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Cohen, Lecture Notes in Mathematics, Vol. 533, Chapter 5, 6, 7, 8, 9, 10, 11, the cohomology algebra $H^*(B(\mathbb{R}^{n+1},p),\mathbb{Z}_p)$, for $p$ prime and $B(\mathbb{R}^{n+1},p)=F(\mathbb{R}^{n … For other manifolds $M$ such as $S^m$ and $S^m\times \mathbb{R}^k$ ($H^*F(\mathbb{R}^{n+1},p;\mathbb{Z}_p)$ is known in these cases), are there any results for the cohomology algebra $H^*(B(M,p);\mathbb …

In the paper Homology fibrations and group completion theorem, McDuff-Segal (www.maths.ed.ac.uk/~aar/papers/mcdsegal.pdf), page 281: Let $M$ be a topological monoid such that $\pi_0M$ is generated by …

In the lecture notes The Homology of $\mathcal{C}_{n+1}$-spaces, $n\geq 0$, F. Cohen, page 225 -226, it is obtained that there is a group completion on homology $$ \alpha_n: C(\mathbb{R}^n;X)\to \Omeg …

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 …

Question: How to compute the cohomology ring $$ H^*(\Gamma(\xi);\mathbb{Z}_2)? $$ My attempt: I want to construct fibrations and use Serre spectral sequence. … Moreover, could the Serre spectral sequence give the explicit ring structure of $$ H^*(\Gamma(\xi);\mathbb{Z}_2) $$ completely, but not partial information about the cohomology? …

Their cohomology rings are expressed in terms of universal Stiefel-Whitney classes $$ H^*(G_k(\mathbb{R}^\infty);\mathbb{Z}_2)=\mathbb{Z}_2[w_1,w_2,\cdots,w_k],$$ $$ H^*(G_k(\mathbb{R}^n);\mathbb{Z}_2) … =\mathbb{Z}_2[w_1,w_2,\cdots,w_k]/(\bar w_{n-k+1},\bar w_{n-k+2}\cdots,\bar w_n).$$ What are the Steenrod operations $Sq^i$ on these cohomology rings? …

Let $M$ be a topological monoid with product $\mu$. Then $H_*(M)$ is a Hopf algebra with product $\mu_*$ and coproduct $\Delta_*$. The group-completion theorem by McDuff-Segal, 1976 gives that as a P …

Suppose all differentials in the cohomology Serre spectral sequence (corresponding to the above fibration) are zero maps. … . $$ Question 1: as cohomology rings with cup products, do we still have the isomorphism $$ H^*(E)\cong H^*(F)\otimes H^*(B)? …

Let $X$ be a topological space and $F$ a field. Let the $n$-th permutation group $\Sigma_n$ act on $$ \prod_n X $$ by $$ \sigma(x_1,\cdots,x_n)=(x_{\sigma(1)},\cdots,x_{\sigma(n)}), \sigma\in \Sig …

Let $M$ by an compact, connected $n$-dimensional manifold without boundary. Are there any other computable examples of the Stiefel-Whitney class $w(M)$ except for $M=S^m, \mathbb{R}P^m,\mathbb{C}P^m, …

Question: are there any references for the cohomology ring $$ H^*(B(M,k,\epsilon);\mathbb{Z}_2)? $$ …

Or even in terms of the cohomology ring $$ H^*(B;\mathbb{Z}_2), H^*(F;\mathbb{Z}_2) $$ And other factors? …

What is the cohomology ring $$ H^*(A_4;\mathbb{Z}/3) $$ and its Steenrod operation $P^i$'s? (2). … Are there general results about the cohomology ring $$ H^*(A_{p+1};\mathbb{Z}/p) $$ for general primes $p\geq 3$? (3). …

Let $M$ be a manifold. The total Stiefel-Whitney class of $M$ is defined to be the Stiefel-Whitney class of the tangent bundle $TM$ $$ w(M)=1+w_1(TM)+w_2(TM)+\cdots $$ I want to find references for $$ …

Let $M,N$ be manifolds whose dimensions may be different. Let $f: M\longrightarrow N$ be a smooth map. What geometric conditions on $f$ can we impose such that the induced homomorphism $$ f^*: H^*(N; …