HOMOTOPY METHODS IN ALGEBRAIC TOPOLOGY

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Let F be a functor from a basepointed category with finite
coproducts to a category of chain complexes over an abelian category.
Such a functor is homologically degree * n *
if its *(n+1)*-st cross effect (in
the sense of Eilenberg and Mac Lane) has trivial homology. We describe a
method for constructing, by means of cotriples associated to the cross
effects of F, a universal tower,

... ---> P_{n+1} F ---> P_n F ---> ... ---> P_1 F ---> P_0 F = F(*),

in which each functor P_n F is homologically degree * n*. This
construction arose from the study of Goodwillie's Taylor tower in the case
of functors of modules over a ring. Using this model, we will
characterize homologically degree *n * functors in terms of modules over a
certain DGA, and discuss some related constructions due to Eilenberg-Mac
Lane, and Dold-Puppe. This is joint work with Randy McCarthy.

We sketch a new cohomology theory of "quantum" differential forms which generalizes Sullivan's work for rings of arbitrary characteristic. This new approach may be extended to other contexts, like the cohomology with values in a sheaf. As applications we propose

1. A much simpler description of cup-i products (and of Steenrod operations)

2. Thanks to the recent work of Kriz, Mandell and May, a new approach to homotopy type through the concept of "neo-algebra" (a generalisation of a commutative DGA in positive characteristic)

3. An explicit computation of the cohomology of iterated loop spaces in terms of this neo-algebra approach.

(joint work with Florian Luca)

Let G be a compact Lie group, R be a commutative G-Mackey functor ring, and R(G) be the value of R at G. There is a topology on the set Spec(R) of Mackey functor prime ideals of R which is an obvious generalization of the Zariski topology on the spectrum of an ordinary commutative ring. This space Spec(R) carries a significant of information about R. In particular, the spectrum Spec(R(G)) of the ordinary commutative ring R(G) is a retract of Spec(R). Moreover, there is a function from the set Spec(R) to the set Conj(G) of conjugacy classes of subgroups of G, which can be used to determine the strongest possible induction theory satisfied by R.

This talk will be devoted to a discussion of the properties of Spec(R). Examples of Spec(R) for various groups G and rings R will also be discussed.

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