty$ X or QX. This space is rightly viewed both as a basic mapping space to study and the underlying space for the “free derived abelian group on X.” In this talk I give a recent Hopf-ring based approach to the mod-two cohomology ring of QX, which was originally calculated by Dung. Our approach is through the finite divided powers of X, whose cohomology rings were previously only implicitly known. We present geometric representatives for the cohomology of DX which if it could be extended to QX (for X = $CP^\infty$ ) would give such for the stable cohomology of mapping class groups.

October 9, Eric Ramos (UO)
Commutative Algebra in the Configuration spaces of Graphs

Abstract: Let G be a graph, thought of as a 1-dimensional simplicial complex. Then the n-stranded configuration space of G is the space F_n(G) = \{(x_1,\ldots,x_n) \in G^n \mid x_i \neq x_j\} / S_n, where S_n is the symmetric group on n letters. While one would hope to say something meaningful about the homology groups H_i(F_n(G)) of these spaces, it is known that they can be quite chaotic. Following recent trends in algebra, we therefore shift our focus to studying all of these groups simultaneously +_n H_i(F_n(G)), sacrificing knowledge about individual homology groups for statements which are more asymptotic in nature. In particular, it can be shown that +_n_i(F_n(G)) can be encoded as the additive group of some finitely generated graded module over an integral polynomial ring. Specializing to the case of trees, we compute the generating degree of this module, and show that it naturally decomposes as a direct sum of graded shifts of square-free monomial ideals. As an application we show that the homology groups of F_n(G), in the case of trees, are only dependent on the degree sequence of G. We conclude the talk by discussing what little is known in the general case, and provide a theorem describing aspects of the Hilbert polynomials of these modules.

October 23, Haofei Fan (UCLA)
Unoriented cobordism maps on link Floer homology

Abstract: We study the problem of defining maps on link Floer homology induced by unoriented link cobordisms. We provide a natural notion of link cobordism, disoriented link cobordism, which tracks the motion index zero and index three critical points. Then we construct a map on unoriented link Floer homology associated to a disoriented link cobordism. Furthermore, we will discuss some potential applications on the involutive upsilon invariants and unoriented four-ball genus.

November 6, Marco Golla (University of Nantes)
Special Time: 9am in 210 Deady.
Smooth concordance bounds on wrapping numbers

Abstract: Suppose that K is a knot in the solid torus T. A natural invariant of K is the wrapping number of K, i.e. the minimal number intersections of K with a meridional disc of T, up to isotopy. I will talk about an analogue 4-dimensional question for knots up to concordance (i.e., essentially, up to an annulus in T x [0,1]). The main tool will come from (twisted) correction terms in Heegaard Floer homology. This is joint work with Daniele Celoria.

November 13, Charles Katerba (Montana State)
Searching for closed essential surfaces in knot complements with character varieties

Abstract: Culler-Shalen theory uses a 3-manifold’s (P)SL(2,C) character variety to construct essential surfaces in the manifold. It has been a fundamental tool over the last 35 years in low-dimensional topology. Much of its success is due to a solid understanding of the essential surfaces with boundary that can be constructed with the theory. It turns out, however, that not every surface with boundary is detected. Moreover, one can also construct closed essential surfaces within this framework. In this talk, we will discuss a module-theoretic perspective on Culler-Shalen theory and apply this perspective to show that there are knot complements in $S^3$ which contain closed essential surfaces, none of which are detected by Culler-Shalen theory. As a corollary, we will construct an infinite family of closed hyperbolic Haken 3-manifolds whose representations into PSL(2, C) have a special number-theoretic property.