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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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Deformation retraction: special case of homotopy equivalence?

Wikipedia says about deformation retraction that it "...is a special case of homotopy equivalence..." I fail to see how this is true. Say $A \subset X$ and $F$ a deformation retraction from $id_X$ to $id_A$. If $F$ was a homotopy equivalence we…
Rudy the Reindeer
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Nullhomotopic loop in a topological space $X$ lifts to a loop in a covering space $\widetilde{X}$.

Suppose $\Gamma : I=[0,1] \rightarrow X$ is a nullhomotopic in a topological space $X$. I want to show that any lift $\widetilde{\Gamma}:I \rightarrow \widetilde{X}$ in a covering space $\widetilde{X}$ must be a loop. I want to verify that my proof…
Tuo
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Two maps inducing the same homotopy groups

Let $X,Y$ be two CW-complexes and $f,g$ two continuous maps from $X$ to $Y$. Suppose that $f_*=g_*:\pi_q(X) \to \pi_q(Y)$ for any $q$, then can we prove that $f$ and $g$ are homotopic?
Totoro
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Free topological $G$-space and G-CW-Complex

Let $G$ be a finite group, and $X$ a free topological $G$-space which admits a CW-structure. Is there a CW-structure on $X$ that compatible with its $G$-action, i.e, a cell structure that turns $X$ to a G-CW-complex? Note, the n-sphere…
123...
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Compute $\pi_2(\mathbb{S}^2,X)$ where $X$ is the figure 8

We have the following short exact sequence from the long exact sequence for a pair $$0\to\pi_2(\mathbb{S^2})=\mathbb{Z}\to\pi_2(\mathbb{S}^2,X)\to\pi_1(X)=F_2\to0.$$ I wanted to construct a section (I guess there is one), so…
ah--
  • 545
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cohomology groups and the pontryagin construction

I'm just beginning to learn about cohomology groups, and have been told that at this point, it definitely behooves me to step away from the crutch of geometric intuition, but I'm going to try to lean on it for a little longer anyway. When learning…
gmoss
  • 922
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Nonhomotopic retractions $S^1\vee S^1\to S^1$ without Seifert-Van Kampen?

Let me give a more precise form of the exercise: Construct infinitely many nonhomotopic retractions $S^1\vee S^1\to S^1$. (Actually this is exercise 1.1.17 in Hatcher's Algebraic Topology) The family of retractions, say $r_n$, which are identity…
cjackal
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What would be the water-line of a Klein bottle afloat in the sea?

If you were on a ship at sea, and were a topologist of sufficient strength to be able to conjure up a Klein bottle on the spot and then toss it into the sea, what would be the water-line of the floating Klein bottle, after all disturbing water waves…
user584285
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Long Exact Sequence to compute $H_m(P^n)$.

I was trying to compute the homology groups of $P^n$ by induction using the long exact sequence of homology (Without using cellular homology, although some cellular reasoning can be used, if it can be justified in a simple way without cellular…
LeviathanTheEsper
  • 3,163
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A question on the deformation retract

I want to show that the subspace $A\cup B$ where $A=\{(x,y);(x-1)^{2}+y^{2}=1\}$ and $B=\{(x,y);(x+1)^{2}+y^{2}=1\}$ is a deformation retract of $X=\{(x,y);x^{2}+y^{2}\leq 4\}-\{(1,0),(-1,0)\}$. For this, I define the function $H:X\times I\to X$ as…
Aliakbar
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Is this space simply connected?

After thinking about it for a while, I believe that the following space is not simply connected, but don't know how to prove it (either way). Let $Q = [0, 1]^2$ be the unit square. The space is $$X = \{ (z_1, z_2) \in Q^2 ~:~ z_1 \neq z_2…
Lee
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Cellular inclusion is a cofibration

I have to show that if $X$ is a CW-complex, $A$ is a subcomplex and $i:A\hookrightarrow X$ is a cellular inclusion, then $i$ is a cofibration. My attempt is as follows. I think my proof, which first proves a lemma and is based on an induction…
Espen Nielsen
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Homology groups of $R^n $ \ a closed subset which is homeomorphic to $R^k$ for a $k$

How can i calculate the Homology groups of $R^n $ \ (a closed subset which is homeomorphic to $R^k$ for a $k$) ? (i mean $R^n $ with a closed subset which is homeomorphic to $R^k$ removed.
user470412
  • 989
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Cohomology Finite covering and Weyl group

Let $G$ a compact Lie group, $T$ a maximal torus in $G$ and $W=N(T)/T$ its Weyl group. Then we have a finite covering (why is a covering?) $ W \rightarrow G/T \rightarrow G/N(T) $ Has $G/N(T)$ a manifold structure? How can I prove (rigorously) that…
ArthurStuart
  • 4,932
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1 answer

Euler characteristic formula of a fibre bundle via a good cover

Let $p: E \rightarrow B$ a fiber fundle with fiber $F$. I'd like to prove that if $U$ is a good-cover of $B$ then the Euler characteristic, that I denote with $\chi$ is $\chi(E)= \sum_{p,q}\sum_{\alpha_{0}, \cdots, \alpha_{p}} (-1)^{p+q}\dim…
ArthurStuart
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