Topology Seminar

Seminars During Academic Year 2002-03

Fall 2002

Wednesday September 11, 2002 in Rawles Hall 104

Ernesto Lupercio, Geometry and Topology of Orbifolds I

Orbifolds are geometric objects that generalize the concepts of manifolds and of finite groups. Just as a Manifold is a topological space with additional structure, an orbifold includes a topological space and the local information of a group action on a manifold. Orbifolds appear naturally in algebraic geometry and in physics, especially in the phenomenon of Mirror Symmetry. In the first talk I will introduce the category of orbifolds, and natural cohomology Theories on them. I will also introduce Gerbes and Twisted K-theory.

Thursday September 12, 2002 in Rawles Hall 104

Ernesto Lupercio, Geometry and Topology of Orbifolds II

In the second talk I will explain how the Loop Orbifold provides a natural framework for the study of topological and algebraic geometrical questions for orbifolds.

Wednesday September 18, 2002 in Rawles Hall 104

Jae Choon Cha, Concordance and rational knots

Wednesday September 25, 2002 in Rawles 104

Scott Baldridge, Seiberg-Witten invariants of 4-manifolds with S^1 actions

Abstract: In this talk we show how to calculate the SW invariant of any b_+>1 closed, smooth, oriented 4-manifold with an effective circle action. Applications of this theorem to questions in symplectic geometry and topology will be discussed.

Wednesday October 2, 2002 in Rawles Hall

Scott Baldridge, Seiberg-Witten invariants of 4-manifolds with S^1 actions, continued

Contuation of Sept. 25 seminar

Thursday October 10, 2002 in Rawles Hall 104

Diarmuid Crowley, Modified L-groups for modified surgery

One of the striking features of the classical surgery obstruction groups is that they depend only upon the fundamental group and its orientation character. The main aim of this talk is to present new L-groups for Kreck's modified surgery theory (at least in the simply connected case) which depend upon higher homotopy groups of the manifolds to be classified. As an application, we show how the new L-groups are used in the classification of simply connected 7-manifolds (with torsion free second homology). A corollary of this classification is that the set of homotopy types of simply connected 7-manifolds with rank(H_2(M)) = r is infinite for r>1.

Wednesday October 23, 2002 in Rawles Hall 104

Chuck Livingston, Knotted Contractible 4-Manifolds

Mazur constructed examples of smooth compact contractible 4-manifolds. His examples (and perhaps all such manifolds) smoothly embed in S^4 with simply connected complement. I will describe Lickorish's construction of large families of such 4-manifolds that possess embedding in S^4 with non-simply connected complement.

Thursday October 31, 2002 in Rawles Hall 104

Alejandro Adem, Fixity and Group Actions

Wednesday November 13, 2002 in Rawles Hall

Vahagn Minasian, THH and TAQ for E_infty - algebras

@algebra, Hochschild Homology (HH) and Andr'e-Quillen Homology (AQ) are homology theories for associative and associative commutative algebras respectively. We discuss the topological analogues of these theories (THH and TAQ), give some comparison results and show how THH and TAQ lead to two notions of 'etale in topological categories. We show an 'etale descent lemma for THH that generalizes the corresponding result from algebra, and use it to prove a topological analogue of the HKR theorem.

Wednesday November 20, 2002 in Rawles Hall

Taehee Kim, New Obstructions to Doubly Slicing Knots

Wednesday December 4, 2002 in Rawles Hall

Pat Gilmer, Integral Bases for TQFT modules

Wednesday December 11, 2002 in Rawles Hall

Ayelet Lindenstrauss, Calculus of Functors, 119

Spring 2003

Wednesday January 22, 2003 in Rawles Hall

Chuck Livingston, Knotted Contractible Manifolds II

In a previous seminar I described Lickorish's result that any perfect group of deficiency 0 arises as the fundamental group of the complement of a contractible manifold in S^4. In this talk I will describe the construction of contractible manifolds with complements having large negative deficiency. Applications to homology 4-spheres will also be presented.

Wednesday January 29, 2003 in Rawles Hall

John Mcatee, Intrinsic knotting and linking of graphs

Summary: A graph G is intrinsically knotted if every embedding of G in R^3 contains a nontrivial knot. G is intrinsically linked if every embedding into R^3 contains a non-split link. We will describe some examples of intrinsically knotted and linked graphs and survey the major results about this subject.

Wednesday February 19, 2003 in Rawles Hall 104

Zhenghan Wang, On TQFT representations of mapping class groups

We will discuss the generic Jones represenations of the braid groups and their relations with TQFT representations of the braid groups. The question whether the generic Jones representationis faithful is still open.

Wednesday April 2, 2003 in Rawles Hall 104

Tibor Macko, Automorphisms of infinite-dimensional real projective space

Useful information about automorphisms of RP^infty can be obtained from the so called block structure space of RP^infty. The block structure space of a compact topological manifold X is a space which has as its k-th homotopy group the group of manifold structures on X x Delta^k relative to the boundary, i.e. the group of homotopy equivalences from a manifold M to X x Delta^k which are homeomorphisms on the boundary . By assigning the block structure space of RP(V) to the finite-dimensional real vector space V we obtain a continuous functor from the category of vector spaces to (based) spaces. In this talk I will present the study of this functor from the point of view of the orthogonal calculus of M. Weiss, which is designed for the functors of this type. More concretely I will show that it is a linear functor. Tools used in the proof are mainly surgery theory and Poincare embeddings. Application of this result is an attractive description of the block structure space of RP^infty, which is a colimit of the block structure spaces of finite-dimensional real vector spaces.

Tuesday April 8, 2003 in Rawles Hall

Andrew Ranicki, Algebraic transversality

Topological transversality of manifolds is a geometric cutting technique which deforms maps on manifolds to ensure that certain preimages are submanifolds. For example, this is the geometric ingredient of the Mayer-Vietoris exact sequence for the homology of a union, and the Seifert-Van Kampen theorem expressing the fundamental group of a union of spaces as an amalgamated free product or HNN extension. Algebraic transversality considers modules and quadratic forms over amalgamated free products and HNN extensions, cutting up such objects by algebraic mimicry of the topological model. This type of algebra has applications in the topology of high-dimensional manifolds with infinite fundamental group.

Wednesday April 9, 2003 in Rawles Hall 104

Stefan Friedl, The eta invariant and slicing obstructions

We classify the metabelian unitary representations of $\pi_1(M_K)$, where $M_K$ is the result of zero-surgery along a knot $K\subset S^3$. We show that certain eta-invariants associated to metabelian representations $\pi_1(M_K)\to U(k)$ vanish for slice knots and that even more eta-invariants vanish for ribbon knots. This sliceness obstruction turns out to be at least as strong as the Casson-Gordon obstruction. It turns out that eta-invariants can in many cases be easily computed for satellite knots. We make use of this to study the relation between the eta-invariant sliceness obstruction, eta-invariant ribboness obstruction and the $L^2$-eta invariant obstruction recently introduced by Cochran, Orr and Teichner. In particular we give an example of a knot which has zero Casson-Gordon invariants, zero eta-invariant sliceness obstruction and zero metabelian $L^2$-eta invariants, but is not ribbon.

Wednesday April 23, 2003 in Rawles Hall 104

Ayelet Lindenstrauss, On the algebraic K-theory of a square-zero extension of ${\bf F}_p$

I will be discussing joint work with Randy McCarthy of UIUC in which we calculate the algebraic K-theory of a square-zero extension of F_p by a finite F_p-vector space after completion at p. The case of a one-dimensional vector space, namely that of the dual numbers over F_p, was done by Hesselholt-Madsen. We use a variant of topological cyclic homology whose construction is motivated by Goodwillie calculus.