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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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Induced map on homology by $f\colon S^4 \to S^2 \times S^2$

Show that $$f_* \colon H_4(S^4) \to H_4(S^2 \times S^2)$$ is the zero map for any $f\colon S^4 \to S^2 \times S^2$. We are working with integral coefficients. I tried applying the naturality of Künneth Theorem, obtaining the following commutative…
Luigi M
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Covering a connected sum

I have the following problem as a part of my homework: Let $S$ be a closed surface (compact and connected). Show that for every $k$ exists a covering map of $k$ folds $p_k:S_k \rightarrow \mathbb{T}\sharp S$, where $\mathbb{T}$ is the torus and…
rf1x
  • 91
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how to compute the de Rham cohomology with compact support of a mobius strip

I am having problem computing the de Rham cohomology with compact support of an open mobius strip,it's aquestion from Bott's book, and Bott said its cohomology is identically zero which can be obtained from MV sequence, I want some suggestions on…
gordon
  • 31
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The fundamental group of some wedge sum

I was wondering how one can compute the fundamental group of the wedge sum of a sphere and 2 circles , i know the fundamental group is Z*Z ,and that the fundamental group of a wedge sum is the free product of each fundamental group but is it…
Butterfly
  • 1,443
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A question about the degree of a map

Consider $f,g: S^{n}\rightarrow S^{n}$ to be continuous maps. Seeing $S^{n}\subset\mathbb{R}^{n+1}$, let's say that $f$ and $g$ are orthogonal at $x\in S^{n}$ whenever $\langle f(x),g(x)\rangle = 0$ for the usual inner product in…
Stanley456
  • 83
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Is a covering space of a topological group a fiber bundle with the structure group the fundamental group of the topological space?

Question. Is a covering space of a topological group a fiber bundle with the structure group the fundamental group of the topological space? Let $p:E\rightarrow X$ be a covering space of X. I know how $\pi_1(X,x_0)$ acts on $p^{-1}(x_0)$. And if…
Babai
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Poincare-Lefschetz duality, universal coefficients, and middle cohomology

Sorry if the question is too simple, algebraic topology is not my strong suit. Let $(M,\partial M)$ be a $2n$-dimensional manifold with boundary, with one-dimensional middle cohomology. By Poincare-Lefschetz duality and universal coefficients…
Philip Engel
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Van Kampen theorem?

So I'm trying to use Van Kampen theorem to prove that a space is null-homotopic. The thing is I got it down to this $\langle a\mid a=1\rangle$, however I'm confused what does this mean. For calculating the the torus you get it down to this $\langle…
thisisnotcool
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Hatcher 3.1.4 What happens if one defines homology groups of the chain complex?

What happens if one defines homology groups $h_n(X,G)$ of the chain complex $\cdots \rightarrow Hom(G,C_n(X)) \rightarrow Hom(G,C_{n-1}(X))\rightarrow \cdots $ ? More specifically, what are the groups $h_n(X,G)$ when $G=\Bbb Z, \Bbb Z_m, \Bbb…
Ri-Li
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Maps between real projective spaces

Say we have a linear map, $f: \mathbb{R}^{m+1} \to \mathbb{R}^{n+1}$, and we define $\mathbb{RP}^{n}$ as $(\mathbb{R}^{n+1} - \{0\})/{\sim}$ with $\sim$ define by $x \sim y$ if $y = \lambda x$ for some $\lambda \neq 0 \in \mathbb{R}$. Then, if we…
Mary
  • 171
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Induced map of homology groups of torus

Let $f$ be a map from $X= S^1 \times S^1$ to itself that is the identity on one factor and a reflection on the other. Then how does the induced map $f_* :H_2(X)\to H_2(X)$ look like? Since $f$ is just a reflection, I guess $f_* = -\mathbb{1}$, but…
JWL
  • 706
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References for sheaf homology

Sheaf cohomology is a well-studied topic with a lot of references available. For example Hartshorne's book. But for a certain paper I am reading now, I have to understand sheaf homology. Could someone tell me some references? This is just sheaf…
Edmond Grigor
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2 answers

Question about Betti's number

In the subject "Algebraic Topology" we define the Betti's number as the greater number $\beta_p$ such that a family $\{z^i_p\}_{i=1}^{\beta_p}$ of $p-$cicles are linearly independent (i.e. there's no exists a family $\{\lambda_i\}_i\subset…
Heracles
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Homology groups of $\mathbb{R}^3\setminus \ S^1$ and $\mathbb{R}^3\setminus E^1_+$

Here is a problem and my attempt at the solution. If my conclusion or the proof is incorrect I would appreciate a pointer in the right direction. Thanks in advance. Let $S^1$ be the unit circle in the xy plane in $\mathbb{R^3}$ and let $E^1_+$ and…
user198182
  • 625
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Singular Homology- Question regarding a theorem

I'm currently trying to learn Singular Homology from Munkres' book- "Elements of Algebraic Topology" . On page 173, theorem 30.7 appears: " If $f,g:(X,A) \to (Y,B) $ are homotopic, then $f _ * = g_* $ " The author finishes the proof by showing…
joshua
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