Kunneth formula is $$ H^\ast (S^K;{\bf Z})\otimes_{{\bf Z}} H^\ast (S^M;{\bf Z}) =H^\ast (S^K\times S^M;{\bf Z}) =\wedge_{\bf Z} [a,b]$$ where $K=2k+1<M=2m+1$, $H^\ast (S^K;{\bf Z}) = \wedge_{\bf Z}[a]$, and $ H^\ast (S^M;{\bf Z}) =\wedge_{\bf Z}[b]$
Note that by definition of tensor poeduct, $$ab=ba \in H^\ast (S^K;{\bf Z})\otimes_{{\bf Z}} H^\ast (S^M;{\bf Z})$$
But in exterior algebra, $$ ab=(-1)^{KM}b\cup a=-ba\in H^\ast (S^K\times S^M;{\bf Z}) =\wedge_{\bf Z} [a,b]$$
Where is wrong point ?
(My reference is 218 page in Hatcher's book " Algebraic topology ")
