# Tagged Questions

Questions about algebraic methods and invariants to study and classify topological spaces: homotopy groups, (co)-homology groups, fundamental groups, covering spaces, and beyond.

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### $H^*(S^1\times S^3;\mathbb{Z})$ and $H^*(S^1\vee S^3 \vee S^4;\mathbb{Z})$ are not isomorphic as rings

I'm stuck to prove that the singular cohomology groups with coefficients in $\mathbb{Z}$, $H^*(S^1\times S^3;\mathbb{Z})$ and $H^*(S^1\vee S^3 \vee S^4;\mathbb{Z})$, are not isomorphic as rings. What ...
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### Help with formalisation of a reasoning about simplicial sets

We briefly mention during lectures that the homotopy relation between two maps of simplicial sets is not necessarily symmetric. So I thought for a counterexample but I don't know how to formalise it ...
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### Given $(X,A)$，$(Y,B)$ such that $X/A$ and $Y/B$ are homotopy equivalent,are their relative homology groups isomorphic?

Suppose that $(X,A)$，$(Y,B)$ are pairs of topological spaces. If $X/A$ and $Y/B$ are homotopy equivalent, are $H_*(X,A;\mathbb{Z})$ isomorphic to $H_*(Y,B;\mathbb{Z})$ ?
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### Simplicial homology for infinite complexes

Simplicial homology can be viewed as a covariant functor from the category of finite simplicial complexes with continuous maps over support polyhedra, to the category of sequences of abelian groups. A ...
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### Is there a long exact cofiber sequence for a homotopy pushout?

Let $f:Z \to X$, $g: Z \to Y$ and let $M(f,g)$ be the corresponding mapping cylinder. Does the homotopy pushout diagram induce a long cofiber sequence? If so what does it look like?
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### How does singular homology work?: $H_1(S^1)$.

This is a fundamental inquiry into the nature of singular homology. Let $\gamma$ and $\sigma$ be the following singular 1-chains on the circle: $$\gamma(t)=2\pi t,~~~~~\sigma(t)=4\pi t,$$ Now, based ...
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### triviality of vector bundles with the reduced homology of base space entirely torsion

Let $\xi$ be an $n$-dimensional vector bundle over a manifold $M$ such that the reduced cohomology $\tilde H^*(M;\mathbb{Z})$ is entirely torsion (every element has finite order under addition). ...
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### Hatcher example 3H.3. Local coefficients via Modules

I'm trying to understand the following example made by Hatcher at page 329 in the section "Local Coefficients via Modules" The problem arises when I start to prove that a basis for $C_n^+(X')$ is ...
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### Action induces action of group ring on singular chain complex. [duplicate]

See here for a related question. Let $X$ be a space that satisfies the hypotheses used to construct a universal cover $\overline{X}$. Let $\pi = \pi_1(X)$ and consider the action of the group $\pi$ ...
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### Classification of Torus bundle over $\mathbb{S}^1$ [closed]

$T^2$ has a fixed free isometry group as $D_8$, is this true that $T^2$ bundle over $S^1$ have three type?
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### What is $H_*(S^m \times S^n)$, without using Künneth theorem?

As the question suggests, what is $H_*(S^m \times S^n)$ for $m \ge 1$ and $n \ge 1$? I would like to see a way without using the Künneth theorem...
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### Difference between homology and integral homology?

What is integral homology? And how does it relate to homology? I can't find a good answer anywhere, so I thought I would ask here.
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### triangle with edges identified

What is the space obtained by identifying the three edges of a triangle in this way: assume the vertex of the triangle is a,b and c, then we identify ab,bc and ca. Also, what is the fundamental group ...
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Let $X$ be a topological space and let $x\in X$. Suppose that $\pi_1(X,x)$ is abelian. Now I know that this means given two closed paths $f,g\in\pi_1(X,x)$ (i.e. start/end at x), then $[f]*[g]=[g]*[... 2answers 56 views ### Homeomorphism of$\Delta^n$into itself that switches interior points Let$\Delta^n$be the standard$n$-symplex. Let$x, y$two interior points of$\Delta^n$. How can I prove that there exists an homeomorphism of$\Delta^n$into itself that maps$x$to$y$? I can see ... 0answers 106 views ### Homotopy classes of functions from a finite CW complex I am given the following problem: taken$X$finite CW complex and$Y$a space such that for every basepoint$y \in Y$the group$\pi_i(Y,y)$is finite$\forall i \leq \text{dim} X$then the set$[X, Y]...
I am working on cofibrations for an assignment and I am looking in the book Topology and geometry by Bredon and he states an inclusion map is a cofibration if and only if $A\times I \cup X\times\{0\}$ ...
### Is there an odd continuous map $f:S^{2}\to GL(2,\mathbb{C})$?
Motivated by this MO question and as an attempt to a possible generalization of the Borsuk Ulam type theorems we ask: Is there an odd continuous map $f:S^{2}\to GL(2,\mathbb{C})$ (Or \$GL(2n, \...