# Seminars & Events for Algebraic Topology Seminar

##### On k-regular maps

The question about the existence of a continuous k-regular map from a topological space X to an N-dimensional Euclidean space R^N, which would map any k distinct points in X to linearly independent vectors in R^N, was first considered by Borsuk in 1957. In this talk we present a proof of the following theorem, which extends results by Cohen--Handel 1978 (for d=2) and Chisholm 1979 (for d power of 2): For integers k and d greater then zero, there is no k-regular map R^d -> R^N for N < d(k-a(k))+a(k), where a(k) is the number of ones in the dyadic expansion of k. Joint work with G. M. Ziegler and W. Lück.

##### Equivariant maps from a configuration space to a sphere

THIS IS A JOINT TOPOLOGY/ALGEBRAIC TOPOLOGY SEMINAR. PLEASE NOTE DIFFERENT TIME AND LOCATION. There are several distinct reasons to ask for the existence of an S_n-equivariant map from the configuration space F(R^d,n) of n labeled points in R^d to a certain S_n-representation sphere of dimension (d+1)(n-1)-1. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### Beyond the Borsuk-Ulam/Dold theorem (in Combinatorial Geometry)

**THIS IS A SPECIAL ALGEBRAIC TOPOLOGY SEMINAR. PLEASE NOTE DIFFERENT DATE, TIME AND LOCATION. **In this talk we present an evolution of equivariant topology methods in Combinatorial Geometry.

We start with: (a) the Topological Radon's theorem, an application of the Borsuk-Ulam theorem, and proceed, via non-planarity of K_{3,3}, to (b) the Topological Tverberg and the Weak Colored Tverberg theorem for primes, which are applications of Dold's theorem, to continue with (c) the Topological Tverberg for prime powers, an application beyond Dold's theorem based on the connectivity and localization theorem for elementary abelian groups, to finally ask: "What needs to be done in the case of Barany-Larman conjecture and Nandakumar & Ramana-Rao problem when all the previously known methods fail?"

##### The space of metrics of positive scalar curvature

Given a closed smooth manifold admitting a Riemannian metric of positive scalar curvature, we are interested in the space of all such metrics. We will give a survey on recent results concerning the homotopy type of this space and of the associated moduli space of positive scalar curvature metrics.

##### Double commutants of multiplication operators on $C(K)$

We consider the following topological property of a compact Hausdorff space $K$: for every operator $M$ of multiplication by a real-valued function from $C(K)$, its double commutant coincides with the norm-closed algebra generated by $M$ and the identity operator $I$. If it is the case, we say that $K \in DC$. The main result states that if $K$ is a metrizable connected and locally connected compact space, then $K \in DC$. We also provide examples of metrizable continua not in $DC$ and examples of continua that are not locally or even arc connected but are in $DC$.

##### The Cayley Plane and String Bordism

I will describe how an affinity between projective spaces and bordism rings extends further than previously known. The well-known manifestations of this affinity are that real projective bundles generate the unoriented bordism ring; that complex projective bundles generate the oriented bordism ring; and that quaternionic projective bundles "almost" generate the spin bordism ring---they generate the kernel of the Atiyah invariant. The new manifestation of this affinity is that Cayley plane bundles---that is, octonionic projective plane bundles---almost generate string bordism after inverting 6---they generate the kernel of the Witten genus.

##### Feynman categories: Universal operations and Hopf algebras

After briefly giving the definition of Feynman categories -a toy example being finite sets and surjections- we will consider other algebraic structures that can be derived for them. The first are universal operations,

the probably most known example being the Gerstenhaber structure for Hochschild cochains. The second type of structure are Hopf algebras. Examples here include the Hopf algebras of Connes--Kreimer, Goncharov and Baues.

##### Complex compact manifolds with maximal torus action

**Please note different time and location. **In the talk we describe a class of manifolds Z constructed via certain combinatorial data, called a complete simplicial fan. In the case of rational fans, the manifold Z is the total space of a holomorphic bundle over a toric variety with fibres compact complex tori. In general, a complex moment-angle manifold Z is equipped with a canonical holomorphic foliation F and a C*-torus action transitive in the transverse direction. Examples of moment-angle manifolds include Hopf manifolds of Vaisman type, Calabi-Eckmann manifolds, and their deformations. Recent results of H. Ishida imply that these manifolds are the only compact complex manifolds with maximal torus action. Explicit construction of the manifold allows precise description of its geometry and topology e.g.

##### Stable homology for moduli spaces of manifolds

THIS IS A JOINT ALGEBRAIC TOPOLOGY / TOPOLOGY SEMINAR. There will be two separate talks: 3:00-4:00 pm (Fine 214) and 4:30-5:30 pm (Fine 314). For a compact manifold $W$, possibly with boundary, we shall let $\mathrm{Diff}(W)$ denote the topological group of diffeomorphisms of $W$ fixing a neighborhood of $\partial W$. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### Topological Actions of Connected Compact Lie Groups on Manifolds

THIS IS A JOINT ALGEBRAIC TOPOLOGY / TOPOLOGY SEMINAR. PLEASE NOTE DIFFERENT TIME AND LOCATION. We survey some new methods and results on existence and on topological classification

of actions of connected, compact Lie groups on manifolds. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### Toric Varieties and Complex Cobordism

After a brief introduction to complex cobordism and toric varieties, we will discuss two problems in which the two are related. First, we will examine which cobordism classes can be represented by smooth projective toric varieties. A complete answer will be given for low-dimensional cobordism. After this, we will examine the role that smooth projective toric varieties play as polynomial ring generators of the complex cobordism ring.

##### Computing Equivariant Cohomology for SL(n,Z)

In a series of papers, Ash, Gunnells and McConnell have studied cohomology groups for congruence subgroups of SL(4,Z), and have verified experimentally that Hecke eigenclasses x for these cohomology groups seem to have attached Galois representations.

##### Scissors congruence and algebraic K-theory

Hilbert's third problem asks the following question: given two polyhedra with the same volume, can we decompose them into finitely many pairwise congruence pieces? The answer, provided by Dehn in 1901 is no; there is a second invariant on polyhedra, now called the Dehn invariant. Classical scissors congruence asks this question in other dimensions and geometries. In this talk we construct an abstract framework for discussing scissors congruence problems using algebraic K-theory.

##### Periodic pseudo-triangulations

We review briefly the notions of periodic frameworks and periodic deformations. Then, we propose a purely geometric criterion for characterizing auxetic one-parameter deformations. Simply phrased, auxetic behavior means becoming laterally wider when stretched and thinner when compressed. Our geometric approach relies on the evolution of the periodicity lattice. A deformation path will be auxetic when the Gram matrix for a basis of periods gives a curve with all tangents in the positive semidefinite cone, analogous to a causal line in special relativity. This concept is valid in arbitrary dimension. Auxetic mechanisms are then compared with expansive mechanisms defined by the stronger property that the distance between any pair of vertices increases or stays the same.

##### Equivariant principal bundles on nonsingular toric varieties

A classification of torus equivariant holomorphic principal G-bundles on a nonsingular toric variety, where G is any holomorphic closed Abelian linear group, will be described. This generalizes the classical result regarding equivariant line bundles. The method is complex analytic in nature.

##### The topology of toric origami manifolds

The topology of a toric symplectic manifold can be read directly from its orbit space (a.k.a. moment polytope), and much the same is true of the (smooth) topological generalizations of toric symplectic manifolds and projective toric varieties. An origami manifold is a manifold endowed with a closed 2-form with a very mild degeneracy along a hypersurface, but this degeneracy is enough to allow for non-simply-connected and non-orientable manifolds, which are excluded from the topological generalizations mentioned above. In this talk we examine how the topology of an (orientable) toric origami manifold, in particular its fundamental group, can be read from the polytope-like object that represents its orbit space. We conjecture that these results hold for the appropriate topological generalization of the class of toric origami manifolds.

##### Expanders and k-theory for group c* algebras

An expander or expander family is a sequence of finite graphs X_1, X_2, X_3,... which is efficiently connected. A discrete group $G$ which contains an expander in its Cayley graph is a counter-example to the Baum-Connes (BC) conjecture with coefficients. Such a group is known as the Gromov monster and is the only known example of a non-exact group. The left side of BC with coefficients ``sees" any group as if the group were exact. This talk will indicate how to make a change in the right side of BC with coefficients so that the right side also ``sees" any group as if the group were exact.

##### Slices of co-operations for KGL

The classic structure of the co-operations KU(KU) of topological K-theory as a Hopf algeboid is due to Adams. In joint work with Pelaez, we determine the analogous structure for algebraic K-theory KGL, regarded as a motivic spectrum. Applying the motivic slice filtration, we solve a problem of Voevodsky.

##### Homological stability of configurations spaces

Church [Homological stability for configuration spaces of manifolds, Invent. Math. 188 (2012),465--504] used representation stability to prove that the space of configurations of distinct unordered points in a closed manifold exhibit rational homological stability.A second proof was also given by Randal-Williams [Homological stability for unordered configuration spaces, Q. J. Math. 64 (2013),303--326] using transfer maps. We give a third proof of this fact using localization and rational homotopy theory. This gives new insight into the role that the rationals play in homological stability. Our methods also yield new information about stability for torsion in the homology of configuration spaces of points in a closed manifold.

##### Explicit triangulation of complex projective spaces

We give explicit construction of some triangulations of complex projective space CP^n with $(4^{n+1}-1)/3$ vertices for all $n$. No explicit triangulation of CP^n is known for $n > 3$.