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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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Showing two topological spaces are homotopy equivalent

Could someone explain, like just a geometric description, how the space $\mathbb{R}^3\backslash A$, where $A$ is the unit circle in the $xy$-plane, is homotopy equivalent to $S^2\vee S^1$, the one point union of a 2-sphere and a circle. I can't see…
908979
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CW structure induced by a group action

Let $X$ be a CW-complex on which a group $G$ acts. How does the CW-complex structure of $X/G$ relates to that of $X$?
palio
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Covering of Graph

Let $X$ be graph and $p:E\rightarrow X $ be covering map. It is followed by Theorem $83.4$, $E$ is graph. Now, assume that $v\in V(X)$ is a vertex with $deg(v)<\infty$ and $w\in p^{-1}(v)$. Can we conclude that $deg(v)=deg(w)$? I've tried…
user108209
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Reduced suspension and unreduced suspension

In May's "A concise course in Algebraic Topology" Chap 14 section 1, the author says $\Sigma (X_+)$ is $\Sigma X\vee S^1$ where $X$ is an unbased space and $X_+$ is the union of a disjoint basepoint and $X$ and $\vee$ is the wedge sum. Obviously,…
HBS
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Why is $(X\times EG)/G\to X/G$ a fibration if $G$ acts freely on $X$?

Suppose that $G$ acts freely on $X$, and let $EG$ be a contractible space on which $G$ acts freely. According to many references, the projection $(X\times EG)/G\to X/G$ is a fibration. However, I can't find a proof for this, nor construct one…
Bashar Saleh
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The fundamental group of the product of a 3-sphere and circle

I know that a torus the product of circles. But what the fundamental group of the product of a 3-sphere and a circle? ie $\pi_1((S^3 \times S^1), (1,1))$? Thanks!
Monafeir
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Proof of an algebraic topological lemma

I have been given the following result without proof, so I would like to show it is true: Let $I=[0,1]$, then: $$H^\bullet(I,\partial I;R)\cong H^\bullet(I/\partial I,*;R)\cong H^\bullet(S^1,*;R)=R\cdot x$$ where $x$ is a generator of degree $1$. So…
Daniel Robert-Nicoud
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Relative homology of disk and any of its subspace is isomorphic to reduced homology of the subspace?

Consider the pair of topological space $(\mathbb{D}^n,X)$, where $X \subset \mathbb{D}^n$ is a subspace. We know that there is a long exact sequence of reduced homology groups, $$\cdots \to \tilde{H}_q(X) \to \tilde{H}_q(\mathbb{D}^n) \to…
Jean Valjean
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Two sheeted disconnected cover of a connected topological space has exactly two components

Let $\tilde{X}$ be a two-sheeted cover of a connected topological space X. If $\tilde{X}$ is disconnected then this has exactly two components. Further each component is homeomorphic to X by the covering projection. Can anyone give the argument for…
user93620
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Is the boundary of a triangulated manifold a subtriangulation?

Let $K$ be a finite simplicial complex with the underlying topological space $|K|=\cup K$. If $|K|$ is also a topological manifold with boundary, does it hold that some subcomplex of $K$ triangulates exactly the boundary $\partial |K|$? Intuitively,…
Peter Franek
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2 answers

Covering through group action and corresponding deck transformations

I'm having a bit of trouble with the following exercise: Let $G$ be a group acting properly discontinuous and continuous on a topological space $E$. Then $p:E\to G\backslash E$ is a covering. Let $N_G(H)$ be the normalizer of $H$ in $G$. Show…
blst
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lifts of continuous map to covering space

The following problem gives me a bit of trouble: Let $p:E\to X$ be a covering map. Let $g_1,g_2$ be two lifts of the continuous map $f:Y\to X$. Show that $T:=\{y\in Y:g_1(y)=g_2(y)\}\subseteq Y$ is clopen. Also show that for $Y$ connected, $g_1$…
blst
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Show that $R^3 - \{x,y\}$ is homotopy equivalent to $S^2\vee S^2$

Let $A$ be two distinct points in $R^3$. How would I go about showing that $R^3\backslash A$ is homotopy equivalent to the one-point union $S^2\vee S^2$? Any help appreciated
al1023
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Dimension of restriction of surjective linear map

I'm trying to understand the proof of Theorem 4.23 (Case 1) in Allen Hatcher's Algebraic Topology. We have a map $f$, for which $f^{-1}(\Delta ^{n+1})$ is a finite union of convex polyhedra, on each of which $f$ is the restriction of a linear…
Kristoffer
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Basic question on cohomology of pairs

This is a screenshot from Hatcher's Algebraic Topology. I can't understand the last sentence. By definition $C_n(X)$ is the free $\mathbb Z$-module generated by all n-simplices. But, why should every singular map go to exactly exactly $A$ or $X-A$…
Dignaga
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