Homotopical methods in manifold theory

- Alexander Kupers (University of Toronto)

- Manuel Krannich (Karlsruhe Institute of Technology)

- Markus Land (University of Munich)

- Andrew Senger (Harvard University)

Below are the topics that will be covered.

(1) An operadic setup for embedding calculus and its variants (4 talks, Kupers)

Several recent applications of homotopy theory to manifold theory rely on Goodwillie—Weiss' embedding calculus, which is a homotopy-theoretic tool to study spaces of smooth embeddings. This series of lectures introduces from scratch a unifying higher-categorical setup for embedding calculus and its variants (such as a version for topological embeddings or Boavida de Brito--Weiss‘ theory of configuration categories), ready to be used for further applications (as exemplified by the accompanying lectures series on Pontryagin-Weiss classes).

(2) Pontryagin—Weiss classes and diffeomorphisms of discs (3 talks, Krannich)

Pontryagin classes were originally considered as invariants of real vector bundles, but it was realised in the 60s that they can be defined more generally for Euclidean bundles, that is, fibre bundles whose fibres are homeomorphic to Euclidean space. This led to the question whether the well-known vanishing of large-degree Pontryagin classes for small-dimensional vector bundles continues to hold in the setting of Euclidean bundles. Surprisingly, Weiss proved a few years ago that this often fails, even for bundles over spheres. In this lecture series we discuss a strengthening of this result: For all k, there exists a 6-dimensional Euclidean bundle over a sphere whose kth Pontryagin class is nontrivial. Through smoothing theory, this yields infinite families of rationally non-zero elements in the homotopy groups of diffeomorphisms of discs.

(3) Grothendieck—Witt theory and relations to geometric topology (4 talks, Land)

I would like to talk about how to determine GW(Z) in more or less all its variants and how to use it to calculate the cohomology of certain isometry groups of forms and determine universally valid divisibility results for local systems of unimodular forms over framed base manifolds. For the last lecture, I would like to focus on the (visible) symmetric signature of a PD complex and the total surgery obstruction, and possibly mentioning our current work (very much in the beginning of progress) on L-theory of manifolds and its aspects for homology manifolds.

(4) Synthetic homotopy theory and classifications of highly-connected manifolds (4 talks, Senger)

All lecturers are held in the August Krogh Building, Auditorium 1, Universitetsparken 13, 2100 København Ø.
Registration is also in August Krogh Bygningen. 

The registration has closed.

Pierre Elis

Branko Juran

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