18.917: Topics in algebraic topology

Time and place: TR 1-2:30 in 8-119.

In this course, I plan to give an introduction to topological Hochschild homology and the de Rham-Witt complex.

Associated to any ring A or, more generally, to any category with cofibrations and weak equivalences C one has the topological Hochschild spectrum T (A) or T (C), which is an object in the 𝕋-stable homotopy category (𝕋 is the circle group). In the equivariant setting the natural notion of cells are (the suspension 𝕋-spectra associated with) the 𝕋-spaces D^{q} ×𝕋/C, where C ⊂ 𝕋 is a closed subgroup. Hence, the natural notion of homotopy groups are maps in the homotopy category from

S^{q} ∧ 𝕋/C_{+} = (D^{q} × 𝕋/C)/(∂D^{q} × 𝕋/C).

We fix a prime p and consider the equivariant homotopy groups

TR_{q}^{n}(A; p) = [S^{q} ∧ 𝕋/C_{p^{n−1}+}, T (A)]_{𝕋}.

There are usually very large abelian groups. But as n and q varies they are related by a number of operators and the combined algebraic structure is rather rigid. We call this structure a Witt complex over A. There is an initial example of this structure called the de Rham-Witt complex, and we use the canonical map

W_{n}Ω_{A}^{q} → TR_{q}^{n}(A; p)

to describe the right-hand groups. This map is similar to the canonical map from Milnor K-theory to algebraic K-theory, and in favorable cases, the latter can be expresed in terms of the former.

After an introduction, here is a tentative route:

(i) Cyclic objects.

(ii) Equivariant stable homotopy theory.

(iii) The topological Hochschild spectrum.

(iv) The de Rham-Witt complex.

(v) The Tate spectrum.