# Seminars & Events for Algebraic Topology Seminar

##### Constructing equivariant spectra

Equivariant spectra determine cohomology theories that incorporate a group action on spaces. Such spectra are increasingly important in algebraic topology but can be difficult to understand or construct. I will discuss recent work with Angelica Osorno, in which we build such spectra out of purely algebraic data based on symmetric monoidal categories. Our method is philosophically similar to classical work of Segal on building nonequivariant spectra.

##### Topological Complexity of Spaces of Polygons

The topological complexity of a topological space X is the number of rules required to specify how to move between any two points of X. If X is the space of all configurations of a robot, this can be interpreted as the number of rules required to program the robot to move from any configuration to any other. A polygon in the plane or in 3-space can be thought of as linked arms of a robot. We compute the topological complexity of the space of polygons of fixed side lengths. Our result is complete for polygons in 3-space, and partial for polygons in the plane.

##### An infinite-dimensional phenomenon in finite-dimensional metric topology

We introduce a notion of “deformation equivalence” for topological manifolds. Deformation equivalent manifolds are homotopy equivalent via a simple-homotopy equivalence inducing isomorphisms on rational Pontrjagin classes, but they need not be homeomorphic. If M^m, m > 6, is a closed simply connected manifold such that pi_2(M) vanishes, then there are manifolds which are deformation equivalent to M if and only if KO_m(M) has odd torsion. Indeed, for each odd torsion class tau in KO_m(M) there is a unique homotopy equivalence f:N -> M which is realized by a deformation and whose signature operator differs from that of M by tau. This gives many simply-connected examples -- for instance between S^3-bundles over S^4.

##### Toric polynomial generators in the unitary cobordism ring

It is well-known that the unitary cobordism ring Omega^* is isomorphic to the polynomial ring with one generator in every even degree. However explicit description of 'nice' representatives of the generators turns out to be a difficult problem. In this talk we aim at constructing connected algebraic toric polynomial generators of Omega^*. Recently large progress in this problem was obtained by Andrew Wilfong: he provided polynomial generators in all odd dimensions and dimensions one less than prime power. We resolve the problem in the remaining dimensions by applying certain birational equivariant modifications to the P^k-bundles over P^{n-k} (generalized 2-stage Bott towers).

##### Schubert Calculus and Positivity

Schubert calculus is the study of certain intersections of varieties in the flag manifold. These intersections are */positive/* in the differential-geometric sense, but they also have “positivity properties” in several associated rings, notably equivariant cohomology and equivariant K-theory. We show how one can get new and old formulas for the structure constants in these rings using Bott-Samelson manifolds, a sequence of projective bundles lying over the flag manifold. Time permitting, we discuss the notion of positivity in the symplectic category, and explore how to extend these ideas in the more general context.

##### Equivariant unitary bordism and equivariant cohomology Chern numbers

By using the universal toric genus and the Kronecker pairing of bordism and cobordism, we show that the integral equivariant cohomology Chern numbers completely determine the equivariant geometric unitary bordism classes of closed unitary G-manifolds, which gives an affirmative answer to the conjecture posed by Guillemin--Ginzburg--Karshon in the book [Moment maps, cobordisms, and Hamiltonian group actions. Appendix J by Maxim Braverman. Mathematical Surveys and Monographs, {\bf 98}. American Mathematical Society, Providence, RI, 2002], where G is a torus. Our approach heavily exploits Quillen's geometric interpretation of homotopic unitary cobordism theory. As a further application, we also obtain a satisfactory solution of [Question (A), \S1.1, Appendix H] of the above book on unitary Hamiltonian G-manifolds.

##### On the topology of a small cover associated to a shellable complex

Small covers are real analogues to quasi-toric manifolds: the fixed points under complex conjugation (i.e., the real toric manifold) in a projective toric manifold is a small cover; Buchstaber and Ray showed that every unoriented cobordism class contains a small cover as its representative. A shelling of a pure simplicial complex $K$ is a special ordering of its facets. If $K$ is a piecewise linear sphere (with a $\mathrm{mod}$ $2$ characteristic function), such a shelling gives a handle decomposition of the associated small cover $M$, which is a piecewise linear manifold. With this decomposition, we analyze the cohomology of $M$ with integer coefficients, using (higher) $\mathrm{mod}$ $2$ Bockstein homomorphisms on the $\mathrm{mod}$ $2$ cohomology ring of $M$.

##### A family of simplicial complexes and the cohomology of their associated polyhedral products

A polyhedral product space, $Z_K(X,A)$, is a subspace of a product of spaces. More specifically, it is the union of products of spaces indexed by a simplicial complex. The most well known examples are moment-angle complexes and their real analogue. The polyhedral join (a generalization of the J-construction) is an analogous construction for simplicial complexes. Most notably, these constructions provide a family of simplicial complexes for which moment-angle complexes and real moment-angle complexes are homeomorphic. In general, statements in one polyhedral product can be translated into statements in another polyhedral product of more complicated spaces $(X,A)$ but over a simpler simplicial complex, or vice versa. I will discuss the effect of the polyhedral join on the cohomology of the associated polyhedral product.

##### "In memory of John Coleman Moore": The exponent of the homotopy groups of an odd primary Moore space

Norman Steenrod once said: "If you are having trouble explaining something, perhaps it is because you have not understood it well enough." Recent developments have led to better understanding of the proof of the best possible exponent for the homotopy groups of an odd primary Moore space. Thus for the first time it has become realistic to lecture on the proof. In so doing, it is necessary to give a quick outline of the work of Cohen, Moore, and Neisendorfer.

##### Product decomposition theorem for simplicial non-positive curvature and applications

Let $h$ be a hyperbolic isometry of a *systolic* complex (a simply connected complex of simplicial non-positive curvature). In joint work with D. Osajda we show that the *minimal displacement set* of $h$ decomposes up to quasi-isometry as the product of a tree and the real line. From this we deduce the following two corollaries. For a group $G$ acting properly on a systolic complex we construct a finite dimensional model for the classifying space * EG*. If the action is additionally cocompact, we prove that the centralizer of any hyperbolic element is virtually free-by-cyclic. Before outlining these results I shall give some background on classifying spaces for families and systolic complexes.