Equivariant Bordism Theory and Applications
This seminar is previous to the event Equivariant Bordism Theory and Applications from BANFF International Research Station scheduled for June 18-23 in 2023 with location Casa Matemática Oaxaca.
The talks of this seminar are usually on Friday 11:00hrs Mexico City time -6GMT
Transmissions by Zoom: 971 5021 3285
In order to obtain the key of the Zoom session please SUSCRIBE. ZOOM Youtube Chanel: YouTube
Previous events: Bordism theory and finite group actions (CLAMVI)
Current events: Lecture seminar in equivariant bordism
Talks 2022:
Title:
Finite Group Actions on 1-Complexes and Homology
Resumen
We explore the extent to which a finite group action on a finite connected 1-complex is determined up to equivariant homotopy equivalence by the induced action on integral homology: How many actions could induce the same representation on homology? What representations arise? What sorts of conditions on the homology representation could guarantee a global fixed point? We discuss the case of semifree actions in detail and obtain rather complete results in the case of a cyclic group of prime order.
TBA
Title:
Resumen
In this talk I will explain the calculation of the equivariant unitary and oriented bordism groups for surfaces and its implication on the equivariant evenness conjecture for unitary bordism.
This is joint work with Andrés Angel, Eric Samperton and Carlos Segovia
Title:
Resumen
I will describe my construction of a certain “weak orientation class” and use it to derive a new relation between Borel and Mackey cohomology. (Some related observations were also made in [Hill-Hopkins-Ravenel: On the non-existence of elements of Kervaire invariant 1].) One application is a new completion theorem for complex cobordism modules that does not involve higher derived functors. Applying this result to Morava K(n)-theory, I will describe my recent counterexample to the homotopical version of the evenness conjecture for equivariant complex cobordism. I will also describe the relationship between this result and previous work on various forms of the evenness conjecture, including the recent theorems by Samperton and Uribe.
TBA
Title:
Cut and paste invariants of manifolds and cobordism
Resumen
Cut and paste or SK groups of manifolds are formed by quotienting the monoid of manifolds under disjoint union with the relation that two manifolds are equivalent if I can cut one up into pieces and glue them back together differently to get the other manifold. Cobordism cut and paste groups are formed by moreover quotienting by the equivalence relation of cobordism. We categorify these classical groups to spectra and lift two canonical homomorphisms to maps of spectra.
Title:
Cut and paste invariants and invertible TQFTs
Resumen
Two smooth, oriented, closed manifolds M and N are cut-and-paste equivalent if one can obtain N by a series of cut-and-paste operations from M. Cut-and-paste controlled invariants (SKK invariants) are functions on the set of smooth manifolds whose values on cut-and-paste equivalent manifolds differ by an error term depending only on the gluing diffeomorphisms. The relation between SKK invariant and invertible TQFTs was first investigated by Kreck, Stolz, and Teichner. In this talk, I will present the construction of a group homomorphism between the group of invertible TQFTs and the group of SKK invariants and describe how these groups fit into a split exact sequence.
Title:
Cut and paste spectrum of manifolds: K_1 invariants, relation to BCob
Resumen
We will take a closer look at K(Mfd) spectrum that was discussed in the previous talks. Recall that the zeroth homotopy group of the spectrum recovers the classical cut and paste group of manifolds SK_n. I will show how to relate the spectrum K(Mfd) to the algebraic K-theory of integers, and how this leads to the Euler characteristic and the Kervaire semicharacteristic when restricted to the lower homotopy groups. Further, I will explain how to construct the maps relating BCob, K(Mfd) and K^{cube}(Mfd) that spectrify the natural group homomorphisms relating SKK, SK, and the cobordism group.
(joint work with Mona Merling and George Raptis)