Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 0708.1575

Symmetric Homology of Algebras

Shaun Ault·Zbigniew Fiedorowicz
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this note, we outline the general development of a theory of symmetric homology of algebras, an analog of cyclic homology where the cyclic groups are replaced by symmetric groups. This theory is developed using the framework of crossed simplicial groups and the homological algebra of module-valued functors. The symmetric homology of group algebras is related to stable homotopy theory. Two spectral sequences for computing symmetric homology are constructed. The relation to cyclic homology is discussed and some conjectures and questions towards further work are discussed.

Explore related subjects

math.AT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Shaun Ault, Zbigniew Fiedorowicz. 2007-11-05. Symmetric Homology of Algebras. https://arxiv.org/abs/0708.1575

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Galois Connections in Persistent Homology

We present a new language for persistent homology in terms of Galois connections. This language has two main advantages over traditional approaches. First, it simplifies and unifies central concepts such as interleavings and matchings. Second, it provides access to Rota's Galois connection theorem -- a powerful tool with many potential applications in applied topology. To illustrate this, we use Rota's Galois connection theorem to give a substantially easier proof of the bottleneck stability theorem. Finally, we use this language to establish relationships between various notions of multiparameter persistence diagrams.

math.AT↗
arXiv2026

The Dold-Kan theorem for paracyclic modules

We study the Karoubi operator on the unnormalized chain complex of a paracyclic module; its restriction to the normalized chain complex has previously been considered by Dwyer and Kan, and in the cyclic case by Cuntz and Quillen. We obtain a direct proof of the Dold-Kan theorem for paracyclic modules of Dwyer and Kan, by directly relating the Karoubi operator to projection to the normalized subcomplex.

math.AT↗
arXiv2026

Equivariant bordism rigidity for toric manifolds

In this paper, we develop a bordism-theoretic approach to rigidity problems for toric and quasitoric manifolds. We prove that two toric manifolds are isomorphic as varieties if and only if they are weakly equivariantly unitary bordant. We also establish a parallel rigidity result for omnioriented quasitoric manifolds satisfying the injectivity condition, showing that their equivariant unitary bordism classes completely determine their omniorientation-preserving equivariant homeomorphism types. Thus, equivariant bordism provides a topological framework for detecting geometric and combinatorial rigidity.

math.AT↗