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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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$H_q(X;\mathbb{Z})=0$ when X spherical complex with $H_q(X;F)=0$ for all $q>0$ and for all $F=\mathbb{Q}$ or $F=\mathbb{Z}/p\mathbb{Z}$

Suppose that X is a spherical complex with $H_q(X;F)=0$ for all $q>0$ where $F=\mathbb{Q}$ or $F=\mathbb{Z}/p\mathbb{Z}$, as $p$ runs over all primes. I want to prove $H_q(X;\mathbb{Z})=0$. I know Singular Homology Theory and some of Singular…
S. Ha
  • 393
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How to compute Euler characteristic from polygonal presentation?

How can I compute the Euler characteristic of a compact surface from its polygonal presentation $\langle S | W_1 , \ldots , W_k \rangle$? I guess that the number of edges is the number of different symbols in $S$, but how to get the number of…
Johannes Wandinger
  • 63
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If Y dominates X and Y is a CW complex, then X has the homotopy type of a CW complex

Let $f\colon X \to Y$ and $g \colon Y \to X$ be maps such that $g \circ f \simeq \mathrm{id}_X$, and suppose $Y$ ix a CW complex. Then show that $X$ has the homotopy type of a CW complex This is an exercise in May's Concise Course in Algebraic…
Carl
  • 3,548
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Is there a "standard" way to compute the fundamental group of the $CP^n$?

I know that $\pi_1(CP^n)=0$ here is a possible proof: Notation: for a CW complex, denote by $X^k$ the $k$-skeleton of $X$. I will show that $\pi_1(CP^n)$ is contained in $\pi_1(S^2)=0$. Let $f:S^1 \longrightarrow CP^n$ be a map. By cellular…
Anthony Conway
  • 1,114
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Question About Proof of Simple Connectedness of Sphere

The question is, does the proof from Hatcher given below still work if we take out the part about $f^{-1}(x)$ being compact? $\hskip 0.5in$
Dilitante
  • 299
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When does a homotopy in $\mathbb{C}$ extend to a homotopy in the Riemann sphere?

I have a polynomial $p(z) = z^n + a_{n-1} z^{n-1} + \ldots + a_0$, and I extend it continuously from $p : \mathbb{C} \to \mathbb{C}$ to a map from $S^2 \to S^2$. $(n > 0)$. In the plane, I have the homotopy $tp(z)$ between $p$ and the constant zero…
Elle Najt
  • 20,740
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2 answers

Universal cover of a torus "pillow"

I was thinking today, what is the universal cover of a torus with the "donut hole" shrunk to a point? I am certain it must include a sphere, but that can't be enough because of the point at the center of the manifold. Is it a wedge of two spheres…
Johnny Apple
  • 4,211
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Proof that homotopy equivalent maps induce equivalent homomorphisms on homology groups....

I am reading Hatcher, theorem 2.10. ( http://www.math.cornell.edu/~hatcher/AT/ATpage.html page 112 ) I mostly understand the proof, but am having trouble verifying that (except for the two), the terms with $i = j$ cancel. Why should the restriction…
Elle Najt
  • 20,740
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1 answer

Definition of a lift in algebraic topology

Definition: A lift of a map $f: X \rightarrow Y$ is a map $\widetilde{f}: X\rightarrow \widetilde{X}$ s.t. $\rho \widetilde{f} = f$. Question: What here is meant by the map $\rho$? In my text (Hatcher) it doesn't seem to be explicitly defined.
user125103
  • 523
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1 answer

Null-homotopy of a map to the circle

Let X be a path connected and locally path connected topological space. Suppose there exists a continuous $f: X \rightarrow S^1$ inducing the trivial map between the fundamental groups of X and $S^1$. How do I show that this implies the…
user18679
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3 answers

Two basic problems about fundamental group and paths

I have two problems that I don't know how to do )=. i) Let X path connected. Let $x_0 , x_1 \in X$. Show that $\pi _1 \left( {X,x_0 } \right)$ is abelian iff for every $\alpha, \beta$ paths from $x_0$ to $x_1$ , we have $\widehat\alpha =…
Daniel
  • 3,053
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2 answers

Constant loop homotopic to circle on the sphere

I want to show that the loop $(\cos 2 \pi s, \sin 2 \pi s,0) \in S^2$ for $s \in [0,1]$ is homotopic to the constant loop with base point $(1,0,0)$. I can contract the loop $(\cos 2 \pi s, \sin 2 \pi s,0)$ to the constant loop using…
MovingUp
  • 41
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Singular cohomology with compact support

If $X$ is a locally compact Hausdorff space, then for any $n \geq0$ is $H_c^n(X) \cong {\tilde H^n}({X^ + })$? ($H_c^n(X)$ is the Singular cohomology with compact support and $X^+$ is the one-point compactification of $X$)
Summer
  • 6,893
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The homology group for $\mathbb{R}^n$ minus two points

This question is somehow related to the generalized Jordan curve theorem. I have already showed that if $h:S^k\to S^n$ is an embedding ($0\leq k
Kaa1el
  • 2,058
3
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1 answer

Hatcher Exercise (4.1.20)

I'm having a little trouble doing exercise $4.1.20$ at page $359$ of Hatcher. It states: Show that $[X,Y]$ is finite if $X$ is a finite connected CW complex and $\pi_i(Y)$ is finite for $i \leq \text{dim }X$", where by $[X,Y]$ we mean the…
arando
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