Homotopy groups of infinite wedge - TEST UPLOAD ONLY NOT FOR INDEXING

Abstract

In {\it Homotopy Theory} (Pure and Applied Mathematics, Vol. VIII, Academic Press, New York--London, 1959), Sze-tsen Hu proved for $X\vee Y$, the wedge sum of pointed spaces $(X,x_0)$, and $(Y,y_0)$ that for $n\geq 2$ there is an isomorphism
\begin{equation}\label{e1}
\pi_n(X\vee Y,u_0)\approx \pi_n(X,x_0)\oplus \pi_n(Y,y_0)\oplus \pi_{n+1}(X\times Y,X\vee
Y, u_0),
\end{equation}
where $u_0=(x_0,y_0)$.

This result was not generalized for an infinite wedge $\vee Y_\om$,\, $\om\in \Om$, of pointed spaces $(Y_\om, y_\om^0)$ in view of the fact that an infinite wedge $\vee Y_\om$ is not a subspace of the direct product $\prod Y_\om$, $\om\in \Om$.

In the present work we prove that for $n\geq 2$ there is an isomorphism
$$
\pi_n(\vee Y_\om, y^0)\approx \sum_{\om\in \Om} \pi_n(Y_\om, y_\om^0)\oplus \pi_{n+1}(L Y_\om, \vee Y_\om, y^0),
$$
where $L Y_\om$ is the weak product of pointed topological spaces $(Y_\om, y_\om^0)$, $\om\in \Om$ (see %\cite{3}.
C. J. Knight, {\it Weak products of spaces and complexes}, Fund. Math. {\bf 53} (1963), 1--12.)