## Topological Hochschild homology of twisted group algebras

##### Author(s)

Vera, Daniel Joseph
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##### Other Contributors

Massachusetts Institute of Technology. Dept. of Mathematics.

##### Advisor

Lars Hesselholt.

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Let G be a group and A be a ring. There is a stable equivalence of orthogonal spectra ... between the topological Hochschild homology of the group algebra A[G] and the smash product of the topological Hochschild homology of A and the cyclic bar construction of G. This thesis generalizes this result to a twisted group algebra AT[G]. As an A-module, Ar[G] = A[G], but the multiplication is given by ag. a'g' = ag(a') gg', where G acts on A from the left through ring automorphisms. The main result is given in terms of a variant THH9(A) of the topological Hochschild spectrum that is equipped with a twisted cyclic structure inherited from the cyclic structure of the cyclic pointed space THH(A)[-]. We first define a parametrized orthogonal spectrum E(A, G) over the cyclic bar construction NCY(G). We prove there is a stable equivalence of spectra between the associated Thom spectrum of E(A, G) and THH(AT[G]). We then prove there is a stable equivalence of orthogonal spectra ... where the wedge-sum on the left hand side ranges over the conjugacy classes of elements of G and the equivalence depends on a choice of representative g E (g) of every conjugacy class of elements in G.

##### Description

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2006. Includes bibliographical references (p. 61-62).

##### Date issued

2006##### Department

Massachusetts Institute of Technology. Dept. of Mathematics.##### Publisher

Massachusetts Institute of Technology

##### Keywords

Mathematics.