# Tagged Questions

41 views

### Question about the Betti numbers

can someone explain me this definition from :http://en.wikipedia.org/wiki/Betti_number The $n^{th}$ Betti number represents the rank of the $n^{th}$ homology group, denoted $H_n$ "which tells us the ...
47 views

### Deck transformation acting properly discontinuously assumed covering space is path-connected

Let $p:E\rightarrow X$ be a covering space, $x\in X$ fixed point, $E$ path-connected and $\Delta(p)$ – Deck transformation group of $p$, that is $\Delta(p) = \{f\in \text{Homeo}(E):pf=p\}$. Let ...
62 views

### Is continuous map from covering space to itself homeomorphism assumed both cover and base path-connected and $pf=p$?

In my topology assignment I came across the following problem: True or false? Let $E$ and $X$ be path-connected. For every covering map $p:E\rightarrow X$ and continuous map $f:E\rightarrow E$ ...
95 views

### Covering through group action and corresponding deck transformations

I'm having a bit of trouble with the following exercise: Let $G$ be a group acting properly discontinuous and continuous on a topological space $E$. Then $p:E\to G\backslash E$ is a covering. Let ...
39 views

### a topological space is the union of its irreducible components

if we define irreducible component of a topological space $X$ as the maximal closed irreducible subset of $X$,prove that we can write $X$ as the union of its irreducible components. how to approach ...
40 views

### a quotient space homeomorphic with $\mathbb{R}\mathbb{P^2}$

prove that if we glue the closed unit disc to the circular boundary of the mobius strip we obtain a quotient space that is homeomorphic with $\mathbb{R}\mathbb{P^2}$. it is my general topology ...
61 views

### lifts of continuous map to covering space

The following problem gives me a bit of trouble: Let $p:E\to X$ be a covering map. Let $g_1,g_2$ be two lifts of the continuous map $f:Y\to X$. Show that $T:=\{y\in Y:g_1(y)=g_2(y)\}\subseteq Y$ ...
44 views

### one point compactification of upper half plane

"prove that the closed unit disc could be considered as the one point compactification of the upper half plane with the X-axis as the bounday" i believe in this statement by the imagination,but i ...
100 views

### $f:X\rightarrow S^1$ a continuous map. $X$ a path-connected topological space.

Let $f:X\rightarrow S^1$ be a continuous map from a path-connected topological space $X$ and let $p:\mathbb{R} \rightarrow S^1$ be the universal covering. What is the condition when $f$ admits a ...
77 views

### (Certain) colimit and product in category of topological spaces

Consider the diagram $$(*):\;\;\;X_0 \stackrel{i_0}\hookrightarrow X_1 \stackrel{i_1}\hookrightarrow X_2 \stackrel{i_2}\hookrightarrow \cdots$$ in category of topological spaces. Denote $I$ the ...
137 views

### mapping torus eqivalent definition

Let $X$ be a topological space and $f:X\to X$ a homeomorphism. I need to find a continuous, properly discontinuous $\mathbb{Z}$-action on $X\times\mathbb{R}$, such that the quotient ...
48 views

### Use Hurewicz Theorem to calculate $\pi_3(\mathbb{R}P^4 \vee S^3)$

Want to calculate $\pi_3(\mathbb{R}P^4 \vee S^3)$, using Hurewicz theorem. This is one of the questions on the previous topology qualifying exams. Any help will be appreciated! I am thinking in stead ...
21 views

### Prove that if $p: Y \to X$ is a covering space and $X$ is path connected, then the cardinality of $p^{-1}(X)$ is constant. [duplicate]

Let $p: Y \to X$ be a covering space and $X$ is connected. I want to show that $\forall x \in X$ the cardinality of $p^{-x}$ is the same. $\textbf{My Attempt:}$ Let us first fix a point $x_0 \in X$ ...
37 views

### Action of SO(n) on Euclidean n-space

I'm asked to describe the orbits of the "natural action" of SO(n) on euclidean n-space as a group of linear transformations, and to identify the orbit space. I'm not sure how to do this except I've ...
47 views

### Prove that if $f:D^2\to D^2$ is a homeomorphism, then $f(S^1)=S^1$

I've already proved that set of points $z\in D^2$ such that $D^2-z$ is simply connected is precisely $S^1$. Now from this, I'm supposed to conclude that if $f:D^2\to D^2$ is a homeomorphism, then ...
22 views

### Find a circle which is a strong deformation retract of $\mathbb{R^2}-x_0$

Let $x_o\in\mathbb{R^2}$. Find a circle which is a strong deformation retract of $\mathbb{R^2}-x_0$ Proof: If $x_0=(a,b)$, Let $S=\{(x-a)^2+(y-b)^2=1\}$. Then $f_t(r,\theta) = (\exp((1-t)r),\theta)$, ...
20 views

83 views

### Homotopy between two functions to a circle.

Suppose $f,g: X\to S^1$ are such that $f(x)\neq -g(x)$ for any $x\in X$. I need to construct a homotopy between these two functions. Now, the fact that $f(x)\neq -g(x)$ guarantees that there is always ...
When I read the book "Algebra Topology-A First Course", I find a problem. It is on the Page 97, Exercise (16.15). Problem We define $f$, $g\colon \mathbb{S}^{n}\to\mathbb{S}^{n}$ to be orthogonal ...
I'm having trouble finishing this homework assignment. I did the first part by showing that the orbits are invariant: every element from the same $(S^1(z_1, z_2) \in S^3/S^1)$ is mapped to the same ...