Topology Seminar

Seminars During Academic Year 2004-05

Fall 2004

Tuesday August 10, 2004 in Rawles Hall 316

Stefan Friedl, New examples of topologically slice knots

Wednesday October 6, 2004 in Rawles Hall 104

P. Kirk, The SU(3) Casson invariant for Brieskorn Spheres

Wednesday October 13, 2004 in Rawles Hall 104

P. Kirk, The SU(3) Casson invariant for Brieskorn Spheres

Wednesday October 20, 2004 in Rawles Hall 104

Jeremy Brookman, Splitting homotopy equivalences along codimension 1 submanifolds

Abstract: In the 1970s, Cappell defined the obstruction groups UNil_n, containing the obstruction to splitting a homotopy equivalence of (n-1)-dimensional manifolds along a codimension 1 submanifold. Separate consideration was given to the cases of n being odd and n being even. I shall describe the algebraic definition of the UNil groups and the construction of the splitting obstruction, including a new definition when n is odd using the algebraic theory of surgery of Ranicki.

Wednesday October 27, 2004 in Rawles Hall 104

Jeremy Brookman, Splitting homotopy equivalences along codimension 1 submanifolds (continued)

In the 1970s, Cappell defined the obstruction groups UNil_n, containing the obstruction to splitting a homotopy equivalence of (n-1)-dimensional manifolds along a codimension 1 submanifold. Separate consideration was given to the cases of n being odd and n being even. I shall describe the algebraic definition of the UNil groups and the construction of the splitting obstruction, including a new definition when n is odd using the algebraic theory of surgery of Ranicki.

Wednesday November 3, 2004 in Rawles Hall 104

Jeremy Brookman, Splitting homotopy equivalences along codimension 1 submanifolds (continued)

Wednesday November 10, 2004 in Rawles Hall 104

Qayum Khan, Stable homeomorphisms of 4-manifolds with infinite fundamental group

I.Hambleton and M.Kreck (1993) showed for closed topological 4-manifolds $X$ and $Y$ with finite fundamental group that: if there exists a homeomorphism between $X # r(S^2\times S^2)$ and $Y # r(S^2times S^2)$ for some $r$, then there exists a homeomorphism between $X$ and $Y$. Such $X$ and $Y$ in the hypothesis are called “stably homeomorphic”. There is an additional hypothesis that $X$ itself is internally a connected sum $X_0 # S^2\times S^2$. I extend their theorem to the case of infinite (good) fundamental group, under certain ring-theoretic conditions. The motivating example comes from understanding the tangential homotopy type of the connected sum of two copies of real projective 4-space along a homotopy 3-sphere, which has fundamental group the infinite dihedral group.

Wednesday December 1, 2004 in Rawles Hall 104

Yuhan Lim, SU(2) Seiberg-Witten Invariants.

Wednesday December 8, 2004 in Rawles Hall 104

Chuck Livingston, Twisted Alexander Polynomials of Periodic Knots

I will describe joint work with Hillman and Naik giving extensions of Murasugi's results on the Alexander polynomials of periodic knots the setting of twisted polynomials.

Spring 2005

Tuesday January 25, 2005 in Rawles Hall 104

Jae Choon Cha, Homology Equivalences and Algebraic Closures

We discuss the relationship between algebraic closures and homology equivalences, and some applications to homology cobordisms of odd-dimensional manifolds and link concordance.

Monday January 31, 2005 in Rawles Hall 104

Slavik Jablan

Monday January 31, 2005 in Rawles Hall 104

Slavik Jablan

Wednesday February 2, 2005 in Rawles Hall 104

Julia Viro, Lines and circles meeting links

We will consider low bounds for the number of lines meeting given 4 disjoint smooth closed curves in the real projective 3-space in a given cyclic order. Similarly, we estimate the number of circles meeting in a given cyclic order given 6 disjoint smooth closed curves in Euclidean 3-space. The estimations are formulated in terms of linking numbers of the curves and based on a study of a surface swept by projective lines meeting 3 given disjoint smooth closed curves and a surface swept by circles meeting 5 given disjoint smooth closed curves. These results admit generalizations to higher dimensions and more complicated patterns of intersections. Lines and circles can be replaced by configurations of curves of other kinds; linking numbers, by Vassiliev invariants.

Thursday February 3, 2005 in Rawles Hall 104

Oleg Viro, Khovanov homology of classical and virtual links

In 1999 Khovanov introduced homology groups for classical links such that coefficients of the Jones polynomial appear as alternating sums of the ranks of these groups. In the talk, down to earth adaptation of this construction will be discussed. It is generalized to a class of virtual links (or abstract Gauss diagrams). The original chain complex defining the Khovanov homology is huge, but can be replaced by its deformation retract which is essentially smaller than the original one.

Wednesday February 23, 2005 in Rawles Hall 104

Jim Davis, A two component link with Alexander polynomial one is concordant to the Hopf link

Wednesday March 2, 2005 in Rawles Hall 104

Chuck Livingston, The Hausmann-Weinberger invariant of Z^n

The Hausmann-Weinberger invariant of a finitely presented group is the minimum Euler characteristic of a closed 4-manifold have that group as its fundamental group. In this talk I will describe my joint work with Paul Kirk computing this invariant for finitely generated free abelian groups.

Wednesday March 9, 2005 in Rawles Hall 104

Paul Kirk, Symplectic four-manifolds with prescribed fundamental groups

Wednesday March 30, 2005 in Rawles Hall 104

Swatee Naik, Realizing Ozsvath-Szabo Invariants

Tuesday April 5, 2005 in Rawles Hall 104

Cornelia Van Cott, Braid Groups and Manifolds

A brief survey talk about braid groups

Wednesday April 27, 2005 in Rawles Hall 104

Jennifer Franko, A Representation Arising from the Yang-Baxter Equation

Thursday July 21, 2005 in Rawles Hall 104

Constance Leidy, Higher order linking forms: theorem and conjecture