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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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Failure of application of excision theorem and Lefschetz duality

I was confused about the following deduction. Let $M$ be a compact manifold and $f: M \to \mathbb R$ be a Morse function. Let $\mathbb k$ be a fixed field. We have \begin{equation} H^*(M, f^{-1}((-\infty, t]); \mathbb k) = H^*(f^{-1}([t, \infty)),…
user72443
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Hatcher Exercise 3.2.3: Cup product on $H^1(\mathbb{R}P^n, \mathbb{Z}_2)$.

This is not a homework problem and I'm just doing it for fun. The problem statement is: Using the cup product structure, show there is no map $\mathbb{R}P^n \to \mathbb{R}P^m$ inducing a nontrivial map $H^1(\mathbb{R}P^m; \mathbb{Z}_2) \to…
nekodesu
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Relation between a universal cover, a fibre, and the fundamental group

As the title states, I'm looking the relationship between the three. From what I've seen from other question, if $X$ is a space, and $X^*$ a cover, then $X = X^*/\pi_1(X)$, but I'm not entirely sure how this relates to the fibres of the cover.
user497174
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Help me to understand this Fundamental Region of a Klein Bottle from M.Armstrong text

I'm a beginner in Algebraic topology and reading Armstrong as it has a nice algebraic flavor. The shaded areas are the fundamental regions in the groups of isometries of the plane. I understand how we get Torus and Sphere but unable to visualize…
MathCosmo
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Homotopy equivalence of mapping telescope of CW complex

On P.138 of Hatcher (P.147 of pdf), Hatcher claims that (paraphrase): Let $X$ be an infinite-dimensional CW complex. Let $X^n$ denote the $n$-skeleton of $X$, i.e. the cells with $n$ dimensions or below. Identify $X^n$ as a subset of $X^{n+1}$. Let…
Kenny Lau
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Lifting of a continuous map from a CW complex

I want to understand the following statement: A continuous map from a CW-complex lifts, up to homotopy, through a weak homotopy equivalence. I don't need a proof I just want to understand the content of the statement. So let $f:X\longrightarrow Y$…
palio
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Intuition for excision

I am studying Algebraic Topology from Hatcher. In Chapter 2 Hatcher has a section on Excision which states that for triples $(X,U,A)$ where $A\subset U\subset X$ and $A\subset X$ is closed and $U\subset X$ is open we have that $H_n(X,U)\cong…
Grobber
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Why homology is dual to cohomology? Action of dual to Steenrod algebra

This question seems to me quite basic, but I cannot find the answer. In order to define coaction of dual Steenrod algebra, we take action on cohomology (here all homology and cohomology are taken with coefficients in a field): $A\otimes…
Igor Sikora
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Relative homology of $D^2$ and $S^1$

I am going crazy showing something that is clearly wrong, but I can't see the error in my logic. I am calculating $H_1(D^2,S^1)$ Now the disc is contractible so $H_n(D^2)=0$ for all $n$. Also $H_n(S^1)=\mathbb{Z}$ for $n=0,1$ and is $0$ otherwise.…
Juan S
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Topology Of Suspension Of Real Projective Space

The suspension $SX$ of a topological space $X$ is defined as follows: $${\displaystyle S(X)=(X\times I)/\{(x_{1},0)\sim (x_{2},0){\mbox{ and }}(x_{1},1)\sim (x_{2},1){\mbox{ for all }}x_{1},x_{2}\in X\}}.$$ As an easy observation, the suspension of…
MathFun
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Homology of $I$-bundles over the Klein bottle.

Let $I=[0,1]$. I would like to compute the homology of $I$-bundles over the Klein bottle $K^2$. As far as I know there are three $I$-bundles over $K^2$: the trivial bundle $K^2\times I$ (I have no problem computing the homology of this one) and two…
Sak
  • 3,866
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Fixed point of a map of wedge product

Does every continuous function $f: \mathbb R P^2 \vee \mathbb R P^2 \to \mathbb R P^2 \vee \mathbb R P^2$ have a fixed point? I don't really have a good feeling as to whether or not this is true. My first thought was to try and get a map…
Ben Tighe
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Example of a quasi-isomorphism in $\operatorname{Top}$ which is not a homotopy equivalence

Can we have a continous map $f: X \longrightarrow Y$ such that $f$ induces an isomorphism on all homology groups i.e. $f_* : H_n(X) \longrightarrow H_n(Y)$ is an isomorphism of abelian groups for all $n \geq 0$ but $f$ itself is not a homotopy…
topology
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Proving that the empty glass is a deformation retract of the full glass

Let me write down the exercise described in the title formally. Let $X=E^2 \times I$ and $A=E^2 \times {0} \cup S^1 \times I$, where $E^2$ designates the ball of radius 1 in $\mathbb{R}^2$. I want to show that $X$ is a deformation retract of $A$.…
aadcg
  • 690
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Compacity in the homological definition of orientation.

For a manifold $M^n$, orientation is often defined as a globally consistent choice of local orientations ie. a choice of generators $\mu_x$ of $H_n(M,M-x;R)$ (this group is isomorphic to R by escision) such that every point $x \in M$ has a compact…
Vincent L.
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