# 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 *H*_{∗}^{Q}(*G*,*A*) the homology of
invariant group chains defined in K. Knudsonās 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
*H*_{∗}^{Q} (*G*,*A*) for some coefficients. When this spectral sequence collapses
this yields an isomorphism H_{∗}^{Q} (*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 *H*_{1}^{Q}(*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.

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