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Questions tagged [algebraic-topology]

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

Algebraic topology is a mathematical subject that associates algebraic objects to topological spaces which are invariant under homeomorphism or homotopy equivalence. Often, these associations are functorial so that continuous maps induce morphisms between appropriate algebraic objects.

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21356 questions
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Homeomorphisms between closed surfaces and induced maps on fundamental groups

I am currently working on a problem and I have solved all the differents points except for one, that I have reduced to the following exercise which I could not solve: Show that two homeomorphisms between closed surfaces (in my case, between tori)…
Skalf
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Question on suspension of $n$-sphere.

Definition $:$ For a given topological space $X$ we define the cone of $X$ denoted by $C(X)$ as follows $:$ $$C(X) : = (X \times I)/ (X \times 0)$$ where $I = [0,1]$ and we define the suspension of $X$ denoted by $S(X)$ as follows $:$ $$S(X) : =…
Anil Bagchi.
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induced isomorphisms from Gysin sequence

Consider the path fibration: $K(\mathbb Z,2r-1)\rightarrow PK(\mathbb Z,2r)\rightarrow K(\mathbb Z,2r).$ Suppose that $H^*(K(\mathbb Z,2r-1);\mathbb Q)=H^*(S^{2r-1};\mathbb Q).$ We want to show that $H^*(K(\mathbb Z,2r);\mathbb Q)=\mathbb…
palio
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a counterexample of covering projection

Let $X=S^1\times S^1\times\cdots$ be a countable product of 1-spheres and for $n\geq 1$ let $\tilde{X_n}=R^n\times S^1\times S^1\times\cdots$.Define $p_n:\tilde{X_n}\rightarrow X$ by…
mathon
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How to show this function defined using a homotopy is continuous?

Suppose there exists a homotopy $H$ between continuous functions $f,g:X\longrightarrow Y$, where $X,Y$ are non-empty topological spaces. Consider the function $h:X\times [-1,1]\times [0,1]\longrightarrow Y\times[-1,1]$ defined by…
tashakinns
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thom space associated to a fibration

let $p:S\rightarrow B$ be a fibration wiht fiber a rational $r$-homology sphere $\Sigma^r$, i.e., $H_*(\Sigma^r;\mathbb Q)=H_*(S^r;\mathbb Q)$. to such a fibration we associate its Thom space $$MS=(S\times I \cup_p B)/S\times\{1\}$$ where we…
palio
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Deformation retract of $X_0\sqcup_F(X_1\times I)$ onto $X_0\sqcup_fX_1$

If $(X_1,A)$ is a CW pair and we have attaching maps $f,g:A\to X_0$ that are homotopic, then $X_0\sqcup_f X_1\simeq X_0\sqcup_g X_1\ \text{rel}\ X_0$. In the proof of this statement, they claim that if $F:A\times I\to X_0$ is a homotopy from $f$…
one potato two potato
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Universal bundle and induced bundle.

Let be $G$ a compact Lie group and $\pi: EG \rightarrow BG$ the associated universal bundle. If $X$ is a compact Hausdorff space there is a one-to-one correspondence between the equivalence classes of principal $G-$bundles on $X$ and the homotopy…
ArthurStuart
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The quotient map $q: S^1 \times S^1 \to S^2$ induces an isomorphism in $H_2$

I know this question has been asked several times, but I'm trying to solve it with a different approach that I didn't find online. I'm just wondering if my approach would work. We look at the quotient map $q: S^1 \times S^1 \to S^2$ by $S^2 \cong…
M. Wang
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A question about book by John Lee

The author first proved the following important theorem - "Unique Lifting Property of the Circle" Then, the author proved the following corollary. My question is: if corollary 8.6's assumption underlined by a red line is wrong. When two lifts…
Hamilton
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Cohomology $H^ {i}(\mathbb{R}P^n, \mathbb{Z}_2)$

I have to compute $H^{i}(\mathbb{R}P^2,\mathbb{Z}_2)$. I know that is $\mathbb{Z}_2$ for $i=0,1,2$ but I'm looking for a proof without universal coefficient theorem. Have you some ideas?
ArthurStuart
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Is there always a homotopy taking a point to another in a connected manifold?

Let $M$ be a connected topological $n$-manifold. Note this implies $M$ is path-connected. Let $a,b \in M$. Must there always exist a continuous $H:M \times [0,1] \to M$ such that $H(m,0)=m$ for all $m \in M$, and $H(a,1)=b$? We can note that this is…
MathematicsStudent1122
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Trivial cup product in a (cohomology) subring

I was going through a proof that $S^n\times S^n$ and $S^n\vee S^n \vee S^{2n}$ are not homotopy equivalent spaces. Both spaces have the same cohomology groups in degree $n$ and $2n$, namely $$H^k(\ \cdot\ ;\Bbb{Z}) = \begin{cases} \mathbb{Z}\oplus…
Zest
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$H_2(\partial V)=0$ if $V$ is a rational homology 4-ball

I’m reading “On knots” by L. Kauffman. In Chapter XVII (which follow the prominent paper of Casson and Gordon,) we have the following lemma. Lemma 17.3. Let $V$ be a $\mathbb{Q}$-homology 4-ball. If the image of $H_1(\partial V)\to H_1(V)$ has order…
NothingInSense
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number of simplices in barycentric subdivision

Let $K$ be a simplicial complex. Is there a way to calculate the number of k-simlices in the barycentric subdivision $K'$ of $K$? Given the number of $l$-simplices in $K$, for any $l$, of course. (I am suposed to show that the Euler characteristic…
Tori
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