On the Symmetric Homology of Algebras

Details On the Symmetric Homology of Algebras On the Symmetric Homology of Algebras

2008-07-28

arxiv.org/abs/0807.4521v1

Mathematics

Algebraic Topology

xi+155 pages. This manuscript represents the author's PhD dissertation thesis at The Ohio State University, August 2008. This

Scientific paper

Symmetric homology is an analog of cyclic homology in which the cyclic groups are replaced by symmetric groups. The foundations for the theory of symmetric homology of algebras are developed in the context of crossed simplicial groups using derived functors and the symmetric bar construction of Fiedorowicz. The symmetric homology of group rings is related to stable homotopy theory. Two chain complexes are constructed that compute symmetric homology, as well as two spectral sequences. In the setup of the second spectral sequence, a complex isomorphic to the suspension of the cycle-free chessboard complex of Vrecica and Zivaljevic appears. Homology operations are defined on the symmetric homology groups over Z/p, p a prime. Finally, an explicit partial resolution is presented, permitting the computation of the zeroth and first symmetric homology groups of finite-dimensional algebras.

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