In the lecture notes [The homology of $\mathcal{C}_{n+1}$–spaces, n ≥ 0. F. Cohen, 1978,][1] 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,][1] 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!

  [1]: http://www.math.uchicago.edu/~may/BOOKS/homo_iter.pdf