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Let $M$ and $N$ be $CW$-complexes. Definition. (different from the isotopy notion in geometry of submanifolds). A (topological) isotopy is a fibre-wise continuous map $$ F: M\times [0,1]\longright …

Suppose we have a compact $m$-dimensional submanifold $M$ of $\mathbb{R}^N$ and we want to know the cohomology ring $H^*(M;\mathbb{Z})$. Let $\epsilon>0$ and a $m$-dimensional finite simplicial com …

Let $X$ be a path-connected manifold (or a CW complex). Let $\pi_1(X)$ be the fundamental group of $X$. Let $\pi: \tilde X\longrightarrow X$ be a covering map. For each $m\geq 0$, let $C_m(\tilde X)$ …

Let $X$ be a manifold or a CW-complex. Let $\pi: \tilde X\longrightarrow X$ be a covering map. Let $\pi_1(X)$ be the fundamental group of $X$ and let $\rho: \pi_1(X)\longrightarrow O(n)$ be an ort …

In Homology of $C_{n+1}$-spaces, $n\geq 0$, F.R. Cohen, Lecture Notes in Mathematics, Vol. 533, page 210 (the preface part before contents): Line 2: ... is used to compute the precise algebra struct …

Let $G$ be a group, either a Lie group or a discrete group. Let a principal $G$-bundle $$ G\to E\to B,$$ then $B=E/G$, the orbit space under action of $G$. Let $BG$ be the classifying space of $G$. …

In the paper GEOMETRY OF TRUNCATED SYMMETRIC PRODUCTS AND REAL ROOTS OF REAL POLYNOMIALS, JACOB MOSTOVOY, Bull. London Math. Soc. (1998) 30 (2): 159-165, Theorem 2. (b): $TP^n(\mathbb{R}P^1)$ is ho …

Let $X$ be an $n$-dimensional cell complex. We attach an $(n+1)$-cell $e^{n+1}$ to $X$ and obtain a new cell complex $X'$. Take the universal cover (or a general covering space) $\tilde X'$ of $X'$. W …

Let $\mathbb{Z}_2=\mathbb{Z}/2\mathbb{Z}$. Let $$ O(\mathbb{Z}_2^{\oplus k})=\{A\mid A \text{ is a } k\times k \text{ - matrix with entries } 0,1, det(A)=\pm 1\} $$ What is $$ H^*(BO(\mathbb{Z}_2^{ …

Let $\Sigma_k$ be the permutation group of order $k$. Let $r: \Sigma_k\to GL(k)$ be the regular representation by permuting the order of the standard basis of $\mathbb{R}^n$. Let $\rho: B\Sigma_k\t …

Let $M$ be a manifold. Then $F(M,k)/\Sigma_k$, the unordered configuration space of $k$ points, is obtained as a quotient of $F(M,k)$, the ordered configuration space of $k$ points, by the group actio …

Let $M$ be a manifold. Let $F(M,n)$ be the configuration space of $n$-tuples on $M$. Let $B(M,n)=F(M,n)/S_n$, where $S_n$ is the symmetric group of order $n$, be the corresponding unordered config …

Let $S_3$ be the symmetric group of order $3$. What is the cohomology ring $$ H^*(S_3;\mathbb{Z})?$$ My attempt: I want to use mathematical induction on $n$ for $S_n$. For $n=1$, $S_1$ is trivial. H …

Let $$ O(\mathbb{Z}^{\oplus k})=GL(\mathbb{Z}^{\oplus k})\cap O(k). $$ What is $$ H^*(BO(\mathbb{Z}^{\oplus k});\mathbb{Z})? $$ If it cannot be computed out, can we get $$ H^*(O(\mathbb{Z}^{\oplus …

Let $F(\mathbb{R}P^n,k)$ be the $k$-th ordered configuration space on $\mathbb{R}P^n$. In http://arxiv.org/abs/1502.04258, the cohomology ring $$ H^*(F(\mathbb{R}P^n,k);R)$$ is obtained for any commu …