# Seminars & Events for Algebraic Topology Seminar

##### v1-periodic homotopy groups of SU(n)

I will survey the various results that have been obtained during the past 22 years on the v1-periodic homotopy groups of $SU(n)$. The most recent work has been combinatorial fine tuning to make the statements more explicit. I will discuss conceptual differences between 2-primary and odd-primary groups and implications for actual homotopy groups

##### Derived functors in unstable homotopy theory

The talk will be about a functorial approach to the problem of computation of homology of Eilenberg-MacLane spaces, homotopy groups of suspensions of classifying spaces, etc.

##### Feynman categories

There is a plethora of operad type structures and constructions which arise naturally in classical and quantum contexts such as operations on cochains, string topology or Gromov-Witten invariants. We give a novel categorical framework which allows us to handle all these different beasts in one simple fashion. In this context, many of the relevant constructions are simply Kan extensions. We are also able to show how in this framework bar constructions, Feynman transforms, master and BV equations appear naturally.

##### The Geometry of Triple Linking

Given a link $L$ in $R^3$ with just two components $X = \{x(s):s \epsilon S^1 \}$ and $Y = \{y(t): t\epsilon S^1\}$, their ** linking number** can be defined as the degree of the

**$f_L: S^1 \times S^1\rightarrow S^2$ , given by $f_L(s,t)=(y(t)-x(s))/|y(t)-x(s)|$, or as the value of the**

*Gauss map***, $1/4\pi\int_{S1\times S1} (dx/ds) x (dy/dt)$ • $(x y)/|x y|^3 ds dt$.**

*Gauss integral*The Gauss map is equivariant with respect to the action of an orientation-preserving isometry $h$ of $R^3$ , that is, $f_h(L) = h\circ f_L$, and the Gauss integrand is invariant with respect to this action, meaning that it is the same for $L$ and for $h(L)$ , attesting to the geometric naturality of both the map and the integral.

##### A combinatorial description of homotopy groups of spheres

The talk is based on the recent results obtained jointly with Jie Wu. For every $n>k>3$, we construct a group given by explicit generators and relations whose center is exactly the $n-th$ homotopy group of the k-sphere.

##### A parametrized version of Gromov's waist of the sphere theorem

Gromov, Memarian, and Karasev--Volovikov proved that any map $f$ from an n-sphere to a k-manifold $(n>=k)$ has a preimage $f^{-1}(z)$ whose epsilon-neighborhoods are at least as large as the epsilon-neighborhoods of the equator $S^{n-k}$, assuming that the degree of f is even in case $n=k$. We present a parametrized generalization. For the proof we introduce a Fadell-Husseini type ideal-valued index of G-bundles that is quite computable in our situation and we obtain new parametrized Borsuk--Ulam and Bourgin--Yang--Volovikov type theorems.

##### Axioms for Differential Cohomology

A differential cohomology theory (DCT) is a type of refinement of a cohomology theory (restricted to the category of smooth manifolds with corners) that contains information that is not homotopy invariant. A detailed definition of a DCT will be given, as well as axioms for a category larger than that of smooth manifolds with corners. Any two DCTs that are defined on this larger category and that refine the same cohomology theory will be naturally isomorphic.

##### Bertini theorems for F-singularities

I will discuss Bertini theorems for F-singularities ( i.e. singularities defined by Frobenius); the proof is based on a slight generalization of Cumino-Greco-Manaresi's axiomatic approach to Bertini theorems. This is a joint work with Karl Schwede.

##### Piecewise Laurent Polynomials and (Operational) Equivariant K-theory of Toric Varieties

For a smooth compact toric variety $X$, results of Bifet-de Concini-Procesi and Brion show that the equivariant cohomology of $X$ is identified with the ring of piecewise polynomials on the associated fan. In 2006, Payne extended this to arbitrary toric varieties, identifying the ring of piecewise polynomials with the operational equivariant Chow cohomology of $X$. It turns out that a similar story holds for K-theory: when $X$ is smooth and compact, Brion-Vergne and Vezzosi-Vistoli show that the equivariant K-theory of algebraic vector bundles on $X$ can be identified with the ring of "piecewise Laurent polynomials" on the associated fan. On the other hand, the bivariant machinery of Fulton-MacPherson can be applied to construct an "operational" equivariant K-theory for singular toric varieties.

##### Cohomological Rigidity Problems in Toric Topology

As is well-known, cohomology ring does not distinguish closed smooth manifolds up to diffeomorphism or homeomorphism in general. However, it does if we restrict our attention to a reasonably small class of objects. For instance, it is known that simply connected closed smooth 4-manifolds are classified up to homeomorphism using their integral cohomology rings. The classification of the toric manifolds up to homeomorphism or diffeomorphism is an interesting open problem. In particular one can ask whether the integral cohomology ring determines the homeomorphism or diffeomorphism type. So far, we do not have any counter example but affirmative partial results. In this talk, we survey results on the topological classifiation of toric manifolds.

##### Rigidity and Origami

Cauchy's famous rigidity theorem for 3D convex polyhedra has been extended in various directions by Dehn, Weyl, A.D.Alexandrov, Gluck and Connelly. These results imply that a disk-like polyhedral surface with simplicial faces is, generically, flexible, if the boundary has at least 4 vertices. What about surfaces with rigid but not necessarily simplicial faces? A natural, albeit extreme family is given by flat-faced origamis. Around 1995, Robert Lang, a well-known origamist, proposed a method for designing a crease pattern on a flat piece of paper such that it has an isometric flat-folded realization with an underlying, predetermined metric tree structure. Important mathematical properties of this algorithm remain elusive to this day.

##### Polyhedral Products, Toric Manifolds, and Twisted Cohomology

I will discuss the cohomology with coefficients in rank one local systems for various polyhedral products, including real Davis-Januszkiewicz spaces and toric complexes. As one application (joint work with Alvise Trevisan), I will show how to determine the Betti numbers and the cup products of real, quasi-toric manifolds. As another application (joint work with Graham Denham and Sergey Yuzvinsky), I will explain why Cohen-Macaulay toric complexes enjoy a certain "abelian duality" property.

##### Intersections of Quadrics: 25 years later

Consider $F:\R^n\rightarrow\mathbb{R}^2$ given by two quadratic forms and let $V=F^{-1}(0)$ and $Z=V\cap S^{n-1}$. In January 1984 I began to study the topology of the generic $Z$ when the quadratic forms are simultaneously diagonalizable. By the end of the year I had an answer, but only around 1986-87 I wrote the details of a proof that left out some cases. Last year I proved the remaining cases and also the non-diagonalizable ones (joint work with Vinicio G\'omez), thus determining the topological type of all the generic $Z$. (Also the \textit{diffeomorphism} type, except for 3 four-dimensional cases). I will describe the proof, with emphasis on some topological constructions and the algebraic topology involved.In the diagonal case $Z$ turns out to be a \textit{generalized moment-angle complex} as defined by A. Bahri, M.

##### Homotopy Theory and Toric Spaces

**Suyoung Choi** of Ajou University, Korea - Toric rigidity of simple polytopes and moment-angle manifolds

##### Stable and Unstable Properties of Real Johnson-Wilson Spectra

I will try to describe the properties of certain spectra known as real Johnson-Wilson spectra, which are obtained as fixed points of involutions on the usual Johnson Wilson spectra. These spectra, that go by the symbol $ER(n)$, have several intriguing properties. For example, they are periodic and they support a self map whose cofiber is the Johnson Wilson spectrum $E(n)$. This makes them computationally amenable. I'll describe how one can use $ER(2)$ to prove some non-immersion results for real projective spaces. Unstably, the spaces in the omega spectra for $ER(n)$ admit product splittings that behave in interesting ways under periodicity. If time permits, I'll go into some interesting questions including the question on $ER(n)$ orientation of bundles. This is ongoing joint work with Steve Wilson.

##### Hermitian K-theory and Cobordism

I will discuss Z/2-equivariant motivic spectra. As an example, I will talk about a Z/2-equivariant motivic spectrum representing Karoubi's Hermitian K-theory, and my joint solution with Kriz and Ormsby of Thomason's homotopy limit problem. As another example, I will talk about motivic Hermitian cobordism, and its topological realization, topological Hermitian cobordism. This is an RO(G)-graded (Z/2 \times Z/2)-equivariant spectrum, whose RO(G)-graded homotopy groups we have computed.

##### A Homotopy-Theorist's View of Knots

Lipshitz and Sarkar recently constructied a stable homotopy refinement of Khovanov homology using Cohen-Jones-Segal Morse theory. In this talk, I will discuss an alternate proof of their result which uses more stable homotopy theory instead (a joint work with Po Hu and Daniel Kriz). I will also relate this to a previous weaker result of ours, and to homotopical realizations of modular functors.

##### Beyond Ellipticity

K-homology is the dual theory to K-theory. The BD (Baum-Douglas) isomorphism of Kasparov K-homology and K-cycle K-homology can be taken as providing a framework within which the Atiyah-Singer index theorem can be extended to certain non-elliptic operators. This talk will consider a class of non-elliptic differential operators on compact contact manifolds. These operators are in the Heisenberg calculus and have been studied by a number of mathematicians. Working within the BD framework, the index problem will be solved for these operators. Corollaries are solutions of the related equivariant and families index problems. This is joint work with Erik van Erp.

##### Loop products and dynamics

A metric on a compact manifold M gives rise to a length function on the free loop space LM whose critical points are the closed geodesics on M in the given metric. Morse theory gives a link between Hamiltonian dynamics and the topology of loop spaces, between iteration of closed geodesics and the algebraic structure given by the Chas-Sullivan product on the homology of LM. Geometry reveals the existence of a related product on the cohomology of LM. A number of known results on the existence of closed geodesics are naturally expressed in terms of nilpotence of products. We use products to prove a resonance result for the loop homology of spheres. I will not assume any prior knowledge of loop products. Mark Goresky, Hans-Bert Rademacher, and (work in progress) Ralph Cohen and Nathalie Wahl are collaborators.

##### An Introduction to Knot Homologies

Knot homology theories associate to a knot or link a complex of graded modules whose graded Euler characteristic is a classical knot polynomial. This type of knot invariant has been increasingly influential in low-dimensional topology in the ten years or so since the first one was developed. This (primarily expository) talk will introduce some knot homology theories with an emphasis on their formal algebraic structures and give examples of their applications. No significant background in low-dimensional topology will be assumed.