What is the difference between multiplication and direct sum on homotopy groups of spheres?

Question

In Allen Hatcher's book Algebraic topology he states that factoring out 2-torsion $\pi_{i}(S^{2n})\cong\pi_{i-1}(S^{2n-1})\times\pi_{i}(S^{4n-1})\:\forall n$ but in his book Spectral Sequences in Algebraic Topology he has $\pi_{i}(S^{n})\cong\pi_{i-1}(S^{n-1})\bigoplus\pi_{i}(S^{2n-1})\:\forall n$ even. Am I correct in assuming these are the same or am I totally missing something?

Answer 1

Finite direct sums and finite direct products are the same thing on groups. The homotopy theory part has nothing to do with it. (Infinite direct sums and products are different, though, so be careful about this.)

Answer 2

The higher homotopy groups are all abelian, so for finite indices you have the canonical isomorphism:

$$ \prod^{n}_{i=1}A_{i}=\bigoplus^{n}_{i=1}A_{i} $$ by mapping $(a_{1}\cdots a_{n})$ to $\sum_{i=1}^{n}a_{i}$.