Decompositions of loop spaces \(\Omega (X\lor Y\lor Z)\): A combinatorial approach
type: Item
instance of: scholarly article
title: Decompositions of loop spaces \(\Omega (X\lor Y\lor Z)\): A combinatorial approach
zbMATH Open document ID: 1188.55004
author: Wen Weng
author: Wenhuai Shen
published in: Topology and its Applications
publication date: 2009-08-20T00:00:00Z
review text: For simplicial groups \(G_\alpha\), \(\alpha \in J\), the authors introduce the groups \[ R \bigg(\coprod_{\alpha \in J}G_\alpha\bigg)= \ker\bigg(q: \coprod_{\alpha \in J}G_\alpha \longrightarrow \prod_{\alpha \in J}G_\alpha\bigg) \] and prove that if \(A_{1}\), \(A_{2}\), \(A_{3}\) are reduced simplicial free groups, then there is an isomorphism of simplicial sets \[ A_{1}* A_{2} * A_{3}\cong A_{1} \times A_{2} \times A_{3} \times R(A_{1}* A_{2} * A_{3}). \] Exploiting the above result and the Kan construction \(GK\) of a reduced simplicial set \(K\) which can be viewed as a loop complex for \(K\) and the Milnor construction \(F[K]\), the authors prove the following main result: if \(X,Y\) and \(Z\) are pointed path connected CW spaces, then there is a natural weak equivalence of spaces \[ \Omega(X \vee Y \vee Z) \simeq \Omega X \times \Omega Y \times \Omega Z \times \Omega\Sigma (W_1 \vee W_2) \] where \(W_1\) is the cofiber of \(i: X\vee Y \vee Z \longrightarrow X\times Y \times Z \) and \(W_2 = \Omega X \wedge \Omega Y \wedge \Omega Z\). By a similar method as the one above, the authors also show that \[ \Omega(X \vee Y \vee Z) \simeq \Omega X \times \Omega Y \times \Omega Z \times \Omega\Sigma W_1 \times \Omega\Sigma W_2 \times \Omega\Sigma ( \Omega\Sigma W_1 \wedge \Omega\Sigma W_2). \]
reviewed by: Younggi Choi
Mathematics Subject Classification ID: 55P35
zbMATH DE Number: 5597557
zbMATH Keywords: Decomposition
zbMATH Keywords: Loop space
zbMATH Keywords: Simplicial group
zbMATH Keywords: Free product of simplicial groups
zbMATH Keywords: Kan construction
zbMATH Keywords: Milnor construction
MaRDI profile type: Publication
full work available at URL: j.topol.2009.04.018
OpenAlex ID: W2026544827
cites work: Simplicial homotopy theory
cites work: Q5797065
cites work: Q4819371
cites work: A combinatorial definition of homotopy groups
cites work: Q4145882
cites work: Q4135772
cites work: Q5550390
cites work: Q5643280
DOI: 10.1016/J.TOPOL.2009.04.018
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description: scientific article; zbMATH DE number 5597557