Moscow Mathematical Journal
Volume 18, Issue 1, January–March 2018 pp. 149–162.
A Spectral Sequence for Homology of Invariant Group Chains
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. 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
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 H1Q(G, ℤ) for some cases. 2010 Math. Subj. Class. Primary: 55N25, 55T05; Secondary: 18G40, 18G35.
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:
Keywords: Bar complex, homology of invariant group chains, spectral sequences.
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