# Seminars & Events for Algebraic Topology Seminar

##### One condition for rational hyperbolicity for moment angle complexes

There is a conjecture in rational homotopy which states "If a finite dimensional space X has two indecomposable classes in the rational cohomology whose cup product is zero, then its rational homotopy has exponential growth , ie, it is hyperbolic." We verify this conjecture for the space X a moment angle complex.

##### Algebraic structures from operads and moduli spaces

Some classical algebraic structures like Gerstenhaber's bracket on the Hochschild complex have an operadic origin. We discuss generalizations of these operations coming from different operadic type settings. This includes geometric constructions from moduli spaces and master equations appearing in string topology and in a "pedestrian" version of string field theory.

##### The EHP sequence and the Goodwillie tower

The EHP sequence and the Goodwillie tower of the identity give two different spectral sequences for computing the unstable homotopy groups of spheres. I will explain how the two can be mixed, so that each provides information about the differentials in the other. To demonstrate the effectiveness of the techniques presented, the methods will be applied to recompute the 2-primary Toda range (first 20 unstable stems).

##### Cohomology of graph products of infinite groups with group ring coefficients

I will explain a computation of the cohomology of any graph product of infinite groups in terms of the factor groups. For example, this gives a calculation for right-angled Artin groups, which are, by definition, graph products of copies of the infinite cyclic group. The method of proof is a simple spectral sequence argument which I don't think has been used previously.

##### Geometry and combinatorics for revolute-jointed robot arms

We present a complete theoretical characterization and a method for calculating the reachable workspace boundary for all serial manipulators with revolute joints having any pair of consecutive joint axes coplanar. The number of joints is arbitrary.

The workspace boundary is a surface of revolution, obtained by rotating a planar bounded real semi-algebraic curve, called the planar workspace boundary, about the first joint axis. We show that the planar boundary is composed of a finite set of circular arcs. Their connectivity is controlled by an underlying combinatorial structure which is fully identified.

##### On spaces of homomorphisms and spaces of representations

The subject of this talk is the structure of the space of homomorphisms from a group $\pi$ to a Lie group $G$ denoted $Hom(\pi,G)$. The space of representations $Hom(\pi,G)/G$ obtained from the adjoint action of $G$ will be considered. In special cases, these spaces can be assembled into a single space analogous to the classifying space of the group $G$. Properties of these spaces will be developed. This talk is based on joint work with A. Adem, E. Torres, and J. Gomez.

##### The Unstable Chromatic Spectral Sequence

##### Nonimmersions of real projective spaces and tmf

We use the spectrum tmf to obtain new nonimmersion results for many real projective spaces RP^n for n as small as 113. The only new ingredient is some new calculations of tmf-cohomology groups. We present an expanded table of nonimmersion results. We also present several questions about tmf.

##### Massey Triple Products

A technique will be discussed to control the indeterminacy in cohomology Massey triple products. A variety of non-vanishing and vanishing results for Massey triple products are proved using this technique. Here are three examples. Many authors have noticed that non-trivial triple products in a submanifold produce non-trivial triple products in the blowup along the submanifold.

##### Equivariant cohomology and orbit structure

Let T be a torus, and let X be a T-space. In this talk I will relate algebraic properties of the equivariant (co)homology of X to the structure of the T-orbits in X. This generalizes a result of Atiyah and Bredon. As applications, I will characterize those compact T-manifolds which admit a Chang-Skjelbred (or GKM) description of their equivariant cohomology, and also those for which the equivariant Poincaré pairing is perfect. This is joint work with Chris Allday and Volker Puppe.

##### An integral lift of the Gamma-genus

The Hirzebruch genus of a complex-oriented manifold $M$ associated (by Kontsevich) to Euler's Gamma-function has an analytic interpretation as the index of a family of deformations of a Dirac operator, parametrized by the homogeneous space $Sp_U$; in more homotopy-theoretic terms, it is the homomorphism $\mu \rightarrow \mu_{MSp} KO$ of ring spectra. It also has an interpretation as a kind of equivariant Euler characteristic of the free loopspace of M, suitably polarized. There are further intriguing connections with the theory of asymptotic expansions, involving the values of the zeta function at odd positive integers.

##### Compact aspherical manifolds whose fundamental groups have center

Classical work of Borel had shown that an action of the circle on a manifold with contractible universal cover yields non-trivial center in the manifold's fundamental group. In the early 70's, Conner and Raymond made further deep investigations which led them to conjecture a converse to Borel's result. We construct counter-examples to this conjecture, i.e., we exhibit aspherical manifolds (in all dimensions greater than or equal to 6) which have non-trivial center in their fundamental groups but no circle actions (and hence no compact Lie group actions). The constructions involve synthesizing rather disparate methods of geometric topology, geometric group theory and hyperbolic geometry. (This is joint work with Shmuel Weinberger and Min Yan.)

##### Moment-angle complexes from simplicial posets

The construction of moment-angle complexes may be extended from simplicial complexes to simplicial posets. As a result, a certain $T^m$-space $Z_S$ is associated to an arbitrary simplicial poset S on m vertices. Face rings $Z[S]$ of simplicial posets generalise those of simplicial complexes, but have much more complicated algebraic structure. These rings $Z[S]$ may be studied by topological methods. The space $Z_S$ has many important topological properties of the original moment-angle complex $Z_K$ associated to a simplicial complex K. In particular, the integral cohomology algebra of $Z_S$ is isomorphic to the Tor-algebra of the face ring $Z[S]$. This leads directly to a generalisation of Hochster's theorem, expressing the algebraic Betti numbers of the ring $Z[S]$ in terms of the homology of full subposets in S.

##### The Geometry of Music

In my talk, I explain how to translate basic concepts of music theory into the language of contemporary topology and geometry. Musicians commonly abstract away from five kinds of musical information -- including the order, octave, and specific pitch level of groups of notes. This process produces a family of quotient spaces or orbifolds: for example, two-note chords live on a Mobius strip, while three-note chord-types live on a cone. These spaces provide a general geometrical framework for understanding and interpreting music. Related constructions also appear naturally in other applied-math contexts, for instance in economics.

##### On the rational homotopy type of Moment-angle complexes

In this talk I will discuss the following theorem, A moment-angle complex is rationally elliptic if it is homeomorphic to a product of odd dimensional spheres and discs.

##### Unstable operations in etale and motivic cohomology

Epstein constructed cohomology operations $P^i$ in etale cohomology, and Voevodsky constructed cohomology operations $P^i$ in motivic cohomology. These differ from the topological operations because of twisting. We modify Henri Cartan's classification of unstable operations to classify all etale operations (they are the expected operations), and classify all weight one motivic operations. Here new operations arise, and we paint a conjectural picture in all weights. This is joint work with Bert Guillou.

##### Integral and Rigid Elliptic Genera

Each Hirzebruch genus is dened by the exponential of the corresponding formal group law. We will describe the formal group law associated with the general Weierstrass model of the elliptic curve with parameters $\mu = (\mu1, \mu2, \mu3, \mu4, \mu6)$ and arithmetic Tate uniformization. We obtain the corresponding general elliptic genus which is $Z[\mu]$-integral.

##### Geometry and combinatorics for revolute-jointed robot arms

(This is joint work with I. Streinu) We present a complete theoretical characterization and a method for calculating the reachable workspace boundary for all serial manipulators with revolute joints having any pair of consecutive joint axes coplanar. The number of joints is arbitrary. The workspace boundary is a surface of revolution, obtained by rotating a planar bounded real semi-algebraic curve, called the planar workspace boundary, about the first joint axis. We show that the planar boundary is composed of a finite set of circular arcs. Their connectivity is controlled by an underlying combinatorial structure which is fully identified.

##### On spaces of homomorphisms and spaces of representations

The subject of this talk is the structure of the space of homomorphisms from a group $\pi$ to a Lie group $G$ denoted $Hom(\pi,G)$. The space of representations $Hom(\pi,G)/G$ obtained from the adjoint action of $G$ will be considered. In special cases, these spaces can be assembled into a single space analogous to the classifying space of the group $G$. Properties of these spaces will be developed. This talk is based on joint work with A. Adem, E. Torres, and J. Gomez.

##### The Unstable Chromatic Spectral Sequence

The chromatic spectral sequence has had a profound influence towards calculation of stable homotopy groups of the spheres. I will outline the changes necessary to set up an unstable chromatic spectral sequence. there will be applications to Hopf invariants.