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

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?

## 2 Answers

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.)

• The difference is this: An infinite sum of groups has elements with only finitely many coordinates non-zero. An infinite product does not have this limitation on its elements (the real, formal difference is category-theoretical, but for groups, this is what the difference manifests as) – Arthur Mar 2 '14 at 18:47

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 $\sum_{i=1}^{n}a_{i}$.

• Abelianness is unnecessary here. (The direct sum still makes sense for nonabelian groups, it just isn't the coproduct anymore. Instead it's the "commutative coproduct.") – Qiaochu Yuan Mar 2 '14 at 19:06
• Yes, thanks for the reminder. I am only aware of direct sum as the coproduct. But I think you mean "non-commutative coproduct" right? – Bombyx mori Mar 2 '14 at 19:09
• No, I mean "commutative coproduct." The direct sum of a family of groups $G_i$ is the universal group equipped with a map from all of the $G_i$ all of whose images commute. – Qiaochu Yuan Mar 2 '14 at 19:41
• I see. Thanks for explaining. This does appear to be the right generalization. – Bombyx mori Mar 2 '14 at 19:55