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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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Is $S^\infty$ contractible? Does it have CW structure?

How can I prove that $S^\infty$ is contractible? Is $S^\infty$ a CW complex?
spencer
  • 89
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3 answers

How equivariant theory (like equivariant cohomology) arise

I understand in mathematics there are many "quotienting " proceduce, is this the only reason that we consider equivariant theory for different "unequivariant" theory? Are there any more applications for equivariant theory?Thanks!
hao
  • 211
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2 answers

Double cover of $GL(n,\mathbb{C})$.

Is there any reference or simple proof to show that the group $\frac{\mathbb{C}\times SL(n,\mathbb{C})}{2\mathbb{Z}}$ is double cover of $GL(n,\mathbb{C})$.
user61135
2
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1 answer

Homology groups of unit square with parts removed -- revisited

I had another go at an exercise that I tried some time ago, the question I asked here. Can you tell me if this is right: Compute the homology groups of the subspace of $I \times I$ consisting of the four boundary edges plus all points in the…
Rudy the Reindeer
  • 46,359
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0 answers

Counter examples of cell complex

I'm reading "Memento on cell complexes" and the following is supposed to be a counter example of a cell complex because $e^1$ is not homeomorphic to the open segment: To me this looks like a deformed disk and is therefore homeomorphic to $D^2$. Why…
Rudy the Reindeer
  • 46,359
2
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2 answers

Are spaces with isomorphic fundamental groups homotopically equivalent?

I know that the converse of this statement is true but I am not sure how to go about finding out the answer to this question.
Asvin
  • 8,229
2
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1 answer

Cellular homology of projective space $\mathbb{R}P^n$

For the projective space the cell decomposition is $e_0 \cup \dots \cup e_n$ and the attaching map is $a_1 a_1 \dots a_k a_k$ for the $k$-th cell. So for $k \leq n$ I thought that this means that the boundary map between the cellular chain groups…
Rudy the Reindeer
  • 46,359
2
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1 answer

Show that $G:=\{g\ |\ g\text{ is a straight line in }\mathbb R^2 \}$ is homeomorphic to the Moebius strip

Show that $G:=\{g\ |g\text{ is a straight line in }\mathbb R^2 \}$ is homeomorphic to the Moebius strip I defined $\bar f:\mathbb R\times S^1/\tilde{}$ where $(t,\theta)\tilde{} (-t,-\theta)$ with $(t,\theta)\mapsto\{x\in\mathbb…
derivative
  • 2,450
2
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1 answer

Follow-up on $H_n(\mathbb{R}^3 - S^1)$

I'm trying to apply Mayer-Vietoris to compute $H_n(\mathbb{R}^3 - S^1)$ as asked by me here. Let $A := X -$"z axis" and $B:= B(0, 0.5) \times \mathbb{R}$ where $B(0, 0.5)$ is the open ball around $0$ with radius $0.5$. The case $n=0$ is clear to me…
Rudy the Reindeer
  • 46,359
2
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1 answer

Degree of map $S^{n-1} \times S^{n-1} \to S^{n-1}$ when restricted to one of the factors

We can define the degree, $d$ of a continuous map $f:S^n \to S^n$ through the induced map, $f_*$, in homology: $x \mapsto dx$. Now consider a map $S^{n-1} \times S^{n-1} \to S^{n-1}$, and let $y \in S^{n-1}$. Let $\alpha$ denote the degree of $g$…
Juan S
  • 10,268
2
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0 answers

Associativity in the fundamental groupoid of a space

Consider the set of all path homotopy classes of paths in $X$ with $[f]\cdot[g]=[f*g]$ defining a binary operation. We have a groupoid with the following conditions: 1) $[c_p][f]=[f]=[f][c_q]$ where $p=f(0)$, $q=f(1)$ and $[c_r]$ denotes a path…
Rahmani
  • 21
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1 answer

Hatcher 2.2.31 Invoke Mayer-Vietoris to wedge sum.

Use the Mayer-Vietoris sequence to show there are isomorphisms $\tilde H_n(X \vee Y) \approx \tilde{H}_n(X) \oplus \tilde H_n(Y)$ if the basepoints of $X$ and $Y$ that are identified in $X \vee Y$ are deformation retracts of neighborhoods $U…
1LiterTears
  • 4,572
2
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1 answer

What is the center of fundamental groupoid?

$C$, $D$ are two categories. $F$, $G$ are functors between $C$ and $D$: $F, G: C\rightarrow D$. Let $Nat(F,G)$ be all the natural transformations between F and G. Like what we do for groups, define the center of the a category $C$:…
Tom
  • 681
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1 answer

Homology of a torus

I am trying to compute the homology of a torus by its chain map, rather than its equivalence to $\oplus H_n (S^1)$. The post Homology groups of torus has been really helpful, but my question is, how can I come up with the boundary maps, so that I…
1LiterTears
  • 4,572
2
votes
2 answers

Computing $H_1(X)$ using Hurewicz

Can you tell me if the following 3 computations are correct: $X = S^1$, $\pi_1 (X) = \mathbb{Z}$, $Ab(\mathbb{Z}) = \mathbb{Z}$ $\implies H_1(X) = \mathbb{Z}$ $X = S^1 \vee S^1$, $\pi_1(X) = \mathbb{Z} \ast \mathbb{Z}$, $Ab(\pi_1(X)) = \pi_1(X) /…
Rudy the Reindeer
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