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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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Prove that the product of a sphere and a torus is parallelizable

The product of sphere and torus, $S^2\times \mathbb T^2$, is parallelizable. How to prove this?
henry
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Retract and Homotopy extension property

See picture below the following picture. According to Hatcher, homotopy extension property implies that for a pair $(X,A)$ where $A$ is a subspace of $X$, $X\times I$ should retract to $X\times\{0\}\cup A\times I$ . My question is whether the…
nonlinearism
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Is there an orientable closed compact $3$-manifold such that its fundamntal group is $\mathbb{Z}$?

Is there an orientable closed compact $3$-manifold such that its fundamental group is $\mathbb{Z}$? How about $\mathbb{Z^2}$?
topology master
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No Embedding From A Torus to A Sphere

I have been trying to prove that there is no embedding from a torus to $S^2$ but to no avail. I am completely stuck on where to start. The proof is supposed to be based on Homology theory. I know how to prove that $S^n$ cannot be embedded in…
Matthew
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What needs to be done to prove that a vector bundle is trivial?

What needs to be done to prove that a vector bundle is trivial? equivalently, This can be thought of as proving that the vector bundle satisfies the criteria of being trivial, then what is this criteria?
melda
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contractible and simply connected

Every contractible space X is simply connected because X is homotopy equivalent to a point. Is there a direct proof of this fact? There obviously is a (free) homotopy between any loop and the trivial loop at the base point. But how to construct a…
Michael C
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$S^3\times \Bbb CP^\infty$ is not homotopy equivalent to $\left(S^1\times \Bbb CP^\infty\right)\big/\left(S^1\times \{x_0\}\right)$

Both $S^3\times \Bbb CP^\infty$ and $\left(S^1\times \Bbb CP^\infty\right)\big/\left(S^1\times \{x_0\}\right)$ have cohomology ring isomorphic to $\Bbb Z[a]\otimes \Lambda[b]$ with $|a|=2$ and $|b|=3$, as can be seen from Künneth and cellular…
iwriteonbananas
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Some Good Algebraic Topology Exercises

I am teaching a topology prep course for first year graduate students taking their qualifying exams. I have been able to think of about ten days' worth of exercises, but am running out of ideas. Does anyone have any good questions or a place to…
J126
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Universal cover of $\mathbb R^2\setminus\{0\}$

We know that a necessary and sufficient condition for a path-connected, locally path-connected space to have a universal cover is that it is semi-locally simply connected. Now since $\mathbb R^2\setminus\{0\}$ is such a space, it must have a…
R_D
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Help with Serre spectral sequences

I'm working through Hatcher's unfinished book Spectral Sequences in Algebraic Topology and have found myself stuck on Exercise 2 on page 23: Compute the Serre spectral sequence for homology with $\mathbb{Z}$ coefficients for the fibration…
Ezy
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Calculating the fundamental group of $\mathbb R^3 \setminus A$, for $A$ a circle

Let $ X = \mathbb R^3 \setminus A$, where $A$ is a circle. I'd like to calculate $\pi_1(X)$, using van Kampen. I don't know how to approach this at all - I can't see an open/NDR pair $C,D$ such that $X = C \cup D$ and $C \cap D$ is path connected on…
Martin
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A question about the proof of Seifert - van Kampen

I'm studying Algebraic Topology using Hatcher's textbook as my main reference, and there is a detail in the proof of the Seifert - van Kampen Theorem (page 43) which is still unclear to me. The heart of the proof of the second part of the statement…
Emilio Ferrucci
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Connection between determinant of $f$ to its induced homomorphism on homology gps

It is my homework from Hatcher's book. It is a problem 7 on section 2.2, stating: For an invertible linear transformation $f:\mathbb{R}^n \to \mathbb{R}^n$ show that the induced map on $H_n (\mathbb{R}^n, \mathbb{R}^n-{0}) \sim H_{n-1}…
Emily
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Covering space is a fiber bundle

How do I show that every covering space $f:X \rightarrow Y$ is a fiber bundle with a discrete fiber? Was wondering is this really trivial So a fiber space is a short exact sequence $ F \rightarrow E \rightarrow B$ So do you just let X=E=F and B=Y.…
simplicity
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a map with contractible domain must be nullhomotopic

Let $f:X\to Y$ be a map between top spaces $X$ and $Y$. Is the following true? If $X$ is contractible then $f$ must be nullhomotopic. Here is an argument: Since $X$ is contractile then $id_X\simeq c$ where $c:X\to X;x\mapsto *_X$. Let $H_t:X\to X$…
palio
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