Topology seminar

The genus of a fibration was introduced by Schwarz in 1962 [1]. Given a map g : A → B, the usual sectional number sec_u(g) is the least integer m such that B can be covered by m open subsets each of which admits a local section of g [1]. Likewise, the sectional category secat(g) is the least integer m such that B can be covered by m open subsets each of which admits a local homotopy section of g [2].

In the case that g is a fibration, the usual sectional number and the sectional category of g coincide with the Schwarz’s genus of g. In this work we introduce a notion of sectional number of a morphism f in a category with covers, sec(f), which extends the usual sectional number and the sectional category (and, of course, the Schwarz’s notion). We study the invariance of sectional number and the behaviour of sec under weak pullbacks and continuous functors. In addition, we introduce a notion of multiplicative cohomology theory on a category and we use it to present a cohomological lower bound for the sectional number.  Our unified concept is relevant in this new setting because it presents a construction of foundations of sectional theory in category theory and build bridges between new areas in mathematics.

G-cyclifications Map(G,X)//G are generalizations of cyclic loop spaces L_c X := Map(S^1, X)//S^1, also known as equivariant or unparameterized free loop spaces. The rational homotopy type of a cyclic loop space has been known since the heyday of rational homotopy theory: a 1985 result of Vigué-Poirrier and Burghelea shows how to construct the Sullivan minimal model of L_c X from that of X. A few years ago, when Berglund gave a talk in this seminar, it was not even clear how to generalize Vigué-Poirrier-Burghelea’s theorem to the toroidification Map(T,X)//T, where T = (S^1)^k is a k-torus. Sati and I needed this generalization, because of a relevance of the toroidifications Map(T,S^4)//T of the 4-sphere to supergravity and a conjectural relevance to del Pezzo surfaces. Now, as we have done this and dramatically simplified the original proof of Vigué-Poirrier and Burghelea, it has opened the way to understanding the rational homotopy theory of a G-cyclification Map(G,X)//G for any compact Lie group G.

VH 570

Classically, the Cartier-Milnor-Moore theorem identifies primitively generated Hopf algebras as (restricted) enveloping algebras of their Lie algebra of primitives.  The Poincaré-Birkhoff-Witt theorem establishes an isomorphism between an associated graded Hopf algebra of these enveloping algebras with a symmetric algebra.  These theorems are fundamental to the study of Hopf algebras and their cohomology.

There is, however, a more general context where one can study Hopf algebra objects in braided monoidal categories.  In this setting, these classical results fail to hold.  I will speak about recent work that provides analogues of these structure theorems using the notion of a braided operad.

VH 570

Abstract not available

VH 570

 Transfer systems are combinatorial objects with connections to both equivariant homotopy theory and model categories. (In particular, transfer systems on the subgroup lattice of a group, G, determine both the N-infinity operads associated to G and the possible model structures on the subgroup lattice.)  In ongoing work (with Balchin, Barrero, Scheirer, Sulyma, Wisdom, Zapata Castro) we investigate several questions related to the generation of transfer systems.  This talk will focus on our results related to the complexity of a finite group, G, which we define to be the maximal size of a minimal generating set for a transfer system on the subgroup lattice of G.  In order to determine the complexity of several families of groups, we introduce rainbow diagrams, a new representation of appropriately symmetric generating sets for transfer systems.  

VH 570