this is a topology question:
True or false: Every simply connected space is contractible
I think it's false, and I saw the counterexample of $S^n$, however the knowledge to show that it is not contractible is (using homology) beyond my course.
Could you provide a counterexample to this statement, and also gives a proof of why it is not contractible?
Thank you in advance for the help!
All the homotopy groups of a contractible space are trivial (because all the homotopy groups of the point you're contracting to are trivial). Consequently, if any $\pi_i \not\cong \{0\}$, your space is not contractible. For all $n > 0$, $\pi_n(S^n) \cong \mathbb{Z}$. This is a standard fact, and you can read more here. (This is proved using degree theory.) In fact, that $\pi_i(S^n)$ is frequently nontrivial for $i > n > 1$ is a wide area of study.
If you are familiar with homology, you could use the standard fact $H_0(S^n) \cong H_n(S^n) \cong \mathbb{Z}$ and the remaining homology groups are trivial. This is normally shown applying the Hurewicz theorem to the above results on homotopy groups and the fact that $S^n$ is $(n-1)$-connected.