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Spring 2026 Algebraic topology seminar

Where:

507 Mathematics

When:

Fridays 9:30am-10:30am (unless otherwise noted)

Date	Speaker (affiliation)	Title & Abstract
January 23	Boris Botvinnik (University of Oregon)	Brunnian Links and Kontsevich Graph Complex Abstract: Recently, Tadayuki Watanabe disproved the Smale Conjecture, which stated that the group $\mathrm{Diff}_\partial(D^4)$ is homotopically trivial. He showed that there are nontrivial homotopy groups $π_q \mathrm{Diff}_\partial(D^4) \otimes \mathbb{Q}$. The first key idea was to exhibit smooth $D^4$-bundles using trivalent graphs equipped with Hopf and Borromean links as their blueprints. The second key in the construction was the use of Kontsevich’s configuration space integral to detect the non-triviality of such bundles. These ideas were generalized by Watanabe for higher-dimensional disks $D^d$, $d \geq 4$. I will explain some of the ideas and constructions of Watanabe’s work. In our very recent joint work, we used Brunnian links to construct a chain map from the Kontsevich graph complex to the rational singular chain complex of $B\mathrm{Diff}_\partial(D^{2k})$ when the dimension $2k$ is sufficiently large. Then we use again Kontsevich’s configuration space integral to detect the nontriviality of such homology elements. In particular, we provide new constructions of non-trivial elements in the homotopy groups $\pi_{8k−10}(B \mathrm{Diff}_\partial(D^{2k})) \otimes \mathbb{Q}$ (for $k \geq 17$) which are derived from well-known cycles in the graph complex.
January 30	Dan Petersen (Stockholm University)	A classifying space for the handlebody group Abstract: We use Teichmüller theory to construct a new geometric model for the classifying space of the mapping class group of a three-dimensional handlebody. Two consequences are obtained: (i) Chan-Galatius-Payne have recently shown that the homology of Kontsevich's commutative graph complex injects into the homology of the mapping class groups of surfaces, producing an enormous amount of highly unstable homology classes. We show that this homomorphism factors through the homology of the corresponding handlebody mapping class groups. (ii) The handlebody mapping class group is a virtual duality group in the sense of Bieri-Eckmann, with dualizing module given by a certain complex of nonsimple disk systems; the analogous result for mapping class groups of surfaces is a theorem of Harer. (Joint with Louis Hainaut and with Ric Wade.)
February 6	TBA	Title Abstract:
February 13	TBA	Title Abstract:
February 20	David Chan (Michigan State University)	Title Abstract:
2:45-3:45pm on Thursday, Feb 26 in Mathematics 520	Manuel Rivera (Purdue University)	Title Abstract:
March 6	TBA	Title Abstract:
March 13	TBA	Title Abstract:
March 20	No seminar	--
March 27	Adela Zhang (University of Copenhagen)	Title Abstract:
April 3	TBA	Title Abstract:
April 10	TBA	Title Abstract:
April 17	Craig Westerland	Title Abstract:
April 24	Alexander Kupers (University of Toronto)	Title Abstract:
May 1	David Gepner (Johns Hopkins University)	Title Abstract: