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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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Continuous map $f:S^1 \to S^1$ with more than $k-1$ fixed points such that induced homomorphism $f_*$ is multiplication with $k$.

For my Algebraic Topology class I have solved the following exercise: Let $x_0 \in S^1$ and let $f: S^1 \to S^1$ be a continuous map with $f(x_0)=x_0$. Suppose moreover that $f_*:\Pi_1(S^1,x_0) \to \Pi_1(S^1,x_0):[g] \mapsto k[g]$ for some natural…
mathstudent
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Homotopy groups of wedge sums

This is an exercise from May's Concise Course In Algebraic Topology Show for $n\geq 2$ that $\pi_n(X\vee Y)\cong \pi_n(X)\oplus\pi_n(Y)\oplus\pi_{n+1}(X\times Y,X\vee Y)$ I feel like I'm missing something obvious. From the relative long exact…
Leon Sot
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Calculate $[\mathbb{S}^n, \mathbb{RP}^{n-1}] \cong \pi_n(\mathbb{RP}^{n-1})/\pi_1(\mathbb{RP}^{n-1})$

I want to calculate $[\mathbb{S}^n, \mathbb{RP}^{n-1}]$. I know that there is a correspondence $[\mathbb{S}^n, \mathbb{RP}^{n-1}] \cong \pi_n(\mathbb{RP}^{n-1})/\pi_1(\mathbb{RP}^{n-1})$ where I mean the quotient by the $\pi_1(\mathbb{RP}^{n-1})$…
CNS709
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Show that if $p: E \to B$ is a covering map and $E$ is path connected while $B$ is simply connected then $p$ is a homeomorphism.

I'm aware of this and this answer but I still don't seem to understand whether my proof is correct. Please point me to any errors or jumps in reasoning. Pick $e_0 \in E$ and let $b_0 = p^{-1}(e_0)$. Since $E$ is path connected, $\pi_1(E, e_0)…
Amin Sammara
  • 67
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Fundamental Group of a finite set with discrete topology

let S be a finite set with say n elements. Give it the discrete topology, Now what can we say about its fundamental group? Atleast can we determine the fundamental group of a set with two elements? Thanks
Dinesh
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Hatcher algebraic Topology $\Delta$-complex

I am having trouble with the following question from Hatcher: Construct a $\Delta$-complex structure on $\mathbb{R}P^2$ as a quotient of a $\Delta$-complex structure on $S^n$ having vertices the two vectors of length 1 along each coordinate axis…
Susan
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Confusion about homology with local coefficients - where is my mistake?

In the Chapter Local Coefficients of his book Algebraic Topology Hatcher writes the following, where $X$ is a path-connected space having a universal cover $\tilde{X}$ and fundamental group $\pi$. For a $\mathbb{Z}[\pi]$-module $M$ let $\pi´$ be…
Frieder Jäckel
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Properties of hole on donut

R2 (i.e. the plane) is a covering map of a donut. R2 is simply connected so different elements in fundamental group of donut will have different lifting correspondence in R2. Now punch a hole on the donut surface and punch holes on the corresponding…
jw_
  • 509
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Algebraic Topology Artin

I am reading algebraic topology by Artin, and he gives the following definition. Let $X$ be a topological space. We define a homomorphism $sd_X: S_p(X) \to S_p(X)$ by induction. If $T: \Delta_0 \to X$ is a singular 0-simplex, we define $sd_XT =…
Susan
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Weak equivalence induces isomorphism on homology via Serre spectral sequence

Via Hurewicz it is easy to see that the homotopy fiber of a weak equivalence has trivial homology. Via the Serre spectral sequence we can then see that the domain and codomain of our weak equivalence have the same homology. Can we deduce from the…
Connor Malin
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$S^2$ and a point are not homotopy equivalent

Is there a way to show that $S^2$ does not have the same homotopy type of a point without using homology groups?
gregory
  • 81
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Determining Fundamental Group

I do not know where to begin on the following exercise. Help is appreciated. Suppose $A$ is a connected $CW$-complex with 4 zero cells $\{a_1, a_2, a_3, a_4\} \in A.$ Let $S^1$ be the unit circle and let $B = S^1\times A\times A/(x_1, x_2,…
Anna
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Show that if $Y$ is contractible, then the canonical projection $\pi: X\times Y\to X$ is a homotopy equivalence

Show that if $Y$ is contractible, then the canonical projection $\pi:X\times Y\to X$ is a homotopy equivalence.
Alex Med
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Can homology groups be used to find the location of holes in addition to their number?

Very new to this subject, apologies if this question is obvious or not. As the title says, we can compute the homology of a chain complex of abelian groups from Smith normal form of an integer matrix. Can we compute their location or does this…
AngusTheMan
  • 541
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Why does there exist a continuous map $f : X \to K(\pi_n(X), n)$ inducing the identity map $1_{\pi_n(X)}$?

Fix $n > 1$ and let's say I have a $(n-1)$-connected space $X$ (not necessarily a CW-Complex) and an Eilenberg-Maclane space $K(\pi_n(X), n)$, I want to show (assuming that this is true in general of course) that there exists a continuous map $f : X…
Perturbative
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