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

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 $M$ 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, …

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 …

Let $S^m$ be the $m$-sphere and $\tau (S^m)$ the sphere bundle consisting of unit tangent vectors in the tangent bundle $TS^m$. Then we have a fibration $$ S^{m-1}\longrightarrow \tau(S^m)\longrightar …

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

What is the mod 2 cohomology ring $$ H^*(O(3)/Sym(P);\mathbb{Z}/2)? $$ …

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; …

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? …

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). …

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

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 …

From the lecture notes INTRODUCTION TO CONFIGURATION SPACES AND THEIR APPLICATIONS, p. 18, I find: Os it possible to derive the cohomology ring $H^*(Conf(S,k)/\Sigma_k;\mathbb{Z}_2)$ from the above theorem … $$ Question 2: Given a group $G=\pi_1(Conf(S,k)/\Sigma_k)$, I find $ K(G,1)=BG. $ Are there any methods to compute the cohomology ring (cup product structure) $$ H^*(BG;\mathbb{Z}_2)? $$ …

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? …