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.

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