# Moscow Mathematical Journal

Volume 18, Issue 1, January–March 2018  pp. 149–162.

A Spectral Sequence for Homology of Invariant Group Chains

Authors:  Rolando Jimenez (1), Angelina López Madrigal (1), and Quitzeh Morales Meléndez (2)
Author institution:(1) Instituto de Matemáticas, Unidad Oaxaca, Universidad Nacional Autónoma de México, León 2, 68000 Oaxaca de Juárez, Oaxaca, México
(2) CONACYT – Universidad Pedagógica Nacional, unidad 201 Camino a la Zanjita S/N, Col. Noche Buena, Santa Cruz Xoxocotlán, Oaxaca. C.P. 71230

Summary:

Let Q be a finite group acting on a group G by group automorphisms, C(G) the bar complex and HQ(G,A) the homology of invariant group chains defined in K. Knudson’s paper The homology of invariant group chains''. In this paper we construct a spectral sequence converging to H(Q,C(G)⊗A) whose second term is isomorphic to HQ (G,A) for some coefficients. When this spectral sequence collapses this yields an isomorphism HQ (G,A) ≅ H(Q, C(G)⊗A), which we use to compute this homology for some cases. The construction uses a decomposition of the bar complex C(G) in terms of the induction from some isotropy groups to the group Q. We also decompose the subcomplex of invariants C(G)Q by Q-orbits and use this to compute the invariant 1-homology H1Q(G, ℤ) for some cases.

2010 Math. Subj. Class. Primary: 55N25, 55T05; Secondary: 18G40, 18G35.

Keywords: Bar complex, homology of invariant group chains, spectral sequences.