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 $Π_n(X)$ denote the set of all continuous maps $a:I^n\to X$ taking $\partial I^n$ to the basepoint. For $a\inΠ_n(X)$, let $[a]\inπ_n(X)$ be its homotopy class. For an open set $E\subset I^n$, let $Π(E,X)$ be the set of all continuous maps $a:E\to X$ taking $E\cap\partial I^n$ to the basepoint. For a cover $Γ$ of $I^n$, let $Γ(r)$ be the set of all unions of at most $r$ elements of $Γ$. Put $r=(n-1)!$. We prove that for any finite open cover $Γ$ of $I^n$ there exist maps $f_E:Π(E,X)\toπ_n(X)\otimes Z[1/2]$, $E\inΓ(r)$, such that $$ [a]\otimes1=\sum_{E\inΓ(r)} f_E(a|_E) $$ for all $a\inΠ_n(X)$.