I read in Algebraic Topology by Allen Hatcher :
1.1.5 Show that for a space X, the following conditions are equivalent:
(a) Every map $S^1 \to X $is homotopic to a constant map, with image a point.
(b) Every map $S^1 \to X $ extends to a map $D^2 \to X$.
now can we prove :
Show that for a space X, the following conditions are equivalent:
(a) Every map $S^n \to X $is homotopic to a constant map, with image a point.
(b) Every map $S^n \to X $ extends to a map $D^{n+1} \to X$.
is it true ?