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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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Two spectra which are not homotopy equivalent but which represent the same reduced cohomology theory

I have to say that I'm very new to this subject and I'm trying to get the general idea of what we're talking about. Starting from the category of spectra, whose objects are sequences of spaces $\{X_i\}_{i\in\mathbb N}$ with structure maps $\Sigma…
Nikio
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The notion of mapping cylinder

This may be a weird question but I really want to know the answer: Hatcher p.2: Why is the name 'cylinder' used in this instance? I don't think that this quotient space, namely the mapping cylinder, is homeomorphic to a cylinder. At the beginning…
Xena
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universal cover of the sphere

what is the map $S^n \longrightarrow S^n$ that defines $S^n$ as the universal cover of $S^n$ ?
studento
  • 155
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Cohomology ring of $\mathbb R P^\infty$ with $\mathbb Z_{2k}$ coefficients

Let $k$ be a positive integer. I am trying to show that as rings, $H^*(\mathbb RP^\infty ; \mathbb Z_{2k}) \cong \mathbb Z_{2k}[a,b]/(2a , 2b , a^2 - kb)$. This is exercise 3.2.5 in Hatcher. The hint is to "Use the coefficient map $\mathbb Z_{2k}…
user15464
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torus by identifying two equivalent points (mod $\mathbb{Z^2}$)

How to visualize the quotient space $\mathbb{R^2}/ \mathbb{Z^2}$ to be a torus? you may also refer me to some books or websites. Because I want to see how the knot torus winds in this case. thank you!
Ronald
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Simultaneous CW Approximation

Given a topological space $X$, we know that there is a CW complex $Z$ with a map $Z\rightarrow X$ inducing an isomorphism on homotopy groups. If we are given two spaces $X_{1}$ and $X_{2}$ with isomorphic homotopy groups (but no continuous map…
bvtran
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Deck Transformations of Branched Covers

Let $p:\tilde{X}\rightarrow X$ be a covering. If this is an unbranched covering, deck transformations are determined by their action on one point. If this is a branched covering, is a deck transformation determined by its action on any point that…
Becca Winarski
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How does gluing work?

If one "glues" together a cylinder $C$ with a cylinder $C'$ the resulting space should be a torus as a subspace of $\mathbb R^3$. Both $C$ and $C'$ are $S^1 \times [0,1]$. If I understand it correctly the identification requires two maps $f: C \to…
goobie
  • 541
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Find $(M,\partial M)$ with injective $H_k(\partial M)\rightarrow H_k(M)$.

I am looking for a compact $2k+1$-dimensional manifold $M$ ($k\ge1$) with boundary $\partial M$, such that a) $H_k(\partial M;\mathbb{Q}) \neq 0$, b) $\iota_*\colon H_k(\partial M;\mathbb{Q})\rightarrow H_k( M;\mathbb{Q})$ is injective. (Where…
Jan Bohr
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Fundamental group calculations

I'm a student taking my first course in algebraic topology. I've stumbled across this exercise: calculate the fundamental group of $S^3-\gamma$, where $\gamma$ is a circumference in $\mathbb{R}^3$ (i.e. $\gamma=S^1$ in $\mathbb{R}^3$) and…
Francesca
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What is the universal cover of the plane minus two points?

I know that the universal cover of the plane minus the origin is the plane with the exponential map, but I can't think of the analogue with two points removed. I figured out what the universal cover of two wedged circles looks like (a sort of…
user363464
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Topological space with fundamental group $\mathbb{Z}/3\mathbb{Z}$.

In my course "Introduction To Algebraic Topology" I had following test problem: Exemplify a topological space with fundamental group $\mathbb{Z}/3\mathbb{Z}$. I was supposed to use this theorem: Let $Y$ be a simply connected topological space. If…
Gleb Chili
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Fundamental group of an orientable surface of infinite genus.

I am trying to calculate the fundamental group of an orientable surface $X$ of countably infinite genus. The $1$-skeleton $Y$ of $X$ is infinite wedge of circles, so its fundamental group is free group on countably infinite generators, but I am not…
kuhu
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Fixed-point-free map on a sphere minus a point

I want to know of a fixed-point-free map from a 2-sphere minus a point to itself. Please also mention why you thought of your example. My example: a sphere minus the north pole is the complex plane via stereographic projection, and consider a…
user345777
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If $X$ deformation retracts to $A$, does it follow that $(X,A)$ has the homotopy extension property?

Let $X$ be an arbitrary topological space and $A$ a subspace such that there exists a (strong) deformation retraction of $X$ to $A$. Does it follow that $(X,A)$ has the homotopy extension property? If not, what are some nice counter-examples? I…
Dejan Govc
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