Abstract
AbstractWe raise the question of the realizability of permutation modules in the context of Kahn’s realizability problem for abstract groups and the G-Moore space problem. Specifically, given a finite group G, we consider a collection $$\{M_i\}_{i=1}^n$$
{
M
i
}
i
=
1
n
of finitely generated $$\mathbb {Z}G$$
Z
G
-modules that admit a submodule decomposition on which G acts by permuting the summands. Then we prove the existence of connected finite spaces X that realize each $$M_i$$
M
i
as its i-th homology, G as its group of self-homotopy equivalences $$\mathcal {E}(X)$$
E
(
X
)
, and the action of G on each $$M_i$$
M
i
as the action of $$\mathcal {E}(X)$$
E
(
X
)
on $$H_i(X; \mathbb {Z})$$
H
i
(
X
;
Z
)
.
Funder
Ministerio de Asuntos Económicos y Transformación Digital, Gobierno de España
Fundação para a Ciência e a Tecnologia
Publisher
Springer Science and Business Media LLC
Reference18 articles.
1. Barmak, J.A.: Algebraic Topology of Finite Topological Spaces and Applications. Lecture Notes in Mathematics, 2032. Springer, Heidelberg (2011)
2. Bouwer, I.Z.: Section graphs for finite permutation groups. J. Combin. Theory 6, 378–386 (1969)
3. Carlsson, G.: A counterexample to a conjecture of Steenrod. Invent. Math. 64, 171–174 (1981)
4. Cianci, N., Ottina, M.: Smallest weakly contractible non-contractible topological spaces. Proc. Edinb. Math. Soc. 63(1), 263–274 (2020)
5. Chocano, P.J., Morón, M.A., Ruiz del Portal, F.: Topological realizations of groups in Alexandroff spaces. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 115(1), Paper No. 25, 20 pp (2021)