Question

Let $Y:=\mathbb{R}^2-\{(0,1),(1,0),(-1,0)\}$. Calculate $\pi_1(Y,y_0)$, where $y_0=(0,0)$.

score 60 · Answer 1 · 2026-04-09 01:32:11

60

Views

Let $Y:=\mathbb{R}^2-\{(0,1),(1,0),(-1,0)\}$. Calculate $\pi_1(Y,y_0)$, where $y_0=(0,0)$.

Published on 09 Apr 2026 - 1:32

score 533 · Answer 2 · 2026-03-25 08:10:57

533

Views

Let $A$ be a finely generated Abelian group. Prove that there is a topological space $X$ with a basic point $x_0$ such that $\pi_1(X,x_0)\cong A$.

Published on 25 Mar 2026 - 8:10

score 86 · Answer 3 · 2026-03-25 08:13:45

86

Views

Let $\mathbb{D}^3=\{(x,y,z)\in \mathbb{R}^3: x^2+y^2+z^2\leq 1\}$. Define $X=\mathbb{D}^3/\sim$ with the relation $(x,y,z)\sim(-x,-y,-z)$

Published on 25 Mar 2026 - 8:13

score 449 · Answer 4 · 2026-03-29 06:55:22

449

Views

Topological space , containing a point whose closure is the whole space, is contractible?

Published on 29 Mar 2026 - 6:55

score 118 · Answer 5 · 2026-04-09 02:00:00

118

Views

Homotopy groups for homogeneous spaces $SU(2n)/Sp(n)$ and $SU(2n)/SO(2n)$

Published on 09 Apr 2026 - 2:00

score 1.5k · Answer 6 · 2026-03-29 08:56:00

1.5k

Views

Why Homology is homotopy invariant

Published on 29 Mar 2026 - 8:56

score 196 · Answer 7 · 2018-03-23 12:41:39

196

Views

closed continuous path in $\mathbb{C}$ homotopic to polygon

Published on 23 Mar 2018 - 12:41

score 508 · Answer 8 · 2026-03-29 08:57:54

loops $\alpha ,\beta$ based at $x_0$ are homotopic then is there another loop such that conjugating $\alpha$ with it is path-homotopic to $\beta$?

score 331 · Answer 9 · 2026-04-09 04:33:31

331

Views

How to prove that $U(n)/O(n)\rightarrow S^1$ is a fibration?

Published on 09 Apr 2026 - 4:33

score 93 · Answer 10 · 2026-03-29 13:40:28

93

Views

The given two spaces are homotopic.

Published on 29 Mar 2026 - 13:40

score 626 · Answer 11 · 2026-03-27 03:09:43

626

Views

Why is homotopy complicated?

Published on 27 Mar 2026 - 3:09

score 63 · Answer 12 · 2026-03-27 10:12:57

63

Views

Let $E$ be a normed vector space and $f,g: X \rightarrow E-{0}$ are two continuous functions where $\|f(x)-g(x)\| < \|f(x)\|$ for any $x \in X$.

Published on 27 Mar 2026 - 10:12

score 154 · Answer 13 · 2026-03-24 23:46:34

154

Views

Question on Madsen's proof of the hairy ball theorem

Published on 24 Mar 2026 - 23:46

score 256 · Answer 14 · 2026-04-10 04:40:44

256

Views

Why this function well defined?

Published on 10 Apr 2026 - 4:40

score 287 · Answer 15 · 2026-03-25 07:52:27

287

Views

Topological manifold in $\mathbb R^n$, of dimension at least 2, minus a countable set

Published on 25 Mar 2026 - 7:52

List Question

Let $Y:=\mathbb{R}^2-\{(0,1),(1,0),(-1,0)\}$. Calculate $\pi_1(Y,y_0)$, where $y_0=(0,0)$.

Let $A$ be a finely generated Abelian group. Prove that there is a topological space $X$ with a basic point $x_0$ such that $\pi_1(X,x_0)\cong A$.

Let $\mathbb{D}^3=\{(x,y,z)\in \mathbb{R}^3: x^2+y^2+z^2\leq 1\}$. Define $X=\mathbb{D}^3/\sim$ with the relation $(x,y,z)\sim(-x,-y,-z)$

Topological space , containing a point whose closure is the whole space, is contractible?

Homotopy groups for homogeneous spaces $SU(2n)/Sp(n)$ and $SU(2n)/SO(2n)$

Why Homology is homotopy invariant

closed continuous path in $\mathbb{C}$ homotopic to polygon

loops $\alpha ,\beta$ based at $x_0$ are homotopic then is there another loop such that conjugating $\alpha$ with it is path-homotopic to $\beta$?

How to prove that $U(n)/O(n)\rightarrow S^1$ is a fibration?

The given two spaces are homotopic.

Why is homotopy complicated?

Let $E$ be a normed vector space and $f,g: X \rightarrow E-{0}$ are two continuous functions where $\|f(x)-g(x)\| < \|f(x)\|$ for any $x \in X$.

Question on Madsen's proof of the hairy ball theorem

Why this function well defined?

Topological manifold in $\mathbb R^n$, of dimension at least 2, minus a countable set

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