3
votes
1answer
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 ...
1
vote
1answer
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 ...
4
votes
1answer
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 $f:E\rightarrow E$ ...
2
votes
2answers
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 ...
0
votes
1answer
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 ...
-1
votes
1answer
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 ...
2
votes
1answer
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 $T:=\{y\in Y:g_1(y)=g_2(y)\}\subseteq Y$ ...
0
votes
0answers
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 ...
2
votes
2answers
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 ...
4
votes
2answers
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 ...
2
votes
1answer
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 ...
2
votes
1answer
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 ...
1
vote
0answers
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 $x_0 \in X$ ...
0
votes
0answers
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 ...
3
votes
0answers
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 ...
0
votes
0answers
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 $f_t(r,\theta) = (\exp((1-t)r),\theta)$, ...
0
votes
1answer
20 views

Prove that $h$ and $p*q$ are homotopic relative to {$0,1$}

Let $0<s<1$. Given paths $p$ and $q$ with $p(1)=q(0)$, define $h$ by the formula $$h(t) = \begin{cases} p(t/s),& \text{if} \quad 0 \leq t \leq s \\ q((t-s)/(1-s)), &\text{if} \quad ...
0
votes
0answers
30 views

prove that if f is a closed path in X then either every lifting of f is closed or none is closed.

A covering is regular if for some $\tilde{x_0}\in\tilde{X}$ the group $p_*\pi(\tilde{X}, \tilde{x_0})$ is a normal subgroup of $\pi(X, x_0)$. In this case, how do I prove that if f is a closed path ...
2
votes
2answers
47 views

homeomorphism $T: X \rightarrow X$

How do i prove that $Tz=\bar{z}+1+i$ defines a homeomorphism $T: X \rightarrow X$ where $X=\mathbb{R}\times[0,1] \subset \mathbb{C}$ ? (how can there be a continuous bijection in this case?) Also, ...
0
votes
0answers
24 views

Euler characteristic and free action

If $K$ is a finite simplicial complex, and $G$ acts simplicialy on $K$ with no fixed points, show $\chi(K) = |G|\cdot\chi(K^2/G)$. Could I have a hint for how to start this question? I was told to ...
0
votes
0answers
33 views

fundamental group of $\mathbb{C^*}/\{e,a\}$

I'm taking an intro to topology course, and am having trouble with this question. What is the fundamental group of $\mathbb{C^*}/\{e,a\}$, where $e$ is the identity homomorphism and $az=\overline{z}$. ...
0
votes
0answers
22 views

Presentations of fundamental groups regarding cones of simplicial subcomplexes

Let $L$ be a simplicial subcomplex of $K$. Let $CL$ be the cone on L. Let $X = CL \bigcup K$. Show that X is a simplicial complex and dscribe a presentation for $\pi_1(|X|,v)$ in terms of ...
1
vote
2answers
87 views

Fundamental Group of Punctured Plane

What is the fundamental group of $(\mathbb{C} \setminus {\{0\}})~/~\{e,a\}$, where $e$ is the identity homeomorphism and $az = -\bar{z}$? Clearly this is homeomorphic to the half cylinder , which is ...
0
votes
0answers
23 views

Edges and Vertices in relation to Free subgroups

Let $\Gamma$ be a finite connected graph (1-dimensional simplicial complex), with $V ( \Gamma)$ vertices and $E(\Gamma )$ edges (1-simplices). Show that $\pi_1(\Gamma, v)$ is a free group with ...
1
vote
1answer
54 views

fundamental group of complex numbers?

Let $\mathbb{C}^*=\mathbb{C}-{0}$. What is the fundamental group $\mathbb{C}/G,$ where G is the group of homeomorphism $\{\phi^n ; n\in \mathbb{Z}\}$ with $\phi(z)=2z$? I think the fundamental group ...
3
votes
1answer
82 views

Homotopy equivalence -> element in outer automorphism of fundamental group

Let $X$ be a path-connected topological space. for $x \in X, G = \pi_1(X,x)$ Show that a homotopy equivalence $f : X \to X$ gives a well-defi ned element $g \in Out(G)$. How might one begin on this ...
2
votes
1answer
67 views

Fundamental group: Attaching a disc to a path connected triangulable space

What's the effect on the fundamental group of a path connected triangulable space X if you attach a disc? So far I've figured out that a disc has trivial fundamental group. So the free product of ...
0
votes
0answers
27 views

a problem about surface $M_{g}$.

1.$M_{g}$ has normal universal cover $\widetilde{X}$ with deck transformation $G(\widetilde{X})=\mathbb{Z}^{n}$ if and only if $n \leq 2g$. 2.for $n=3,g \geq 3$ explain such covering. 3.show that ...
0
votes
3answers
49 views

if $p:\widetilde{X}\rightarrow X$ is a covering space and $\widetilde{X}$ is path connected ,show that $p^{-1}(A)$ is path connected.

if $p:\widetilde{X}\rightarrow X$ is a covering space and $\widetilde{X}$ is path connected ,also $A\subset X$ is a path connected subset,show that $p^{-1}(A)$ is path connected. I suppose that ...
0
votes
0answers
80 views

find a necessery and enough condition just using $\pi_{1}$.

suppose $p:\widetilde{X} \rightarrow X$ will be a covering space and $X$ is path connected and locally path connected, also $\widetilde{X}$ is connected, then find a necessary and enough condition ...
2
votes
0answers
36 views

$H_1(A\cap B)\to H_1B$ is 1-1 implies $H_1A\to H_1(A\cup B)$ is 1-1

Let $A,B$ be two open set of a topological space, $H_1(A\cap B)\to H_1B$ is 1-1 implies $H_1A\to H_1(A\cup B)$ is 1-1, where the homomorphisms is induced by inclusion. I feel that using the ...
3
votes
2answers
89 views

Show that the union of the spheres of radius $\frac{1}{n}$ and center $(\frac{1}{n},0,0)$ is simply-connected.

Show that the subspace of $\mathbb{R}^{3}$ that is the union of the spheres of radius $\frac{1}{n}$ and center $(\frac{1}{n},0,0)$ for $ n=1,2,3,...$ is simply-connected. for showing it is ...
2
votes
0answers
54 views

showing that $\Phi:\Pi_{1}(X,x_{0})\rightarrow [S^{1},X]$ is onto if $X$ is path connected.

We can regard $\Pi_{1}(X,x_{0})$ as the set of basepoint-preserving homotopy classes of maps $(S^{1},s_{0})\rightarrow(X,x_{0})$ . Let $[S^{1},X]$ be the set of homotopy classes of maps ...
-1
votes
1answer
69 views

Relative homology of ball and sphere

What is the result of $H_k(B^n,S^{n-1}; \mathbb{A })$ and in any book can i found the proof ? And what about $H_n(S^{n};\mathbb{A})$ (sigular homology of the sphere )?? Please help me. Thank you
4
votes
2answers
68 views

If the function $\varphi \colon Z\rightarrow C(X,Y)$ is continuous then $F\colon Z\times X\rightarrow Y$, $F(z,x)=\varphi (z)(x)$ will be continuous.

If the function $\varphi :Z\rightarrow C(X,Y)$ ($C(X,Y)$ with compact-open topology) is continuous and $X$ is locally compact, then $$F\colon Z\times X\rightarrow Y$$ $$F(z,x)=\varphi (z)(x)$$ will be ...
1
vote
1answer
63 views

Determining the generators of cohomology (as a ring)

I am working on a problem to show that the cohomology graded rings of $\mathbb{C}P^3$ and $S^2$ x $S^4$ are not isomorphic (unreduced with integer coefficients) I have already calculated the graded ...
2
votes
0answers
45 views

Find Group Action

I'm asked to find a group action G on the unit cylinder C such that $C/G$ is homeomorphic to the torus. Would $\pm1 \cdot (x,y,z) = (x,y,\pm z)$ work? The only problem here is that G should only act ...
1
vote
1answer
58 views

Queston on the definition of singular homology

From the Hatcher's can someone told me why $\sigma$ has singularities ? Thank you
0
votes
1answer
93 views

Showing that two spaces are homotopy equivalent

Let $x_0 \in S^1 \times S^1$. I want to show that $(S^1 \times S^1) - \{x_0\}$ and $S^1 \vee S^1$ are homotopy equivalent. We have to show that $\exists$ maps $f: X \rightarrow Y$ and $g: Y ...
5
votes
1answer
206 views

Hatcher Problem 2.2.36

I am struggling with the following question (2.2.36) from Hatcher for quite some time now: Show that $H_i(X\times S^n) \simeq H_i(X) \oplus H_{i-n}(X)$. I don't know how to use the hint given by ...
1
vote
0answers
68 views

prove that after removing countable infinite point from $S^{2}$,it will remain path-connected.

prove that after removing countable infinite point from $S^{2}$,it will remain path-connected. it was the question that arise in the algebraic topology course where I have this term.I thought ...
2
votes
1answer
106 views

Why is the graph $K_{3,3} $ not one skeleton of the sphere?

I know that someone has already asked the same question here, but there is no solution for part two of the question. And I'm interested in the second part. Here the question: Suppose we build $S^2$ ...
4
votes
1answer
91 views

Existence and homotopies of embeddings between simplicial complexes

Let $K$ and $L$ be simplicial complexes, $m=\dim K$, and $h:|K|\rightarrow |L|$ be a continuous map. Show that $h$ is homotopic to a map carrying $K$ into $L^{(m)}$, the $m$-skeleton of $L$. I'm ...
0
votes
1answer
53 views

Prove that $S^n$ is a strong deformation retract of $\mathbb{R} \backslash\{(0,…,0)\}$

Prove that $S^n$ (n-dimensional sphere with unit radius) is a strong deformation retract of $\mathbb{R} \backslash\{(0,...,0)\}$ This is my attempt: Consider $f:S^n \rightarrow \mathbb{R} ...
5
votes
2answers
289 views

Homology of connected sum of real projective spaces

Let $A_k=RP^2\sharp RP^2\sharp \cdots \sharp RP^2$ be a connected sum of $k$ copies of real projective space. With coefficients in $\mathbb{Z}$, it is clear $H_n(A_k)=0$ when $n\geq2$ and ...
2
votes
1answer
87 views

Show the existence of a certain subgroup of F2

This is a homework question: I need to find three subgroups of the free group with two generators, $F_2$, with certain properties. I have found the other two by constructing covering spaces of $S^1 ...
0
votes
1answer
66 views

Homology and Exact Sequence

I have this exact sequence: $$0\stackrel{f}{\rightarrow} H_k(X,C)\stackrel{g}{\rightarrow} H_k(X,A)\stackrel{h}{\rightarrow} 0$$ Can I say that $H_k(X,A)=H_k(X,C)$ and why? Please; Thank you.
5
votes
1answer
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 ...
1
vote
1answer
124 views

a Problem about Degree of Map between Spheres

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 ...
3
votes
1answer
65 views

Orbit space of S3/S1 is S2

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 ...