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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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How to prove the square lemma?

Let $F:I \times I \to X$ be a continuous map, and let $f,h$ and $k,g$ be paths in $X$ defined by $$f(s) = F(s,0)$$ $$g(s) = F(1,s)$$ $$h(s) = F(0,s)$$ $$k(s) = F(s,1)$$ Then $f \cdot g$ is homotopic to $h \cdot k$ I tried like $F(s,t)F(1−t,s)$, but…
Keith
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Why aren't these loops homotopic?

Let $S^1 = \{z \in \mathbb{C} : |z| = 1\}$. Take the loops $f,g : [0,1] \rightarrow S^1$, $f(t) = 1$, $g(t) = e^{2\pi it}$. I know these represent different elements in $\pi_1(S^1, 1)$, but I don't see why $F(t,s) = e^{2\pi its}$ isn't a homotopy…
Dog
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The morphisms of cellular (co-)homology groups induced by cellular maps

Let $X$ and $Y$ be CW-complexes and $\varphi\colon X\to Y$ be a cellular map. How can we describe the induced morphisms of cellular homology and cohomology groups explicitly? I suppose that $\varphi_*$ maps the homology class of a $n$-dimensional…
anna abasheva
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Hawaiian Earring vs Wedge Sum - A question of definition

Several earlier articles have explained the topological difference between a countably infinite bouquet of circles vs. a countably infinite Hawaiian Earring of circles. But my question is purely one of definitions. What has not been adequately…
PossumP
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Relative homotopy groups for $n$-skeleton of $K(G,1)$

So I'm trying to solve one of Hatcher's exercises and it seems that my proof would work if $\pi_n(X^n,X^{n-1})\simeq \pi_n(X^n/X^{n-1})$ for $n\geq 2$. I tried applying proposition $4.28$, but it doesn't work in this case. All I can gather is that…
Enigma
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Algebraic Topology: Hatcher, Example 1.43, Page 74

In Hatcher's book on Algebraic Topology, in the first sentence of example 1.43 he states: $$ \text{The Antipodal map of $S^n$, $x\to -x$, generates an action of $\mathbb{Z}_2$ on $S^n$ with orbit space $\mathbb{R}P^n$.} $$How do we know that the…
Bob
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Prove that the fundamental group of the circle is non-trivial if the fundamental group of a space $X$ is non-trivial

Prove that if $\pi_1(X,x_0)\neq 1$ for some topological space $X$, then $\pi_1(S^1,1)\neq 1$ I don't quite know how to proceed with this. I know that any path $f:I\to X$ has to factor through $S^1$. I was trying to map non-homotopic paths in $X$,…
freebird
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Equivalences of $S^n$ vs. $\Omega^nS^n$

Let $H(n)$ be the group of self-homotopy-equivalences of $S^n$ preserving the basepoint. I read that $H(n)$ may be identified with ''two components of $\Omega^nS^n$''. What does this mean and how can I see it?
user34987
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Torsion subgroup of closed connected manifold

So there is something that I don't quite understand in Hatcher's proof. I would appreciate it if something could tell me why $H_n(M:\mathbb{Z}_p)$ would be larger than the $\mathbb{Z}_p$ coming from $H_n(M;\mathbb{Z})$. Thanks.
Enigma
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Poincare Duality: Cohom. gp. of $M = \cup U_i$ (increasing) is inverse limit of cohom. gp. of $U_i$?

The question is to prove Poincare Duality of the form $(H^p_c (M))^* = H^{n-p} (M)$ using direct limits and inverse limits, where $H^p (M)$ denotes the $p$-th de Rham cohomology group of $M$. The outline of the proof has been given, which is to be…
ladrine
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1 answer

Isomorphism between $S^n$ and $S^n - \{x\}$

I wanted to understand the local degree of a map $f: S^n \to S^n$ and by doing this I faced a problem I couldn't solve by myself, although it might be very easy to see: I wanted to show, that for $n>0$, $x \in S^n$ there is an isomorphism $H_n(S^n)…
Donut
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1 answer

Showing $H_i(M, M - \{x \}) \cong H_i( \mathbb{R}^n, \mathbb{R}^n - \{0\} )$ via excision

I want to show $H_i(M, M - \{x \}) \cong H_i( \mathbb{R}^n, \mathbb{R}^n - \{0\} )$ via excision and can't quite figure out how to choose my subspaces. For $Z \subset A \subset X$, excision gives the following isomorphism $$ H_i(X -Z, A-Z) \cong…
Yuugi
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Homotopy versus Isotopy

First of all, I apologize for the crudeness of my question. Consider the construction of the homotopy groups. We mod out the space of "loops" at point by the equivalence relation generated by homotopy equivalence then give the new space a group…
Mathmonkey
  • 51
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How do we obtain the following identification

I don't understand geometrically why the identification below let us generate the shape on the right can someone explain or give me some intuition ?
user111750
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1 answer

How is identification is done in the definition of CW complexes

Consider the definition of CW complex from hatcher I am trying to understand the issue with the identification, because I feel there is something I don't understand. I decided to do an example and do all the detailed calculation. Suppose we want…
user111750
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