HOMOTOPY METHODS IN ALGEBRAIC TOPOLOGY

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Spectral algebra or the algebra of spectra is the study of algebra in the context of stable homotopy theory. In this talk I'll discuss some techniques of enriched model category theory which have applications in spectral algebra. Examples of enriched model categories include enrichments over simplicial sets, symmetric spectra, and chain complexes. Time permitting, I'll discuss applications which include a classification of monogenic stable model categories, Morita contexts and tilting objects in stable homotopy, and general methods for constructing Quillen equivalences of model categories. This talk is based on joint work with Charles Rezk and Stefan Schwede.

Deligne conjectured that the little two cubes operad acts on the Hochschild complex of a DGA. In joint work McClure and Smith prove a generalization: If a cosimplicial object (the objects can be spaces or spectra or DG modules) has a cup product and "circle i" products then its Tot has an action of the little two cubes operad. It is well known that the Hochschild complex of a DGA or of a ring spectrum has such products.

In the study of positive scalar curvature metrics on a manifold M, a central role is played by a family of abelian groups R_n, which depend only on the fundamental group of M and its first two Stiefel-Whitney classes. It is analogous to C.T.C. Walls surgery obstruction group L_n (which depends on pi_1(M) and w_1(M)). If M has dimension n greater than or equal to 5, then M admits a positive scalar curvature metric if and only if an obstruction in R_n vanishes; moreover, the group R_{n+1} acts freely and transitively on the set of concordance classes of such metrics (provided of course this set is non-empty). So far, a purely algebraic definition of the groups R_n is sadly lacking; we discuss an "assembly map" with target R_n that is closely related to the topological K-theory assembly map.

Gross and Hopkins have shown that in the K(n)-local stable category, Spanier-Whitehead duality is the same as Brown-Comenetz duality up to twisting by an invertible spectrum which we will call I^, and they identify the Morava module of I^ with a certain algebraically defined object. The proof can be roughly divided into an algebraic half (which has been published) and a topological half (which has not). We will present a new and simpler argument for the topological half of the proof, which does not use any results about A-infinity or E-infinity ring spectra.

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