Chains and cochains on spectra

Abstract: The primary goal of this talk will be to give an appropriate definition of chains and cochains on the spectra of stable homotopy theory, generalizing the usual chains and cochains on spaces. First, we will discuss operads, which encode homotopy coherent algebraic structures, and, in particular, the Eilenberg-Zilber operad \(\mathcal{EZ}\) and the McClure-Smith operad \(\mathcal{MS}\). Given a simplicial set \(X\), the normalized cochains \(\text{N}^{\bullet}(X)\) naturally form an algebra over both \(\mathcal{EZ}\) and \(\mathcal{MS}\), and, under suitable conditions, this algebraic structure on the cochains on \(X\) is enough to remember the homotopy type of \(X\). I will discuss a notion of suspension of operads, and, using this and certain stabilization maps \(\Sigma\mathcal{EZ} \rightarrow \mathcal{EZ}\) and \(\Sigma\mathcal{MS} \rightarrow \mathcal{MS}\), I will define stable analogues \(\mathcal{EZ}_{\text{st}}\) and \(\mathcal{MS}_{\text{st}}\) of the Eilenberg-Zilber and McClure-Smith operads. Finally, as an application, I will define a notion of cochains on spectra and show that they naturally form algebras over both \(\mathcal{EZ}_{\text{st}}\) and \(\mathcal{MS}_{\text{st}}\). This talk will be aimed at a general audience and will not assume any previous knowledge of operads or spectra. This is joint work with Igor Kriz.