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In the lecture notes The homology of $\mathcal{C}_{n+1}$–spaces, n ≥ 0. F. Cohen, 1978, page 228-231, the cohomology ring $$ H^*(\text{Map}_*(S^n, S^n\wedge X);\mathbb{Z}_p) $$ is obtained for any primes $p\geq 2$.

Question: I want to know the cohomology ring $$ H^*(\text{Map}_*(M, S^n);\mathbb{Z}_2) $$ for some manifolds $M$ other than $S^n$, for example, $M=\mathbb{R}^n, \mathbb{R}P^n, \mathbb{C}P^n, \mathbb{T}^n, S^{n-1}\times\mathbb{R}, $ etc. Are there any such generalizations or references? Thanks.

I also do not understand the proof in The homology of $\mathcal{C}_{n+1}$–spaces, n ≥ 0. F. Cohen, 1978, page 228-231, the part proving $H_*\Omega^{n+1}_\phi\Sigma^{n+1}X=AT_nH_*X$ ($\Omega^{n+1}_\phi\Sigma^{n+1}X$ denotes the path-component of $\Omega^{n+1}\Sigma^{n+1}X$ containing the base-point, i.e., the collection of maps in Map$_*(S^{n+1},S^{n+1}\wedge X)$ of degree $0$). Could you explain it more? Thanks!

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    $\begingroup$ I'll just list some basics. Since the mapping space is only dependent on the based homotopy class of $M$, $M=\mathbb{R}^n$ is trivial. For $M=S^{1}\times \mathbb{R}$, then $Map_*(S^1, S^n)=\Omega S^n$, so the cohomology is $\mathbb{Z}/(2)$ in degrees divisible by $n$, and $0$ otherwise. The case of higher tori gets extremely complicated, see here, but the Serre spectral sequence associated to $Map(\mathbb{T}^{m}, S^n)\to *\to Map(\mathbb{T}^{m-1}, S^n)$ can, for example, that for $m=2$ and $n=2s+1$, it is a polynomial ring. $\endgroup$
    – Pax
    Commented Jul 14, 2015 at 12:07

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