Prove that $S^1\times S^3,S^2\times S^2,S^4,S^1\times S^1\times S^1\times S^1$ are not homeomorphic to each other by using fundamental group or homology group.
I have known that the fundamental group of $S^n$ is $$\pi(S^1)=\mathbb Z,\pi(S^n)=\{e\}(n>1),$$ and the homology group of $S^n$ is $$H_i(S^n)=\mathbb Z,(i=0,n);H_i(S^n)=0,(i\ne0,n),$$ But now I don't know what to do. Thanks for any help.
On
All of these spaces can be distinguished by their fundamental groups (using the property that the fundamental group respects products - $\pi_1(X\times Y)\cong\pi_1(X)\times\pi_1(Y)$) except for the spaces $S^2\times S^2$ and $S^4$. To distinguish these spaces, it suffices to calculate homology groups. In particular, you already know that $H_2(S^4)=0$, and so you now just need to calculate $H_2(S^2\times S^2)$ which, by the Kunneth formula, you will find is non-trivial.
Hint: The fundamental group and homology group respect products, i.e. $\pi(A\times B)=\pi(A)\times \pi(B)$ and $H(A\times B)=H(A)\otimes H(B)$. Furthermore if two spaces are homeomorphic they have the same fundamental and homology groups.