List questions from homology-cohomology

Published on 25 Mar 2026 - 14:26

163

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Confusion with a step in the construction of chain homotopy from a homotopy

Published on 26 Jun 2019 - 18:16

102

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Build an example of a topological space $A$ such that $H_4(A;\mathbb{Z})=\mathbb{Z}/2$

Published on 26 Jun 2019 - 19:42

481

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homology of $RP^\infty$ is a coalgebra.

Published on 27 Mar 2026 - 0:06

72

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Splitting induces isomorphisms

Published on 27 Mar 2026 - 6:45

88

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Boundary operator preserved under isomorphism so relative homology isomorphic.

Published on 25 Mar 2026 - 23:16

35

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$H_p(K)=H_p(K^{(p+1)},K^{(p-2)})$?

Published on 25 Mar 2026 - 23:08

76

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Why is $H_1(S^0)=0$?

Published on 26 Mar 2026 - 4:50

39

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Why is $H_i(E_+^n,S^{n-1})\cong H_{i-1}(S^{n-1})?$

Published on 30 Jun 2019 - 16:57

65

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If $\mathbb{Z}≅\widetilde{H_0}(\mathbb{R}P^2)⊕\mathbb{Z}$ then $\widetilde{H_0}(\mathbb{R}P^2)=0$?

Published on 30 Jun 2019 - 20:24

279

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Interesting topological spaces to calculate the homology groups.

Published on 01 Jul 2019 - 18:23

57

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$\Theta$ is induced by the inclusion map $Z_p\otimes Z_q' \to (C\otimes C')_{p+q}$?

Published on 25 Mar 2026 - 7:45

420

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Suppose that $X$ is a topological space and $x_0\in X$. Prove that $\widetilde{H_n}(X)=H_n(X,x_0)$ for all $n\geq 0$.

Published on 02 Jul 2019 - 22:38

361

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If $f,g:(X,A)\to (Y,B)$ are homotopic functions under a homotopy $H:X\times [0,1]\to Y$ then $f_=g_:H_n(X,A)\to H_n(Y,B)$.

Published on 02 Jul 2019 - 23:07

137

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Why $H_1(SX)\cong \widetilde{H_0}(X)$?

Published on 03 Jul 2019 - 22:10

List Question

Isomorphism in all homology and cohomology groups

Confusion with a step in the construction of chain homotopy from a homotopy

Build an example of a topological space $A$ such that $H_4(A;\mathbb{Z})=\mathbb{Z}/2$

homology of $RP^\infty$ is a coalgebra.

Splitting induces isomorphisms

Boundary operator preserved under isomorphism so relative homology isomorphic.

$H_p(K)=H_p(K^{(p+1)},K^{(p-2)})$?

Why is $H_1(S^0)=0$?

Why is $H_i(E_+^n,S^{n-1})\cong H_{i-1}(S^{n-1})?$

If $\mathbb{Z}≅\widetilde{H_0}(\mathbb{R}P^2)⊕\mathbb{Z}$ then $\widetilde{H_0}(\mathbb{R}P^2)=0$?

Interesting topological spaces to calculate the homology groups.

$\Theta$ is induced by the inclusion map $Z_p\otimes Z_q' \to (C\otimes C')_{p+q}$?

Suppose that $X$ is a topological space and $x_0\in X$. Prove that $\widetilde{H_n}(X)=H_n(X,x_0)$ for all $n\geq 0$.

If $f,g:(X,A)\to (Y,B)$ are homotopic functions under a homotopy $H:X\times [0,1]\to Y$ then $f_=g_:H_n(X,A)\to H_n(Y,B)$.

Why $H_1(SX)\cong \widetilde{H_0}(X)$?

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