分野別セミナー

Steenrod's Problem asks which graded commutative algebras (usually over the integers) can be realized as the cohomology of a space. One of the reasons Quillen introduced rational homotopy theory was to solve this problem over the rationals. The Hopf invariant one problem can also be considered a special case of this problem. Around 10 years ago Anderson and Grodal classified which polynomial algebras can be realized. More recently work of Trevisan, Bahri-Bendersky-Cohen-Gitler, So-Stanley and Takeda studied the problem of which polynomial algebras modulo monomial ideals can be realized.

We are interested in a slightly different variant of the problem, we look at algebras that become exterior algebras after tensoring with the rationals. To attack this problem we construct weighted polyhedral products which generalize polyhedral products. Given a sequence of pairs of spaces indexed by a set S and a simplicial complex K with vertex set S, we can construct the polyhedral product. This generalization of moment angle complexes has been a subject of intense study. Our construction of weighted polyhedral products depends additionally on some set of natural numbers c which we call a weight. In many cases we are able to compute the cohomology algebra of the weighted polyhedral product and we apply this computation to Steenrod's Problem.

This is joint work with Larry So and Stephen Theriault.