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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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Fibration with contractible fibre

Consider $f \colon E \rightarrow B$ a Serre fibration with contractible fibre (assume $B$ is path connected, so all fibres are weakly homotopy equivalent). Can we conclude from this that $f$ must be an homotopy equivalence? I know that if we take…
N.B.
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Prove $e^{2 \pi i k s}$ is not homotopic to constant loop at $1$ in $S^1$.

Let $e^{2 \pi i k s} = f(s)$, $f \colon [0,1] \to S^1$ subset of Complex numbers ($S^1$ = unit circle at origin). So $f$ is a loop at the basepoint $1$ in $S^1$. Show that it is not homotopic to the constant loop $c(s) = 1$. This is obvious, but…
Dan Donnelly
  • 303
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The $n-1$ homology group of $U\setminus\{x\}\subset \mathbb{R}^n$ where $U$ is open.

Let $U$ be a open subset of $\mathbb{R}^n$ where $n\geq 2$ and let $x\in U$. Show that $H_{n-1}(U\setminus\{x\})$ is not the trivial group. What I know is that $H_{n-1}(\mathbb{S}^{n-1})=\mathbb{Z}$ and that $U$ is homeomorphic to an open subset of…
UserA
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Why do $P^2$#$T$ and $P^2$#$P^2$#$P^2$ have different genus?

I have seen that the genus of the connected sum of closed surfaces is suppose to be sum of the genus of the individual surfaces thus $P^2$#$T$(connected sum of torus and real projective plane) is suppose to have genus $2$ and $P^2$#$P^2$#$P^2$ is…
TheGeometer
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topology of $S^{2}$-bundle with a section

Let $B$ be an even dimensional closed manifold. Suppose that $\pi: E \rightarrow B$ is a fibre bundle with fibre $F = S^{2}$. Suppose furthermore that there is a section $s : B \rightarrow E$ (i.e. such that $\pi \circ s = id_{B}$). Is it true that…
Nick L
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What about the action on this space?

I think it is true that there is no free-action of $(\mathbb{Z}_2)^3$ on product of $\mathbb{S}^{m}$ and $\mathbb{CP}^n(n$ is odd). I don't know how to prove it. A detailed proof will be very much helpful.
Shivani Sengupta
  • 611
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Connection between two groups related to covering

How the the monodromy group and deck transformation group of a covering $p\colon Y\rightarrow X$ are related in the cases when covering is branched and covering is unbranched?
user8186
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2 answers

Retraction from torus to linked circle

Possible Duplicate: Why is this entangled circle not a retract of the solid torus? I am stuck with exercise 16 (c), pag.39 of Hatcher's Algebraic Topology: prove that there is no retraction from $S^1\times D^2$ onto the set $A$, which is described…
random
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Prove that such homeomorphisms have fix points

If a homeomorphism $f:R\rightarrow R$ satisfies $f^2=1$, prove that it has at least one fix point. What if we set $f^n=1$ instead of $f^2=1$?
Ash GX
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2 answers

The fundamental group of $S^3-S^1$

How can I calculate the fundamental group of $S^3-S^1$? I believe it might be similar to the fundamental group of $S^2-S^0$ which is $ \Bbb Z$ but I'm having trouble showing it. Is this the right direction (showing it's $\Bbb Z$) or am I way off?
Amontillado
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An example of Vietoris–Rips complex

The below figure is from https://arxiv.org/abs/1710.04019. I'm new to the concept of the simplicial complex and studying it through the article. The given definition of Vietoris–Rips complex $\text{Rips}_{\delta}(X)$ of the points in $X$ is "the set…
rioneli
  • 85
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Computing number of path components.

Let $H \subset \mathbb{R}^n$ be a closed subset homeomorphic to $\mathbb{R}^{n - 1}$. Could you give me an idea of how to prove that $\mathbb{R}^n - H$ has exactly two path components. I am expecting an answer using singular homology.
topology
  • 81
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Building a space with given homology groups

Let $m \in \mathbb{N}$. Can we have a CW complex $X$ of dimension at most $n+1$ such that $\tilde{H_i}(X)$ is $\mathbb{Z}/m\mathbb{Z}$ for $i =n$ and zero otherwise?
topology
  • 81
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Finding fundamental group

What is the fundamental group of $\mathbb R^3 \setminus$ two linked circles? This is an example from Hatcher, he says that this space deformation retracts onto the wedge product of$S^2$ and a torus separating the two circles, but I have not really…
Mel
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Summing two simplices in singular homology.

Let $\sigma: \Delta^n \to X$ be a singular simplex such that $\partial(\sigma)=0$. I was wondering if given $k\geq 0$ there is a way of producing another singular simplex $$\theta: \Delta^n \to X \enspace \text{such that } \theta=k\sigma \in…
Abellan
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