Mappings of the sphere to a simply connected space

S. S. Podkorytov

Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 9, Tome 329 (2005), pp. 159-194

Résumé

Fix an $m\in\mathbb N$, $m\ge2$. Let $Y$ be a simply connected pointed CW-complex, and let $B$ be a finite set of continuous mappings $a\colon S^m\to Y$ respecting the marked points. Let $\Gamma(a)\subset S^m\times Y$ be the graph of $a$, and let $[a]\in\pi_m(Y)$ be the homotopy class of $a$. Then for some $r\in\mathbb N$ depending on $m$ only, there exist a finite set $E\subset S^m\times Y$ and a mapping $k\colon E(r)=\{\,F\subset E:|F|\le r\,\}\to\pi_m(Y)$ such that for each $a\in B$ we have $$ [a]=\sum_{F\in E(r):F\subset\Gamma(a)}k(F). $$