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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 $$ …

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

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)? …

Question: as a manifold, what is the cohomology ring (with cup product) $$ H^*(G;\mathbb{Z}_2) $$ and the Steenrod square $Sq$'s acting on the cohomology ring? …

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 $M$ 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 where I can find the explicit expression of the cohomology algebra $$ H^*(\Omega^nS^n;\mathbb{Z}_2)? …

Then according to Theorem 1.6, The Cohomology of BSO n and BO n with Integer Coefficients, Proceedings of the American Mathematical Society 1982 Vol 85-2, Edgar H.Brown JR., $H^*(G_n(\mathbb{R}^\infty …

Let $M$ be a manifold and $k$ be a positive integer. Let $F(M,k)$ be the $k$-th ordered configuration space over $M$, consisting of all ordered $k$-tuples of distinct points in $M$. Let $\Sigma_k$ be …

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

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 …

. $$ Question: Could the theorem 1.1 be strengthened to the statement that for some $k$, $$ (\alpha_k)^*: H^*(\Gamma_k(M;S^0))\to H^*(F(M,k)/\Sigma_k) $$ is a ring isomorphism of cohomology rings? …

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 …

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). …