Let $f \colon S^p \to S^p$ and $g \colon S^q \to S^q$ be continuous maps and consider a map $f*g \colon S^p * S^q \to S^p * S^q$ given by $(1-t)x+ty \mapsto (1-t)f(x) + tg(y)$. Here $*$ denotes join. Since $S^p * S^q = S^{p+q+1}$, it makes sense to calculate Brouwer's degree of $f*g$. I think this should be equal to $(\deg f)(\deg g)$, but I don't have an idea how to prove this: how do I express a generator of $H_{p+q+1}(S^p * S^q)$ in terms of generators of homology of individual spheres? Or maybe my intuition is just wrong?
Degree of a map $f * g \colon S^p * S^q \to S^p * S^q$
2
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algebraic-topology