Schedule for: 16w2679 - 56th Cascade Topology Seminar

Note: the Lecture rooms are available after 16:00.

Beverages and a small assortment of snacks are available in the lounge on a cash honor system.

A buffet breakfast is served daily between 7:00am and 9:00am in the Vistas Dining Room, the top floor of the Sally Borden Building. Note that BIRS does not pay for meals for 2-day workshops.

In algebraic geometry actions of the torus \( (C^∗)^n \) on algebraic varieties with nice properties produce bridges between geometry and combinatorics. We see a similar bridge called moment map for Hamiltonian action of compact torus on symplectic manifolds. In particular whenever the manifold is compact the image of moment map is a simple polytope, the orbit space of the action. A topological counterpart called quasitoric manifolds were introduced by Davis and Januskiewicz in 1991. They also initiated the topological idea of toric orbifolds. Inspired by this idea, Poddar and Sarkar formalized the definition of (quasi)toric orbifolds. A class of examples of quasitoric orbifolds are weighted progective spaces. In this talk I will discuss the following: 1) Some relations between quasitoric orbifolds and polytopes. 2) Several combinatorial properties of simple polytopes and a combinatorial question. 3) A sufficient condition to compute integral homology of quasitoric orbifolds. This is a joint work with Tony Bahri and Jongbaek Song.

Knots and links play an important role in 3-manifolds and the equiva- lence relation of concordance of knots and links plays an important role in 4-manifolds. We will discuss our work that shows, loosely speaking, that we cannot hope to classify knot concordance without simultaneously classifying link concordance for links of an arbitrary number of components. Cochran- Friedl-Teichner considered generalized satellite operations \( R: SL(m)\to AS \), called “infection by a string link”, where \( SL(m) \) is the set of concordance classes of m-component links, \( AS\) is the set of concordance classes of algebraically slice knots, and the “pattern” knot R is some ribbon knot R. They proved that, for any such knot K there exists some \( R\), \( m\) and \( L\) such that \( R(L)=K\). We show that one cannot put an upper bound on \( m\). Links arise from knots since the spine of a Seifert surface is essentially a link. Our obstructions are related to the Alexander polynomials of such links.

A buffet lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building. Note that BIRS does not pay for meals for 2-day workshops.

This is joint work with K.Hess, E.Riehl and B.Shipley. In this talk I will introduce a class of accessible model structures on locally presentable categories, which includes, but is more general than, combinatorial model structures. An accessible model structure is particularly good if one wants to left or right induce it along an adjunction - by a theorem of Burke and Garner the induced weak factorization systems always exist, so one needs to check only a compatibility condition. If it holds then the resulting model structure is again accessible. One example of an accessible model structure is the Hurewicz model structure on \( Ch_R \) (the category of unbounded chain complexes over a ring \( R\) ), which can be induced to many categories of interest, like algebras, coalgebras, comodules, comodule algebras, coring comodules and bialgebras. I will discuss ideas behind some of the proofs for induced model structures and give examples.

I will present joint work with Brooke Shipley, in which we have defined a model category structure on the category of \( \Sigma^{\infty}X_+\)-comodule spectra such that the K-theory of the associated Waldhausen category of homotopically finite objects is naturally weakly equivalent to the usual Waldhausen K-theory of \( X\), \( A(X)\). I will describe the relation of this comodule approach to \( A(X)\) to the more familiar description in terms of \( \Sigma^\infty \Omega X_+\)-module spectra. I will also explain the construction and properties of the topological coHochschild homology of \( X\), which is a potentially interesting approximation to \( A(X)\).

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building. Note that BIRS does not pay for meals for 2-day workshops.

The Picard group is an important invariant of a symmetric monoidal category. In the homotopy category of spectra, these are precisely the isomorphism classes of the n-spheres and the Picard group is a copy of the integers. However, after K(n)-localization, the Picard group can become much more complicated. The K(n)-local categories thus provide examples of interesting Picard groups. Their importance in chromatic homotopy theory is highlighted by the fact that the dualizing object for Brown-Commenetz duality comes from an invertible element. The K(n)-local Picard groups have been computed at all primes when n=1 and all odd primes when n=2. Mahowald predicted that the K(2)-local Picard group at the prime 2 would be very large in comparison to the situation at other primes. In this talk, I will explain why he was right and explain our current, although incomplete, understanding of the structure of this group. This project is joint work with Bobkova, Goerss and Henn.

Suppose K is a finite CW complex with finite fundamental group G whose universal cover is homotopy equivalent to a product of spheres X. Suppose the G-action on the cover is trivial on homology. I will prove the following theorem using classical techniques from geometric topology. Theorem: G acts freely, smoothly, and homologically trivially on X x S^n whenever n is greater than or equal to the dimension of X. Unlu and Yalcin have constructed such a K for any finite fundamental group. Thus the title of the talk is a corollary.

2-day workshop participants are welcome to use BIRS facilities (Corbett Hall Lounge, TCPL, Reading Room) until 15:00 on Sunday, although participants are still required to checkout of the guest rooms by 12 noon. There is no coffee break service on Sunday afternoon, but self-serve coffee and tea are always available in the 2nd floor lounge, Corbett Hall.