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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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What is Cech homology?

I got into a discussion today about how, just as with all the other ways of computing singular homology, there should be an internal sort of integration pairing for Cech cohomology with "Cech homology", whatever it is. My sense is that probably a…
Aaron Mazel-Gee
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Presentation of the fundamental group of a manifold minus some points

I recently noticed a few things in some recent questions on MO: 1) the fundamental group of $S^2$ minus, say, 4 points, is $\langle a,b,c,d\ |\ abcd=1\rangle$. 2) The fundamental group of a torus minus a point is $\langle a,b,c\ |\…
user641
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Homology of real projective space... I'm not satisfied with the argument in hatcher.

In example 2.42 Hatcher computes the homology of real projective space. I follow his argument, but I would be uncomfortable believing the details of the degree computation if I didn't see it in his text. How does Hatcher conclude that the degree of…
Elle Najt
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Showing that a zigzag space is contractible

I'm trying to solve part (b) of exercise 0.6 in Hatcher's Algebraic Topology: (b) Let $Y$ be the subspace of $\mathbb{R}^2$ that is the union of an infinite number of copies of $X$ arranged as in the figure below. Show that $Y$ is contractible but…
PeterM
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Non-orientable 3-manifold has infinite fundamental group

I'm doing past papers for a first course in algebraic topology. The question is: Let $M$ be a 3-dimensional, closed, connected, non-orientable manifold. Show that $M$ has infinite fundamental group. Is there any way of answering this question…
Earthliŋ
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Fundamental group is abelian iff the fundamental group isomorphisms (a-hat) coincide

I want to show that if $X$ is a non-empty path connected space, then the fundamental group is abelian if and only if given any points $y, z\in X$ and paths $\alpha, \beta$ from $y$ to $z$, $\hat{\alpha} = \hat{\beta}$, where $(\hat{\alpha}([f]) =…
thobanster
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2 answers

Why isn't $\mathbb{CP}^2$ a covering space for any other manifold?

This is one of those perhaps rare occasions when someone takes the advice of the FAQ and asks a question to which they already know the answer. This puzzle took me a while, but I found it both simple and satisfying. It's also great because the…
Aaron Mazel-Gee
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Homology groups of torus

I computed the homology groups of the torus, can someone tell me if this is correct? The calculation, not the result that is. Thanks! The cells of $T^2$ are $e^0, e^1_a, e^1_b, e^2$ The chain groups are $$ C_0(T^2) = \{ k e^0 | k \in \mathbb{Z} \} =…
Rudy the Reindeer
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2 answers

Fundamental group as a functor

Is it right to consider assigning a fundamental group to a topological space the same as having a functor from $\mathbf{Top}$ to $\mathbf{Grp}$ ? Are there any other examples of such functors ?
JeanThiviers
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composition of covering maps

The origin of my question arose from a problem: Let $q: X \to Y$ and $r: Y \to Z$ be covering maps, let $p= r \circ q$. Show that if $r^{-1}(z)$ is finite for each $z \in Z$, then $p$ is a covering map. I am just wondering is there an easy counter…
Harry
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Showing that every finitely presented group has a $4$-manifold with it as its fundamental group

Wikipedia: For any finitely presented group it is easy to construct a (smooth) compact 4-manifold with it as its fundamental group. Question: How do we do this? EDIT: Below is a proof sketch found elsewhere with some targeted questions of my own…
user1770201
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A homotopy sphere

My question is part of an exercise in Hatcher's 'Algebraic Topology'. Consider a CW complex $X$, constructed from a circle and two 2-disks $e_2$ and $e_3$, attached to that circle by maps of degree 2 and 3, respectively. Can someone show that $X$ is…
Alexander Shamov
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Classification of covering spaces of $\Bbb{R}\textrm{P}^2 \vee \Bbb{R}\textrm{P}^2$.

I have spent some days now trying to understand how to classify all covering spaces of $X = \Bbb{R}\textrm{P}^2 \vee \Bbb{R}\textrm{P}^2$, and I think it boils down for me to just understanding how the fundamental group $\pi_1(\Bbb{R}P^2 \vee…
user38268
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3 answers

Why does every 7-manifold bound an 8-manifold?

I'm glancing over Milnor's paper on exotic 7-spheres, and one of the first few lines says, `every closed 7-manifold $M^7$ is the boundary of an 8-manifold $B^8$. Here's what I don't understand: The unoriented cobordism ring is isomorphic to a…
Dylan Wilson
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Confusion about covering projection

This is quite a detailed question: I'm struggling to understand a few parts of a proof of the following Lemma. I've placed stars ($\bigstar$) where I'd like to draw your attention. Lemma: Let $P:Y\to X $ be a covering projection and $ f: Z \to X $…
Martin P
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