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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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Excision and induced fibration

I've been stuck in the following small detail which is part of the calculation of the $E^2$ term of the Serre Spectral Sequence. Let $p: E\to B$ be a fibration where $B$ is a CW-complex. Denote by $E^p=p^{-1}(B^p)$ where $B^p$ is the p-skeleton of…
Manuel
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The "geometry" behind relative homology

On p.33 of Sergei Matveev's "Lectures on Algebraic Topology", the relative chain group of a pair $(K,L)$, where $L \subset K$ is a subcomplex, is defined as the free abelian group on simplices with interiors in $K \setminus L$. How is this different…
Tony
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Descriptive explanation of the term "homotopic"

Can someone explain me in descriptive words or maybe with an image, what homotopic actually means and what its relevance is? Thank you for your time, Chris
Chris
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Further Generalization of Jordan Curve Theorem

Recently I have read the proof of the Jordan Curve Theorem in Munkres' Topology, I wonder whether there are some generalizations and corollaries on this theorem as follows: I know any simple closed curve separate $\mathbb R^2$ into two components,…
Y.H. Chan
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$X$ is contractible if and only if it is a retract of any cone over $X$

The first exercise given in Spanier's, Algebraic Topology is: $X$ is contractible if and only if it is a retract of any cone over $X$. I have proven the first implication, however I am stuck on the second implication. In Spanier he defines a cone…
user1058860
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Why does "dimension" of the generator matter for cohomology rings?

For $\mathbb{R} P^{n}$ we have, $$ H^*( \mathbb{R} P^2 ; \mathbb{Z}_2) \cong \mathbb{Z}_2 [ \alpha ] / (\alpha^3)$$ where $|\alpha|=1$. Let us assume there exists $X$ such that, $$ H^*( X ; \mathbb{Z}_2) \cong \mathbb{Z}_2 [ \beta ] / (\beta^3)…
Yuugi
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A question about the proof $\pi_1(S^1,1) \cong \mathbb{Z}$

I am working through the proof that the fundamental group of $S^1$ is isomorphic to $\mathbb{Z}$ from the book Basic Topology by Armstrong. There they are defining a map $\pi: \mathbb{R} \to S^1$ by $x \mapsto e^{2\pi i x}$. What bothers me is the…
Sayantan
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Of what is the Hopf map the boundary?

Consider a generator $x$ of the singular homology group $H_3(S^3)$. I think of this (perhaps wrongly?) as something like the identity on $S^3$, cut up into simplices. Now we have the Hopf fibration $\eta: S^3 \to S^2$, which gives us $\eta_*x \in…
Mees de Vries
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Exercise V.4.1. in Massey

This is about exercise V.4.1. in Massey's A Basic Course in Algebraic Topology. Section V.4. is about the fundamental group of a covering space. Massey assumes that all spaces involved are both arc-connected and locally arc-connected. Theorem 4.1.…
Lennart
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Removing zero section of tautological line bundle

I'm reading through a computation of the Chern number $c_1$ of the complex tautological line bundle $$ \tau = \{(l, p) \in \mathbb{CP}^1 \times \mathbb C^2 \mid p \in l\}\\ \downarrow \\ \mathbb{CP}^1. $$ One of the first steps is to notice that…
William Stagner
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Calculation kernel of boundary operator. (Easy example in Hatcher)

Consider the graph as given in the third page (pg.99) in the link below. https://www.math.cornell.edu/~hatcher/AT/ATch2.pdf We define a homomorphism $\partial: C_1 \to C_0$ by sending each basis element $a,b,c,d$ to $y-x$, the vertex at the head of…
Yuugi
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A quotient space homeomorphic to $\mathbb{S}^{m-1}\times\mathbb{S}^1$.

This is the last question on a homework set from my Introduction to Algebraic Topology class. Consider the action of $\mathbb{Z}$ on $\mathbb{R}^m\setminus\{0\}$ given by $n.x=2^nx$. We have to show that the quotient space…
B. Pasternak
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Diagram in Hatcher's Algebraic Topology - what do the arrows mean?

Here is a picture from Chapter 2 in Hatcher's Algebraic Topology: The context is that the space on the left does not have the structure of a $\Delta$-complex, whereas the shape on the right does. On the left he says that we identify the sides in…
Tim
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$f:X \to Y$ is a homotopy equivalence if there exists $g,h : Y \to X$ with $fg$ and $hf$ homotopy equivalences

Let $X$ and $Y$ be topological spaces, and let $f: X \to Y$. I'd like to show that if there are maps $g,h : Y \to X$ such that $fg$ and $hf$ are homotopy equivalences, then $f$ is a homotopy equivalences. $fg$ is a homotopy equivalence means there…
Matt
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fundamental group of two circles which intersect at two common points

I am trying to solve this by Van-Kampen's theorem. What I do is just move a point at left side and a point at right side. Then I get two open sets whose intersection deformation retracts to a circle. Each open set deformation retracts to a figure 8.…
user198206
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