1
vote
2answers
45 views

Homeomorphism S^1 x Y

Does there exist a topological space $Y$ such that $S^1 \times Y$ is homeomorphic to $\mathbb{R}P^2$ or to $S^2$?
1
vote
1answer
22 views

Number of cells in a minimal cell structure for a non-simply connected manifold?

I have obtained a cell structure of a connected (but not simply connected) manifold using Morse theory. Is there any way for me to know whether this cell structure is minimal?
4
votes
2answers
89 views

Homeomorphism Compact Subsets

Are there compact subsets $A,B \subset \mathbb{R^2}$ with $A$ not homeomorphic to $B$ but $A \times [0,1]$ homeomorphic to $B \times [0,1]$?
7
votes
1answer
105 views

Isomorphism Finite Topological Space

Does there exist a finite topological space with fundamental group isomorphic to $\mathbb{Z_2}$?
3
votes
2answers
27 views

the mapping class group of the disk is trivial proof

Proof : Identify $D^2$ with the closed unit disk in $\mathbb{R}^2$. Let $\phi : D^2 \rightarrow D^2$ be a homeomorphism with $\phi_{\partial D^2}$ equal to the identity. We define, $F(x,t) = ...
1
vote
0answers
29 views

Computing fundamental groups.

How do I compute the fundamental groups of these spaces: (a) $\{(x,y)\in\mathbb{R}^2|x^2+y^2>1\}$; (b) $\mathbb{R}^2$ with two points deleted; (c) $\mathbb{C}$P$^n$, the complex projective ...
1
vote
1answer
45 views

Is it true there exists $f:S^{2n}\longrightarrow S^{2n}$ making the diagram commutative?

Let $g:\mathbb R\mathbb P^{2n}\longrightarrow \mathbb R\mathbb P^{2n}$ be a continuous map where $\mathbb R\mathbb P^{2n}=\mathbb S^{2n}/\{\pm x\}$. Is it true there exists $f:\mathbb ...
2
votes
1answer
48 views

Covering map of $\mathbb R \mathbb P^2$

The question I am trying to answer is: Does the quotient map $ q:[0,1] \times [0,1] \to \mathbb R \mathbb P^2$ extend to a covering map $\mathbb R^2 \to \mathbb R \mathbb P^2$ I know that the ...
0
votes
1answer
51 views

Open and Closed Covering [on hold]

Suppose $p:\widetilde{X} \mapsto X$ is a covering with $f,g:Y \mapsto \widetilde{X}$ continuous such that $pf = pg$. Why is the set of points in $Y$ for which $f=g$ open and closed in $Y$?
3
votes
0answers
47 views

does the pullback of a covering space correspond to the pullback of the corresponding representations of $\pi_1$?

Say you have a covering space $C \rightarrow X$ corresponding to some homomorphism $\pi_1(X)\rightarrow S_n$. Suppose you have an arbitrary (continuous) map $f : Y\rightarrow X$. Then we may pull back ...
4
votes
2answers
81 views

Fundamental group of the Poincaré Homology Sphere

I'm working on the Poincaré Homology Sphere $P_3$ and would like to compute it's Homology $H_1$ and fundamental group. I would like to identify it's fundamental group with the binary icosahedral group ...
1
vote
1answer
44 views

Fundamental Group of Klein Bottle?

Let $C^{*}=C \setminus \{ 0 \}$. What is the fundamental group of $C^{*}/H$, here $H=\{\psi^n;n \in \mathbb{Z}\}$ with $\psi(z) = 2 \bar{z}$?
0
votes
1answer
36 views

Question about degrees of maps from $S^1 \rightarrow S^1$

Note: Since the degree of a map is independent of the base-point I'll speak loosely and just say $\pi_1(S^1)$. One definition of the degree of a map $f$ from the circle to itself is the number $k$ ...
1
vote
1answer
29 views

Show that if $h,k: S^1\rightarrow S^1$ are homotopic, they have the same degree.

We define the degree of a continuous map $h: S^1 \rightarrow S^1$ as follows: Let $b_0$ be the point $(1,0)$ of $S^1$; choose a generator $\gamma$ for the infinite cyclic group $\pi_1(S^1,b_0)$. If ...
5
votes
4answers
525 views

Topological groups, why need them?

I'm reading through Munkres and Armstrong's books on topology. However, I find topological groups to be really complicated objects! I feel they are twice as hard to deal with then just groups and ...
1
vote
2answers
102 views

Let $p: E\to B$ be a covering map. If $B$ is compact and $p^{-1}(b)$ is finite, then $E$ is compact. [duplicate]

So I start off and assume that some $\{U_\alpha\}$ is a cover of $E$. I want to reduce this cover to a finite subcover of $E$. Since $p$ is a covering map it is also an open map, therefore ...
1
vote
1answer
50 views

Mapping $\mathbb R^n - \{0\}$ to $S^{n-1}$

How might one map $\mathbb R^n - \{0\}$ to $S^{n-1}$ ? I found this in a primer on homology where it is proved that the to spaces are homotopy equivalent, as an example of removing a single point ...
0
votes
0answers
30 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 $az=\overline{z}$. ...
1
vote
2answers
70 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
2answers
39 views

Simply Connected Points in Disk

Why is the set of all points $z \in D^2$ for which $D^2 \setminus \{z\}$ is simply connected just $S^2$?
3
votes
1answer
37 views

functoriality of $K(G,1)$ spaces in a particular situation involving complex elliptic curves

I apologize if the subject doesn't accurately describe my question. Let $F_2$ denote the free group on two generators. Suppose you have some group homomorphism $A : ...
1
vote
1answer
50 views

If $X \times \{0\} \cup A \times I$ is closed in $X \times I$. Then, is $A$ closed in $X$?

Let $X$ be a Hausdorff topological space, $A$ subset of $X$. Let $I$ be an interval $[0,1] \subset {\mathbb R}$. Suppose that $X \times \{0\} \cup A \times I$ is closed in $X \times I$. Then, is $A$ ...
0
votes
1answer
25 views

Mapping Class Group of $S^3$

I am wondering if we can compute $\pi_0(Homeo(S^3))$ (i.e. the group of hoemomorphisms of the three-sphere mod isotopy) or if anyone has a reference where I could find such information.
5
votes
1answer
113 views

What exactly is a dimension?

Maybe this is too broad a question, maybe I need to be more specific. I am just clearing my head here, feel free to ignore at your pleasure. In Linear Algebra, we learned that the dimension of a ...
4
votes
2answers
400 views

How to think about a homeomorphism?

Two disjoint circles in the Euclidean space are homeomorphic to two circles interlocked without touching each other. My professor said that to a topologist they are the same thing. I don't understand ...
1
vote
0answers
27 views

Identification topology and disjoint unions

I was reading the book Basic Topology by M.A. Armstrong and I came across something I couldn't understand. I have uploaded the relevant pages- (1) What exactly is a disjoint union and what is the ...
0
votes
1answer
27 views

Showing that a map can be deformed into the identity.

Suppose $F((a,b), k) = (ae^{\pi i k}, be^{\pi i k})$ where $0 \leq k \leq 1$. Now would $g(a,b)$ = $(-a,-b)$ if $g : S^{1} \rightarrow S^{1}$?
0
votes
0answers
31 views

Question about singular homology

in order to prove that $H_0(X)\simeq \mathbb{F}$, $\mathbb{F}$ is the unitary commutative ring we have to prove that $C_0(X)/B_0(X)\simeq \mathbb{F}$ since we have that $C_0(X)$ is generated by the ...
1
vote
1answer
32 views

Full (or universal) group C*-algebra of discrete group $\Gamma$

There is a quotation below (C*-Algebras and Finite-Dimensional Approximations): $ \qquad$We denote the $group~ring$ of $\Gamma$ by $\mathbb{C}[\Gamma]$. By definition, it is the set of formal sums ...
0
votes
2answers
42 views

A question on discrete group

There is a quotation below: For a discrete group $\Gamma$ , $f\in l^{\infty}(\Gamma)$ and $s, t\in \Gamma$. We let $s.f \in l^{\infty}(\Gamma)$ be the function $s.f(t)=f(s^{-1}t)$. My question is ...
0
votes
2answers
60 views

Prove Simply Connected

If $X = U \cup V$ with $U,V$ open and simply connected and $U \cap V$ is path connected, why is $X$ simply connected?
0
votes
1answer
34 views

Deformation retraction onto the boundary

If I have a square and I remove an open disc from its interior, there exists a deformation retraction onto its boundary. Is this also the case, if I remove a closed disc from its interior? Does the ...
2
votes
0answers
43 views

What does base point by us for algebraic topology?

This may be a vague quesion. I am confusing between base pointed case and non base pointed case in algebraic topology. Is there any convinience in base pointed case? For example, it leads to the ...
3
votes
0answers
44 views

homomorphism of fundamental groups induced by a map of two surfaces

I am trying to find another proof of the following theorem Theorem Let $X$ and $Y$ be two compact surfaces of genus greater than $2$. Then every homomorphism $π_1(X,x_0)→π_1(Y,y_0)$ is induced by a ...
0
votes
0answers
23 views

History of vectorial bundles in articles or papers?

I'm looking for an article or book that gives a thorough and interesting history of bundles and vectorial bundles in algebraic topology. I'm looking for it for my own learning, please help its ...
1
vote
1answer
41 views

Why does the intersection change to a union in $r^{-1}(\bigcap r(V_i\cap W_i))=\bigcup V_i\cap W_i$?

Let $q: X\to Y$ and $r:Y\to Z$ be covering maps, $p=r\circ q$. If $r^{-1}(z)$ is finite for each $z$ in $Z$, $p$ is a covering map. There is a proof on ask a topologist, but I can't follow why ...
2
votes
1answer
25 views

Density, Irreducible Topological Space

Let X is a irreducible noetherian topological space, and $U \subseteq X$ is a nonempty open subset. If $B \subseteq U$ is dense, then so is $B \subseteq X$. Is this true or false, and why? Here ...
0
votes
2answers
42 views

Topological dimension and derham cohomological dimension

If G is a compact complex manifold then does the topological dimension bound the deRham cohomological dimension below? By derham cohomological dimension, I mean the largest extended natrual number ...
3
votes
1answer
66 views

Questions on “simple-connectedness-like” property

I wanted to know if there's any notion which is very similar to the simple connectedness, but defined "purely" in terms of points and sets. Here's my attempt to do it. Let $X$ be a topological space. ...
1
vote
1answer
44 views

If $X$ is homeomorphic to $Y$, does every map $X \to X$ factor through a map $Y \to Y$?

Let $X$ and $Y$ be topological spaces, and $h:X \to Y$ a homeomorphism. For every continuous map $f:X \to X$, is there a continuous map $g:Y \to Y$ such that $f=h^{-1} \circ g \circ h$? This came up ...
1
vote
0answers
23 views

Open Book Decompositions of 3-manifold and Associated Heegard Splittings

In page 13 of the paper: http://arxiv.org/pdf/math/0510639v1.pdf It is stated that "An open book decomposition (S,h, K) , gives rise to a special Heegard decomposition of M ". Here, S is a surface , ...
0
votes
1answer
39 views

Is a Covering Space of a Topological Space always Hausdorff?

Is a Covering Space of a Topological Space always Hausdorff? I can separate two different points from the same fiber, but what about two arbitrary points?
1
vote
2answers
46 views

A more general definition of branched covering.

If $f:X\longrightarrow Y$ is a holomorphic map between two compact Riemann surfaces, then $f$ is called also a branched covering map. This because the branched points of $f$ form a finite set ...
10
votes
1answer
102 views

How can I understand the three-dimensional space forms?

Here is what I know: A space form is defined as a manifold admitting a Riemannian manifold of constant sectional curvature A classical result of Cartan states that a manifold is a space form if and ...
7
votes
2answers
118 views

Is there any point-set definition of simple connectedness?

The definition of path-connectedness refers to the set of real numbers, $\mathbb{R}$. (More precisely, the interval $[0,1]$) On the other hand, connectedness is defined "purely" in terms of points and ...
0
votes
1answer
50 views

Manifolds Resulting from Gluing Tori

I'm trying to show that if solid tori $T_1, T_2; T_i=S^1 \times D^2$ ,are glued by homeomorphisms between their respective boundaries, then the homeomorphism type of the identification space depends ...
1
vote
0answers
31 views

Fundamental groups of configuration spaces

In a previous answer see here by Samuel Reid, I read the following: "The configuration space of $n$ points in a topological space $X$ is usually defined to be, $$C_{\hat{n}}(X) = \{(z_{1},...,z_{n}) ...
0
votes
1answer
26 views

Is it possible to compute homology groups of a space given the Pontryagin ring?

Or similarly, given the cohomology ring of a space, is it possible to compute its cohomology groups? I'm mainly interested in integer and mod 2 homology and cohomology.
2
votes
1answer
50 views

If M is a non-orientable closed connected 3 manifold prove H1(M) is an infinite group.

This is an example from a question sheet (non-assessed) of a university class. If M is a non-orientable, closed, connected 3 manifold, prove $H_1(M;\mathbb{Z})$ is an infinite group. I know that since ...
0
votes
0answers
40 views

How to deform a curve in specific manner

I am wondering whether we can deform a path in specific ways continuously i mean if there is a closed piece wise $C^1$ smooth path which has to be deformed to another piece wise $C^1$ smooth path. Let ...