Schedule for: 23w5138 - Equivariant Bordism Theory and Applications

In this talk I will summarize some of the properties of the equivariant unitary bordism groupthat have been shown lately. I will also present some results that seem to be related to the equivariant bordism group, as well as some open problems in the area.

Fix a finite group $G$ and an oriented surface $S$. Given an action of $G$ on $S$, when does this action extend to an action on a 3-manifold $M$ with boundary $S$? What can we say about $M$ or the qualities of the action of $G$ on $M$? I’ll quickly review the concrete answer that Angel, Segovia, Uribe and I gave to the first question. I’ll then report on various examples and counterexamples regarding the second question; this part of the talk is based on joint work with Marco Boggi and Carlos Segovia.

Problem 1. Let $G$ be a compact Lie group and $M$ a closed smooth $G$ manifold. Does there exist a nonsingular real algebraic $G$ variety $X$ equivariantly diffeomorphic to $M$? so that all equivariant vector bundles over $X$ are strongly algebraic (classified by entire rational maps)? For the trivial group this problem was posed by J. Nash and resolved positively by Tognoli (including the bundle question by Benedetti and Tognoli). The problem reduced to a bordism question that had been solved previously. In the equivariant setting the algebraic realization problem reduces to the following equivariant bordism problem. Problem 2. Does every class in $\mathcal N_*^G(\mathfrak G)$ have an algebraic representative? Here $\mathcal N$ denotes unoriented bordism and $\mathfrak G$ is a product of equivariant Grassmannians. An {\em algebraic representative} of a class in $\mathcal N_*^G(\mathfrak G)$ is a nonsingular real algebraic $G$ variety $X$ with an equivariant entire rational map $f:X \to \mathfrak G$. Solutions to the bordism problem provide answers to the algebraic realization problem. There are some efficient tools that help analyze equivariant bordism problem, such as blow--ups and reduction to $2$ groups. There are some computations of equivariant bordism groups that come in handy, like for elementary abelian $2$ groups. We did our own computations that yield algebraic realization results for cyclic groups, and groups with cyclic Sylow $2$ subgroups. On the other hand, manifolds of positive dimension have an uncountable number of birationally inequivalent algebraic models. In the equivariant setting only homogeneous spaces have unique algebraic structure.

I will discuss the properties of orientation classes in Mackey cohomology and their relationship to Borel cohomology. As an application, I will describe a non-derived completion theorem for certain equivariant ring and module spectra. I will also discuss an application of these results to finding examples of finite groups G whose G-equivariant complex cobordism Thom spectra coefficients are not flat modules generated in even degrees over non-equivariant complex cobordism. This gives counterexamples to the homotopical version of the evenness conjecture for equivariant complex cobordism.

Compositional Quantum Field Theory (CQFT) is an axiomatic framework for quantum theories that focuses on spacetime locality and compositionality. I will explain what the axioms for CQFT mean, how they can be encoded as involutive symmetric monoidal functors with extra structure, and I will explain how every CQFT contains a criterion for when the action by homeomorphisms of a group on a closed oriented hypersurface extends to an action on a region bounding it.

In this talk, we will be concerned with a relation between TQFTs and the controlled cut-and-paste invariants introduced by Karras, Kreck, Neumann, and Ossa. The controlled cut-and-paste 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. I will present a natural 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. We conclude in particular that all positive real-valued SKK invariants can be realized as restrictions of invertible TQFTs.

Cut and paste or SK groups of manifolds are formed by quotienting the monoid of manifolds under disjoint union by the relation that two manifolds are equivalent if I can cut one up into pieces and glue them back together to form 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 of groups to maps of spectra. This is joint work with Mona Merling, Laura Murray, Carmen Rovi and Julia Semikina.

We will take a close look at the K(Mfd) spectrum whose zeroth homotopy group 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.

Drinfeld doubles are certain braided tensor categories associated to finite groups. 3-cocyles of group cohomology can be used to modify )or twist) Drinfeld doubles. We use this categorical interpretation of 3-cocyles to define a filtration on the third cohomology, measuring the strength of "non-triviality" of a cocycle.

Given a Lie group $G$ acting on a manifold $M$, one can consider manifolds built from this local model for rigid geometries with isometry group $G$; i.e. one can look at manifolds with an atlas of charts into open subsets of $M$, where the transition functions for these charts are given by the $G$-action on $M$. One use for this category of manifolds equipped with a rigid geometry is as input data for factorization algebras, a model of observables of a field theory. I show that factorization algebras on the category of all manifolds equipped with a rigid geometry given by the pair $(M, G)$ are equivalent to equivariant factorization algebras on $M$. This is related to work of Dwyer-Stolz-Teichner on bordism categories of manifolds equipped with rigid geometry and topological quantum field theories constructed from these. I will briefly sketch the relationship to this work.

The rationality problem in algebraic geometry asks whether a given variety is birational to a projective space. We will gently introduce the problem and take a look at some recent advances, principally in the direction of negative examples constructed via cohomological obstructions. Special focus will be set on quotient varieties by linear group actions.

We present the non-orientable version of the Schur and Bogomolov multiplier associated with a finite group $G$. They serve as obstructions to extending finite free actions from non-orientable surfaces to 3-manifolds. We provide the Miller description of the Schur multiplier regarding universal relations for squares. This allows us to define the non-orientable Bogomolov multiplier as the quotient of the Schur multiplier by the group generated by 1-tori, 1-Klein bottle, and 1-projective space,. We show that every finite free action over non-orientable closed surfaces, different from the trivial $G$-bundle over the projective space, always extends for abelian and dihedral groups.

We present calculations for the non-orientable Schur multiplier, particularly for the cyclic and dihedral groups. We explicitly give each case's elements and what this means in the non-orientable Bogomolov Multiplier. Finally, we show an example of a non-trivial element in the Bogolomov multiplier that is trivial in the non-oriented case.

We find an analogous of Hopf formula for the non-oriented Schur multiplier $\mathcal{N}(G)$. We also give an interpretation of $\mathcal{N}(G)$ in terms of a certain class of group extensions.

Let $S\to S'$ be a finite, possibly ramified, cover of closed oriented topological surfaces. A problem which recently has received much attention is how to generate the first homology group of $S$ in terms of $1$-cycles supported on elevations of curves on $S'$ via the given covering map. I will explain recent advances (joint work with A. Putman and N. Salter) and the connection with the problem of extending finite group actions from surfaces to handlebodies (joint work with E. Samperton and C. Segovia).

Let Mod(S) be the mapping class group of a connected surface S of finite type with negative Euler characteristic. In joint work with León Álvarez and Sánchez Saldaña, we show that the commensurator of any abelian subgroup of Mod(S) can be realized as the normalizer of a subgroup in the same commensuration class. As a consequence, we give an upper bound for the virtually abelian dimension of Mod(S). In this talk we will introduce the necessary definitions and explain how these results are obtained.

Binary $G$-actions are a generalization of usual (left) $G$-actions on topological spaces. In this cases a group $G$ acts through invertible binary continuous operations on a topological space. Usual notions as orbits are not easily generalized to this context. We will show how to give a universal construction for these spaces in whenever there exists a universal space for usual $G$-actions. Also it will be shown some concrete examples of types of orbits of such spaces. This is joint work with Pavel S. Gevorgyan.

Given a discrete group G acting on a compact CW-complex M, we introduced the notion of a derivation of line bundles over M, which can be used to define a projective bundle over the Borel construction. There is a spectral sequence converging to the twisted Borel equivariant K-theory of M in terms of the K-theory of M, the derivation and group cohomology. In this talk I will begin describing the work of Harju-Mickelsson on twisted K-theory for decomposable twistings, which motivated this work. A generalization for mapping tori of homeomorphisms will be presented next, which will allow me to connect with the general case. This is joint work with Alffer G. Hernández.

Quantum invariants are certain topological invariants of knots and 3-manifolds built from representation theory, more precisely, the theory of quantum groups, Hopf algebras and monoidal categories. However, what geometric/topological information of a given knot these invariants contain is still mysterious. In this talk, we will explain how to add group actions/equivariance into the usual picture to obtain knot invariants that do carry some geometric information, namely, lower bounds to the Seifert genus. As a corollary, we get genus bounds for “non-semisimple” quantum knot polynomials. This is joint work with Roland van der Veen.

The James spectral sequence is a generalization of the Atiyah-Hirzebruch spectral sequence and is used for computing generalized homology groups of Thom spectra. This spectral sequence was first introduced by P. Teichner in 1993 and later it has been used in many geometric applications due to its edge homomorphism from the baseline. For a compact lie group $G$, a $G$-equivariant stable vector bundle $\xi:X \rightarrow BO_G(U)$ and an equivariant homology theory $h$, we give the construction of the equivariant $RO(G)$-graded version of the James spectral sequence by expressing the Thom spectra $M\xi$ as the homotopy colimit of a suitable functor and using the homotopy colimit spectral sequence. Later we discuss some consequences and applications of this spectral sequence. This is a joint work with Özgün Ünlü.

In this talk I wanna present some historical background about the Steenrod representation problem and how it relates bordism and singular homology. I present the concept of stratifold developed by Mathias Kreck as a way to resolve the representation problem. After that, I introduce $\mathbb{Z}_k$-stratifolds to resolve the representation problem with $\mathbb{Z}_k$-coefficients.

Freed-Hopkins propose a model for the classification of invertible phases of matter with a symmetry group acting on space using Borel-equivariant bordism. In this talk, I'll discuss a generalization of their ansatz using twisted equivariant bordism to account for cases where the symmetry type mixes nontrivially with the spatial symmetry, such as crystalline phases with spin-1/2 fermions. Using this ansatz, one recovers as a theorem the "fermionic crystalline equivalence principle" predicted in the physics literature; I will discuss this theorem and consequences in some examples.