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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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When is the $i$-th homology group of the $p$-skeleton of a complex isomorphic to the $i$-th homology group of that complex?

For what values of $i$ is it true that $H_i(K^{(p)})\simeq H_i(K)$? My guess is that this is true for $i>dim K$. Otherwise, we can use $n$-simplex for a counterexample.
Hashirama
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Hatcher Exercise 0.19 Attaching $2$-cells to $S^2$

Hatcher has the following exercise in chapter $0$: Show that the space obtained from $S^2$ by attaching $n$ 2 cells along any collection of $n$ circles in $S^2$ is homotopy equivalent to the wedge sum of $n + 1$ 2-spheres. So, looking at the case…
slowlylearning
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Definition of orientable as given in Hatcher's Algebraic Topology

A $\textbf{local orientation}$ of a manifold $M$ at a point $x$ is a choice of generator $\mu_x$ of the infinite cyclic group $H_n(M, M- \{x\} )$. For example, in the case of $M= \mathbb{R}^n$, $H_n(\mathbb{R}^n, \mathbb{R}^n - \{x \}) \cong…
Yuugi
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If $p:E\to X$ universal compact covering map then any continous $f:X\to S^1$ is homotopic to a constant.

Let $p:E\to X$ be a universal covering map. Suppose that $E$ is compact and $X$ is path connected. Show that any continous $f:X\to S^1$ is homotopic to a constant. Can you give me some hints?
Giulio
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Openness condition in Seifert-van Kampen Theorem

I'm starting to learn some algebraic topology now, and came across the "classical" version of the Seifert-van Kampen theorem, whose statement is given in Theorem $4.5.2,$ on page $69$ here. If $X = U_1 \cup U_2$ with $U_1, U_2$ open, and $U_1, U_2,…
Math Maniac
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Is $X$ a subset of $CX$?

In Spanier's, Algebraic Topology, he writes: "A topological pair $(X,A)$ consists of a topological space $X$ and a subspace $A \subset X$." In a question at the end of the section he asks a question about the topological pair $(CX,X)$, where $CX$…
user1058860
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Question: Corollary 2.25 in Hatcher's algebraic topology

click here to see I can prove easily in the case of $n$ is over zero but failed to prove when $n$ equals 0. I tried like below. Since $(X_a, x_a)$ is a good pair, a pair of disjoint union of $X_a$ and disjoint union of $x_a$ is also good. So one…
user301120
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Problem understanding the external cup product

The external cup product is defined to be the map $$ H^{k}( X ; R) \times H^{l}(Y ; R) \overset{\times}{\to} H^{k+l}(X \times Y; R)$$ where $a \times b = p_1^*(a) \smile p_2^*(b)$ where $p_1$ and $p_2$ are projections of $X \times Y$ onto $X$ and…
Yuugi
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How to prove "zigzag comb" is contractible (Hatcher ex. 0.6(b))

I am doing a self-study in algebraic topology and have a question about Hatcher's Ex.0.6(b) and contractibility on p18. There have been several other posts concerning why the "zigzag comb" (space Y) is not deformation retractable, and that part is…
PossumP
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Algebraic Topology by Rotman Exercise 6.8

So I've been working through some of the suggested exercises through Rotman and I have one problem that took longer than I expected. Most of the starred exercises seem to have a short quick proof except this one,I was wondering if someone could help…
Enigma
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Why is the subdivision an extension?

Bredon defines, in page 224, the "subdivision" operator by induction on the affine chains of a simplex as $$\Upsilon(\sigma)= \underline{\sigma}(\Upsilon\big(\partial \sigma)\big) \quad \text{if deg($\sigma$)>0}$$ and $$\Upsilon(\sigma)=\sigma…
Aloizio Macedo
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Description of normal subgroup when using Seifert van Kampen

I am doing an exercise where I am supposed to compute the fundamental group of $\mathbb{S}^1\times[0,1]$ using Van Kampen's theorem with the open cover $A=\mathbb{S}^1\times[0,3/4)$ and $B=\mathbb{S}^1\times(1/4,1]$. I know the answer is…
user194469
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Question about group of covering/deck transformations and lifting

Suppose $E$ is path-connected and $p:E\to B$ is a covering map. Then the group $G$ of covering transformations is the group of homeomorphisms $f:E\to E$ where $p\circ f=p$. The question is asking me to show that $f$ is uniquely determined by its…
grayQuant
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computing fundamental group of $S^1 \times S^1$

How to find $\pi_{1}(S^{1}\times S^{1})$ ? I know $\pi_1(S^1)$ but ho to do this ?
user289509
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How can I complete this proof of a algebraic topology theorem?

The theorem is: Let be f, g : X→Y two continuous maps between topogolical spaces and H : X$\times$[0, 1] a homotopy such that H(x,0)=f(x) and H(x,1)=g(x). Given $x_0\in X$ let be $y_0=f(x_0)$, $y_1=g(x_0)$ and $\tau(t)=H(x_0,t)$ a path in Y from…
Aarón David Arroyo Torres
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