An iterated sum formula for a spheroid's homotopy class modulo 2-torsion

Abstract

Let $X$ be a simply connected pointed space with finitely generated homotopy groups. Let ${\Pi }_n(X)$ denote the set of all continuous maps $a:I^n\to X$ taking $\partial I^n$ to the basepoint. For $a\in {\Pi }_n(X)$, let $[a]\in {\pi }_n(X)$ be its homotopy class. For an open set $E\subset I^n$, let $\Pi (E,X)$ be the set of all continuous maps $a:E\to X$ taking $E\cap \partial I^n$ to the basepoint. For a cover $\Gamma$ of $I^n$, let $\Gamma (r)$ be the set of all unions of at most $r$ elements of $\Gamma$. Put $r=(n-1)!$. We prove that for any finite open cover $\Gamma$ of $I^n$ there exist maps $f_E:\Pi (E,X)\to {\pi }_n(X)\otimes Z[1/2]$, $E\in \Gamma (r)$, such that $$[a]\otimes 1=\sum _{E\in \Gamma (r)}{f}_E(a{|}_E)$$ for all $a\in {\Pi }_n(X)$.

13 pages