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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.

Please use more specific tags like , ,, or whenever appropriate.

21356 questions
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What is the relationship between $\pi_{2}\overline{X}$ and $\pi_{2}(X)$?

From Harvard qualification exam, 1990. Consider the space $X=\mathbb{S}^{1}\wedge \mathbb{S}^{2}$, alternatively viewing it as a sphere with north and south poles connected. I was asked to: 1): the relationship between $\pi_{2}X$ and…
Bombyx mori
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Relation between path-homotopy classes and path-components

Let us have a topological space $X$. What is the relation between path-homotopy classes and path-components? For example, can we somehow define a map from path-components to path-homotopy classes?
Ninja
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Free groups on simply connected space : application of Van-kampen theorem

This comes from the application of Van-Kampen theorem. Note Van-kampen theorem, states for $U$, $V$ and $U \cap V$, are open and path connected space we have \begin{align} \pi_1 (U \cup V) =\pi_1 (U) *_{\pi_1(U\cap V)} \pi_1(V) \end{align} Here…
phy_math
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CW complex structure on a disk with 2 smaller disks removed

For some reason, I'm having trouble visualizing how to put a CW structure on a disk with 2 smaller disks removed. What I'd like to do is have three 0-cells, five 1-cells, and a single 2-cell. Three of the one cells would be glued to the three…
John
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Show that if $\widetilde{H}^n(X;\mathbb{Q})$ and $\widetilde{H}^n(X;\mathbb{Z_p})$ are zero, then $\widetilde{H}_n(X;\mathbb{Z})$ is zero.

Show that if $\widetilde{H}^n(X;\mathbb{Q})$ and $\widetilde{H}^n(X;\mathbb{Z_p})$ are zero for all $n$ and all primes $p$, then $\widetilde{H}_n(X;\mathbb{Z})$ is zero for all $ n $. My Try: So I have the following corollary and wanted to use it…
Extremal
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Borsuk theorem using degree theory

watching an online lecture I found the following short proof of the Borsuk theorem using degree theory. Assuming the following statement: Let $M^k,N^k$ be two smooth $k$-dimensional manifolds with boundary and $f:M \rightarrow N$ a continuous map,…
Near
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Show that $\deg f$ is even when $n$ is odd.

A map $f:S^n\rightarrow S^n$ satisfying $f(x)=f(-x)$ for all $x\in S^n$ is said to be an even map. Show that if $f:S^n\rightarrow S^n$ is an even map, then $\deg f=0$ when $n$ is even and $\deg f$ is even when $n$ is odd. Moreover show that when…
Extremal
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two non-diffeomorphic manifolds with the same cohomoly classes.

Question: Calculate the de Rham cohomology groups of: $U=\mathbb{R}^3 - (L \cup C)$ and $V= \mathbb{R}^3 - (L' \cup C)$, where $L' = \{x = 2, y = 0\}$, $L = \{x = y = 0\}$ and $C=\{ x^2 + y^2 = 1, z=0\}$. Then conclude that they are not…
Jude
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Orientation on the boundary of a manifold

Let $M$ be a manifold with boundary. Hatcher writes that a compact manifold with boundary is $R$-orientable if $M - \partial M$ is $R$-orientable. That is there exists a function $x \to \mu_x \in H_n(M \vert x)$ that satisfies the local consistency…
user7090
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Homotopy groups of compact surfaces

I want to calculate the higher homotopy groups of $\Sigma_g$ and $\mathbb{R}P^2\# \mathbb{R}P^2\#\cdots\# \mathbb{R}P^2$. But I haven't found the methods to calculate the homotopy groups of connected sum. So how to solve this question?
346699
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Whitehead product $[\alpha, \beta]=0$ iff there exists an extension.

I am having a lot of difficulty understanding the Whitehead product so I think proving an "obvious" result would help with my understanding. However, I am still completely at a loss as how to show that $[\alpha, \beta]=0$ if and only if there exists…
user7090
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Hatcher 2.1 Problem 17 (a)

I'm only interested in the case where $A$ is a single point and $X=S^2$ for the following question from Hatcher's book on Algebraic Topology. Question: Compute the homology of groups $H_n(X,A)$ when $X$ is $S^2$ or $T^2$ and $A$ is a finite set of…
Bob
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Homology of $\Sigma_{2}\times S^{1}$?

I'm quite at a loss with this...I want to use Mayer-Vietoris with open covers $A=\Sigma_{2}\times (S^{1}\setminus \{p\})$ and $B=\Sigma_{2}\times (S^{1}\setminus \{q\})$ so that $A$ and $B$ both deformation retract to $\Sigma_{2}$ and $A\cap B$…
subfallen
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Questions about complex bordism.

I have some questionss about the construction of the complex bordism ring MU and would appreciate every answer: I have read that the multiplication in MU is given by the tensor product of vector bundles $BU(n) \times BU(m) \rightarrow BU(n+m)$. How…
nick
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$H_c^n(X\times \Bbb R;G)$ is isomorphic to $H_c^{n-1}(X;G)$

I am trying to $H_c^n(X\times \Bbb R;G)$ is isomorphic to $H_c^{n-1}(X;G)$ for all $n$. Here $H_c^n(X;G)$ is just $n$-th cohomology with compact support. At first I tried to find a compact supported version for Kunneth formula, but quickly I give up…
user198206
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