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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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Cup product of cohomology and the Kunneth formula

I'm wondering if there's anything that can be said about the cup product when computing the cohomology ring of a product space? An example would be $S^2\times S^4$, where the generator of $H_2(S^2\times S^4)$ is essentially contributed by $H_2(S^2)$…
asdf
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Application of Seifert-van Kampen Theorem

I am trying to wrap my head around the following problem: I have three objects lined up horizontally, a $2$-sphere, a circle, and another $2$-sphere. It is the wedge sum $S^2 \vee S^1 \vee S^2$. I am trying to find the fundamental group of this…
Chammershield
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How to compute homotopy groups of torus?

The results are listed here: http://topospaces.subwiki.org/wiki/Homotopy_of_torus Is there an intuitive way to understand these results? In particular, why would the higher homotopy group be the trivial group?
Taiben
  • 320
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Existence of weak homotopy equivalence not a symmetric relation

I am having trouble coming up with an example of spaces where there exists a weak homotopy equivalence in one direction but not the other. Any hints or references are greatly appreciated! Note: This is an instance of stagnating autodidactic…
Jawid Ahmad
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The groups $[\Sigma^nX,Y]$ versus the homotopy groups

If $X,Y$ are pointed spaces, denote by $[X,Y]$ the pointed homotopy classes of pointed maps $X\to Y$. The sets $[\Sigma^nX,Y]$ actually have the structure of a group for $n\geq 1$. Here $\Sigma$ denotes reduced suspension. If we take $X=S^0$, then…
Bruno Stonek
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Klein bottle homology by Mayer Vietoris

I'm still applying Mayer Vietoris, this time to the Klein bottle. I'm using the decomposition as on Wikipedia here and I've calculated $H_0$ and $H_n$ for $n \geq 2$ correctly. Now I'm struggling with $H_1$. My sequence: $$ 0 \xrightarrow{} H_1(…
Rudy the Reindeer
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Kunneth formula on product of spheres

Kunneth formula is $$ H^\ast (S^K;{\bf Z})\otimes_{{\bf Z}} H^\ast (S^M;{\bf Z}) =H^\ast (S^K\times S^M;{\bf Z}) =\wedge_{\bf Z} [a,b]$$ where $K=2k+1
HK Lee
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Galois correspondence of covering spaces of spaces not necessarily semilocally simply-connected

I've been trying to solve the following exercise (1.3.24) from Hatcher's Algebraic Topology: Given a covering space action of a group $G$ on a path-connected, locally path-connected space $X$, then each subgroup $H$ in $G$ determines a composition…
PeterM
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Topological Join and Wedge Sum of Spheres

Let $S^{n}$ be an $n$-sphere. I'd like to compute the reduced homology (with $\mathbb{Z}$-coefficients) of the space $\bigvee^{r}_{i = 1} S^{n_{i}} * \bigvee^{s}_{j = 1} S^{m_{j}}$, where $r, s, n_i, m_j \geq 0$ and $*, \vee$ denote the join and…
user02138
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Why doesn't the circle retract to a point?

OK, this appears to me like perhaps a dumb question. I am reading Allen Hatcher's Algebraic Topology. I've seen bits and pieces of further material here and there before, now I'm restarting from the beginning. OK, visually I can see why, say, the…
user80309
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Homology of $\mathbb{R}^2$ under the equivalence $x \sim 2x$

I was computing some examples of homologies of quotient spaces and I thought of the following. Does anyone know how to compute the homology groups of $\mathbb R^2/\sim$, where $\sim$ is the equivalence $x \sim 2x$. Thank you
caley
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Orientability determined by top homology group

Let $M$ be a compact, connected $n$-manifold. Say that $M$ is orientable if there is a class $\alpha$ in $H_n(M)$ such that the reduced homology map $H_n(M)\rightarrow H_n(M,M\setminus\{p\})$ takes $\alpha$ to a generator of…
user15464
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How to show $\mathbb{CP}^{2}\#\overline{\mathbb{CP}^{2}}\not\cong \mathbb{S}^{2}\times \mathbb{S}^{2}$?

I was asked to prove that $$\mathbb{CP}^{2}\#\overline{\mathbb{CP}^{2}}\not\cong \mathbb{S}^{2}\times \mathbb{S}^{2}$$ as fibre bundles over $\mathbb{S}^{2}$ with fibre $\mathbb{S}^{2}$. Since the above connected sum is as manifolds instead of as…
Bombyx mori
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How to visualize Homology groups?

I've been studying Algebraic Topology recently (Hatcher). I have always been very good at Topology, since I am chiefly a visual type of person. I've found that most things in Topology can be thought of visually, and then translated into rigorous…
Doctor Grizz
  • 113
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Connected Partitions of Spheres

Let $U,V$ be disjoint non-empty connected open subsets of the sphere $S^2$ such that $\partial U=\partial V$ and $\operatorname{cl}(U\cup V)=S^2$. Must $U$ and $V$ be simply connected? This seems intuitively obvious, but I'm not sure how to best…
Milo Brandt
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