Topology Seminar talks at Southampton

The talks are all happening at University of Southampton (in person), on Highfield Campus (SO17 1BJ). For the building and the room, see the individual postings. Email the organiser ( d (dot) kasprowski (at) soton.ac.uk ) if you have questions.

More Pure Maths talks at Southampton: we run Pure Maths Colloquia, Pure Lunchtime Seminar, and Pure Postgraduate Seminar (link soon).

Upcoming talks

• Mon 3 Feb 2025 14:00 (in 54 / 10031 (10C)): Jan Steinebrunner (Cambridge): TBA
  No abstract yet.
• Mon 10 Feb 2025 14:00 (in 54 / 10031 (10C)): Jérôme Scherer (EPFL): TBA
  No abstract yet.
• Mon 17 Feb 2025 14:00 (in 54 / 10031 (10C)): Daniel Galvin (MPIM Bonn):TBA
  No abstract yet.
• Mon 17 Mar 2025 14:00 (in 54 / 10031 (10C)): Sebastian Chenery (Bristol): Gyration Stability for Projective Planes
  unroll abstract

  Gyrations are operations on manifolds that first arose in geometric topology. A given manifold M may exhibit different gyrations depending on the chosen twisting, prompting the following natural question: do all gyrations of M share the same homotopy type regardless of which twisting we choose? Inspired by recent work of Duan, which demonstrated that the quaternionic projective plane is not gyration stable (but with respect to diffeomorphism) in this talk we will explore our question for projective planes in general, resulting in a complete description of gyration stability for the complex, quaternionic, and octonionic projective planes up to homotopy. Moreover, we will also see that these results connect to several seemingly distinct contexts. This is joint work with Stephen Theriault.

• Mon 24 Mar 2025 14:00 (in 54 / 10031 (10C)): Lukas Brantner (Oxford): TBA
  No abstract yet.

Recent talks

• Mon 9 Dec 2024 14:00: Csaba Nagy (Glasgow): The classification of 3-connected 8-manifolds
  unroll abstract

  In 1962 Wall classified (n-1)-connected 2n-manifolds up to the action of homotopy 2n-spheres. To obtain a complete diffeomorphism classification based on Wall's result, the stabiliser of each manifold, called the inertia group, has to be determined. I will present the computation of the inertia groups of 3-connected 8-manifolds. This relies on Crowley's Q-form conjecture, describing a sufficient condition for two manifolds to be diffeomorphic, which is proved using a new, extended version of surgery theory.

• Mon 2 Dec 2024 14:00: Kamen Pavlov (Southampton): On the homotopy classification of 4-manifolds with fundamental group ZxZ/p
  unroll abstract

  The quadratic 2-type introduced by Hambleton and Kreck in 1988 has been shown to determine the homotopy type of closed, oriented 4-manifolds for certain classes of fundamental groups, including ZxZ/2, though it fails in general. A counterexample is given by products of 3-dimensional lens spaces with the circle. In this talk we will sketch the proof of the result that, for general manifolds with fundamental group ZxZ/p, fixing the quadratic 2-type imposes the upper bound of p on the number of possible homotopy types. Along the way we will also give a generalization of a result of Hambleton and Kreck on the homotopy classification of oriented Poincaré 4-complexes.

• Mon 25 Nov 2024 14:00: Jelena Grbic (Southampton): Steenrod operations on polyhedral products
  unroll abstract

  I will describe the action of the mod 2 Steenrod algebra on the cohomology of various polyhedral products and related spaces. I'll illustrate this out for Davis-Januszkiewicz spaces and their generalizations, for moment-angle complexes as well as for certain polyhedral joins. By studying the combinatorics of underlying simplicial complexes, some consequences will be deduced for the lowest cohomological dimension in which non-trivial Steenrod operations can appear.

  In addition, I will explain the idea of ``propagating'' the Steenrod algebra action from a simplicial complex K to polyhedral joins over K. Naturally, I will tie the propagation of the Steenrod algebra actions on polyhedral joins to those on moment-angle complexes.

• Mon 18 Nov 2024 14:00: Julie Rasmusen (Warwick): THR of Poincaré ∞-categories
  unroll abstract

  In recent years, work by Calmés-Dotto-Harpaz-Hebestreit-Land-Moi-Nardin-Nikolaus-Steimle has moved the theory of Hermitian K-theory into the framework of stable ∞-categories. I will introduce the basic ideas and notions of this new theory, but as it is often the case when working with K-theory in any form, this can be very hard to describe. I will therefore introduce a tool which might make our life a bit easier: Real Topological Hochschild Homology. I will explain the key ingredients that goes into constructing in particular the geometric fixed points of this as a functor, generalising the formula for ring spectra with anti-involution of Dotto-Moi-Patchkoria-Reeh, as well as providing a suitable version of Morita invariance in this new setting.

• Mon 4 Nov 2024 14:00: Miguel Barrero (Aberdeen): Global transfer systems and global N_\infty-operads.
  unroll abstract

  In this talk I will start by introducing the notion of a global transfer system, which is a collection of subgroup pairs satisfying some simple conditions. I will then show how they can be explicitly classified in the case of abelian compact Lie groups.

  In the second part, I will explain the motivation behind the definition of a global transfer system, coming from global homotopy theory. Unstable global homotopy theory is the study of global spaces, which have compatible actions by all compact Lie groups. Global N_\infty-operads parametrize multiplications on global spaces with intermediate levels of commutativity. This intermediate commutativity is seen in the existence of certain transfer maps between the homotopy groups of the global space, and this collection of transfer maps forms a global transfer system. In fact, I have proven that global N_\infty-operads are completely classified by their associated global transfer systems.

• Mon 21 Oct 2024 14:00: Ian Leary (Southampton): Chern classes for p-local finite groups.
  unroll abstract

  The cohomology ring of a finite group is fairly well-understood, largely due to
  Quillen's theorem describing its prime ideal spectrum. Many years ago, David
  Green and I gave a similar result for various subrings of the cohomology ring,
  including the subring generated by all Chern classes.

  A `p-local finite group' is a topological object that looks like a finite group
  `from the point of view of the prime p'. The analogue of Quillen's theorem is
  folklore, written up by Markus Linckelmann. Recently Jason Semeraro and I have
  established the analogue of my theorem with David Green in this context.

  I will give a brief description of some of this material, including a crash
  course on cohomology of finite groups and the statement of our new theorem.

• Mon 14 Oct 2024 14:00: Maximilian Hans (Southampton): A Light Bulb Theorem for Multi-Disks
  unroll abstract

  Given an embedded link in the boundary of a 4-manifold, a notoriously difficult question to answer is whether the link bounds an embedded disjoint union of disks in the 4-manifold, called a slice multi-disk. The presence of dual spheres simplifies this problem tremendously - each component of the link just needs to be null-homotopic. In this setting, I aim to give a sketch on how to classify the isotopy classes of such slice multi-disks by studying the homotopy type of the corresponding embedding space. This is built upon work of Danica Kosanović and Peter Teichner.

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