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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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Showing two spaces are not homotopy equivalent

I just started learning algebraic topology about 1 week ago. Today, eager to test what I've learned I tried the following exercise from Rotman's Algebraic Topology : Let $X = \{0\} \cup \{1,1/2,1/3,\ldots,1/n,\ldots\}$, while let $Y$ be countable…
user38268
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A question about intersection number

I'm trying to understand the geometric meaning of (symmetric) bilinear forms. I'm reading parts of "Symmetric Bilinear Forms", in particular, the appendix mentions what I'm interested in: on page 100 they write "Let $M = M^{2n}$ be a closed…
Rudy the Reindeer
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Loopspace adjunction: when are unit or counit equivalences?

For (nice?) pointed spaces, the reduced suspension $\Sigma$ is left adjoint to the loop space $\Omega$. This adjunction is given by the unit maps $\eta_X : X \to \Omega \Sigma X$, $x \mapsto (t \mapsto [x,t])$ and the counit maps $\varepsilon_X :…
Martin Brandenburg
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Is there a covering map with uncountably many slices?

The following is from the Topology by Munkres: Let $E$ and $B$ be two topological spaces and $p:E\to B$ a continuous surjective map. The open set $U\subset B$ is said to be evenly covered by $p$ if the i nverse image $p^{-1}(U)$ can be written as…
user9464
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If $f : S^1 \to S^1$ is odd, then $f$ cannot be nullhomotopic

The problem is: Let $f:S^1 \to S^1$ be a continuous map that has the property that $f(-z) = -f(z)$ and $f(1) = 1$. Prove $f$ cannot be null-homotopic. Any ideas? The first step is to show that by defining a map $p: S^1 \to S^1$, $p(z) = z^2$, there…
ThinkConnect
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Showing higher homotopy groups of $S^1$ are trivial

I'm trying to prove $\pi_{i} (S^1) \cong 0$ if $i>1$. Is this correct. You have a short exact sequence, $\mathbb{Z} \rightarrow \mathbb{R} \rightarrow S^1$ (from the fiber bundle of the covering space), then you deduce from some magic I don't…
simplicity
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Path lifting theorem

http://www.maths.manchester.ac.uk/~jelena/teaching/AlgebraicTopology/PathLifting.pdf I'm trying to generalize this theorem. But, was wondering in the proof given here and similarly in Hatchers. Can you replace the $S^{1}$ with a general space X. As…
simplicity
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An isomorphism in relative De Rham cohomology

Let $E$ be a smooth oriented vector bundle over a smooth manifold $M$ and let $E^0$ be the complement of the zero section in $E$. I would like a reasonably explicit isomorphism between the relative De Rham cohomology group $H^p(E,E^0)$ and…
Paul Siegel
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Constructing a compact manifold with chosen homology groups

Let $G_1,G_2,\ldots,G_k$ be $k$ finitely presented abelian groups. It's possible to construct a $(2k+3)$-dimensional manifold $X$ s.t $H_i(X) = G_i$ in following way: consider $k$ copies of the Moore space $X_1,\ldots,X_k$ with $H_i(X_j)=…
Anubhav Mukherjee
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The fundamental group of a plane without a finite number of points.

How can I calculate de fundamenta group of $\mathbb{R}^2$ without a finite number of points? I know that the answer should be $S^1\vee S^1 \vee \cdots \vee S^1 $ where the product repeats as many as points removed. Thanks for helps!
EQJ
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Torsion of homology group

If $U$ is an open connected subset of $\mathbb{R}^n$ where $n\ge 2$, is it true that $H_1(U,\mathbb{Z})$ is torsion-free? Or in general, $H_i(U)$ is free? I am thinking whether it has deformation retract to some nice manifold in…
Ben
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Orientation in Connected Sum

When define the connected sum of two oriented manifods, one gluing along the reversed orientation of the boundary spheres. I am wondering what is the connected sum of $S^2$ connected sum with $S^2$ with the reversed orientation? (suppose to be the…
Sun
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Van Kampen theorem to prove all spheres have trivial fundamental group

I'm trying to prove that $\pi_1(S^n)=0$ when $n>1$. I've noticed that if you just take the north pole and south pole from each sphere, you can apply Van Kampen theorem. This doesn't fail as when $n>1$ it's still simply connected as it's homotopic…
simplicity
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Computing fundamental groups with Van Kampen

The following exercise is drawn from Ch.14 of Fulton's "Algebraic Topology: A First Course." Use the Van Kampen theorem to compute the fundamental groups of: (1) the sphere with $g$ handles; (2) the complement of $n$ pts. in the sphere with $g$…
Vulcan
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Example of closed orientable manifold with a nonzero vector field, but not parallelizable

Can you give a simple example of a closed orientable manifold with an everywhere nonzero section of its tangent bundle, but where the tangent bundle is not trivial? If the manifold is not orientable, it is not too hard to come up with examples (e.g.…
Holographer
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