Decompositions of loop spaces \(\Omega (X\lor Y\lor Z)\): A combinatorial approach
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 \[ \Omega(X \vee Y \vee Z) \simeq \Omega X \times \Omega Y \times \Omega Z \times \Omega\Sigma W_1 \times \Omega\Sigma W_2 \times \Omega\Sigma ( \Omega\Sigma W_1 \wedge \Omega\Sigma W_2). \]
scientific article; zbMATH DE number 5597557
scientific article