Tagged Questions

1
vote
2answers
43 views

Topology - Dunce Cap Homotopy Equivalent to $S^2$

So I'm trying to find two spaces with isomorphic homology groups but where the spaces aren't homotopy equivalent. From my work so far, taking the Dunce Cap as a triangle with the edges identified as ...
2
votes
1answer
49 views

What's the fundamental group of $E^2\setminus Q^2$

Here $E^2$ is the two-dimensional Euclid space and $Q$ is the set of all rational numbers. Regard $E^2\setminus Q^2$ as a subspace of $E^2$. So what's its fundamental group and how to represent it? I ...
1
vote
0answers
70 views

Compactness property

Let $\Omega \subset X$, X: Banach space. Given $\varepsilon \ge 0$, we define the set of $\varepsilon-normals$ to $\Omega$ at $\bar{x}$$\in \Omega$ by:$\widehat N_\varepsilon(\bar ...
3
votes
1answer
61 views

Infinite products of a (finite) group

So I'm having a little trouble understanding the concept of infinite (cartesian) products of a group -- specifically, my notes (and, of course, homework questions) have concepts of, say ...
2
votes
1answer
83 views

Is $\operatorname{Aut}(\mathbb{I})$ isomorphic to $\operatorname{Aut}(\mathbb{I}^2)$?

Is $\def\Aut{\operatorname{Aut}}\Aut(\mathbb{I})$ isomorphic to $\Aut(\mathbb{I}^2)$ ? ($\mathbb{I},\mathbb{I}^2$ have their usual meaning as objects in $\mathsf{Top}$). I show some of one of my ...
1
vote
3answers
60 views

Disc with two points identified

Is a disc $D^2$ with two points on the boundary identified, same as $D^2 \vee D^2$ ? They both have boundary $S^1 \vee S^1$. I am confused because an exercise in Hatcher seems to ask the same question ...
1
vote
2answers
40 views

Complex Solutions to Polynomials

I'm trying to use topology to prove that: $z^{n} + a_{n-1}z^{n-1} + ... + a_{1}z + a_{0} = 0$ has a solution in $\mathbb{C}$ if and only if, for each positive real number $c$, the equation $z^n + ...
4
votes
1answer
48 views

studying the topology of a real algebraic set

Let $f_1,\ldots,f_n \in \mathbb{R}[x_1,\ldots,x_m]$ be polynomials with real coefficients and let $I$ be the ideal that they generate. Denote by $V_{\mathbb{R}}(I)$ the corresponding real variety, ...
4
votes
2answers
55 views

Looking for a (nonlinear) map from n-dimensional cube to an n-dimensional simplex

I am looking for a (nonlinear) map from n-dimensional cube to an n-dimensional simplex; to make it simple, assume the following figure which is showing a sample transformation for the case when $n=2$. ...
3
votes
1answer
79 views

Question on Good Pairs

$(X,A)$ is a good pair if $\exists V\subset X$ s.t. $V$ is a neighbourhood of $A$ that deformation retracts to $A$. Prove that if $(X,A)$ is a good pair, then $(X/A,A/A)$ is also a good pair. How ...
0
votes
0answers
17 views

Form a space X by identifying the boundary of M with C by a homeomorphism. Compute all the homology groups of X. [duplicate]

Let T denote the torus S1×S1 and let M denote the Möbius band. Let C be a simple closed curve in T which bounds a 2-disk. Form a space X by identifying the boundary of M with C by a homeomorphism. ...
4
votes
1answer
65 views

Hatcher 2.2 Exercise 33

The following is a question from Hatcher's "Algebraic Topology": Let $X$ be a space such that $X$ is the union of $n$ open sets $A_i$ with the property that every intersection $A_{i_1}\cap \dots ...
2
votes
2answers
44 views

If two Lie Groups are homomorphic, does that mean that they are homeomorphic?

I am studying Lie groups, and I had a simple question If two Lie Groups are homomorphic, does that mean that they are homeomorphic? I appreciate any help. Thanks in advance.
4
votes
2answers
72 views

Prove rigorously that for two points $x, y \in M$, the spaces $M \ \backslash \{x\}$ and $M \ \backslash \{y\}$ are homeomorphic.

Let $M$ be a connected topological manifold. Prove rigorously that for two points $x, y \in M$, the spaces $M \ \backslash \{x\}$ and $M \ \backslash \{y\}$ are homeomorphic. I am not sure the best ...
1
vote
0answers
39 views

Form a space $X$ by identifying the boundary of $M$ with $C$ by a homeomorphism. Compute all the homology groups of $X$.

Let $T$ denote the torus $S^1\times S^1$ and let $M$ denote the Möbius band. Let $C$ be a simple closed curve in $T$ which bounds a 2-disk. Form a space $X$ by identifying the boundary of $M$ with $C$ ...
1
vote
2answers
47 views

Closed sets in $R^2$ with $d(A,B)=0$ but $A\cap B=\emptyset$

Let $(X,d)$ be a metric space and $A$ and $B$ subsets of $X$. Define the distance $d(A,B)$ to be $d(A,B)=\inf\{d(p,q)\mid p\in A, q\in B\}$. Give an example of two closed subsets $A$ and $B$ of the ...
-1
votes
0answers
71 views

A question about homotopy equivalent [closed]

if $X$ is contractible, for any topological space $Y$ is the product $X\times Y$ homotopy equivalent to $X$.
1
vote
0answers
47 views

Is $Y$ open in $X\cup_f Y$?

Let $X,Y$ be topological spaces, $A\subset X$ - a subspace and $f:A\rightarrow Y$ - a continuous map. Then we can define $X\cup_f Y = X\sqcup Y/\{a\sim f(a)\quad a\in A\}$ Then the composition ...
4
votes
0answers
98 views

Show that $f$ is a homeomorphism of $X$ onto $f(X)$

I am having trouble on the following question. Some help would be much appreciated. Let $X$ be a Hausdorff space and let $f: X \rightarrow \mathbb{R}^n$ be a proper injective continuous function. ...
0
votes
0answers
35 views

Which of the following spaces nontrivially cover themselves?

I am having some difficulties with a qualifying exam question. I would appreciate if someone could give me a little help. Which of the following spaces nontrivially cover themselves? (a) $S^3$ (B) ...
1
vote
1answer
41 views

Suppose one glues a mobius band to the boundary of a disk. What familiar space is this homeomorphic to?

I was doing a problem in algebraic topology and I need to gain knowledge of the following fact to procede. Suppose one glues a mobius band to the boundary of a disk. I want to calculate the ...
6
votes
4answers
67 views

What does it mean for a space to nontrivially cover itself?

I am going through qualifying exam questions and I came to a concept involving covering spaces of whose definition I did not understand. What does it mean for a space to nontrivially cover itself? ...
2
votes
3answers
47 views

Klein Bottle discrete harmonics?

Studying discrete representations of a function, on a $2$ Dimensional compact surface, brings to the use of spherical harmonics for the 2-sphere, and discrete Fourier transformations for the 2-torus. ...
7
votes
0answers
80 views

Existence of a map in a Hilbert space

Let $H$ be an infinite-dimensional Hilbert space, $B$ be its unit ball: $B=\{x\in H: \, \|x\|\leq 1\}$. Does there exist a continuous map $f:H\to H$ such that $f(f(x))=x$ $\forall x\in H$, $f$ has no ...
-1
votes
0answers
28 views

Strong deformation retract, fibration related question [closed]

How to solve the following problem: A is a strong deformation retract of X such that exists a continuous function $\alpha:X\rightarrow I$, $\alpha^{−1}(0)=A$. If $p:E\rightarrow B$ is a fibration ...
2
votes
1answer
45 views

Compact subset in a $\Delta$-complex structure

Suppose that $X$ is a topological space equipped with a delta-complex structure. Suppose that $K\subset X$ is compact. Prove that $K$ meets only finitely many open simplices. I managed to find a ...
3
votes
1answer
66 views

Understanding homotopy types

I am currently self studying algebraic topology from the book "topology and groupoids". I don't understand classifying spaces up to homotopy type. By "I don't understand" I don't mean that I don't ...
2
votes
1answer
44 views

Problem on induced maps in cohomology

I am trying the solve the following problem: Let $g:\mathbb{C}P^\infty\longrightarrow \mathbb{C}P^\infty$ and suppose the induced homomorphism $$g^*:H^2(\mathbb{C}P^\infty)\longrightarrow ...
2
votes
1answer
86 views

Prove that a topological space equipped with a delta-complex structure is Hausdorff

Suppose that $X$ is a topological space equipped with a delta-complex structure. Prove that $X$ is Hausdorff. I can actually "see" why it should be Hausdorff after the hint from Hatcher asks to ...
2
votes
1answer
51 views

Quotient space about identity component

Let $T=\mathbb R/\mathbb Z$ be the circle group, $\mathscr A=C(T)$ be set of continuous function on $T$. $G(\mathscr A)$ denote set of invertible elements in $\mathscr A$, $G_{0}(\mathscr A)$ denote ...
0
votes
0answers
75 views

Homotopy, retraction related question

How to solve the following problem: $A$ is a strong deformation retract of $X$ such that exists a continuous function $α:X\to I$, $\alpha^{−1}({0})=A$. (a) If $H:X\times I\to X$ is a homotopy (rel ...
6
votes
0answers
79 views

Visualize Fourth Homotopy Group of $S^2$

I know $\pi_4(S^2)$ is $\mathbb{Z}_2$. However, I don't know how to visualize it. For example, it is well known that $\pi_3(S^2)=\mathbb{Z}$ can be understood by Hopf Fibration. Elements in ...
3
votes
1answer
127 views

Strong deformation retract

Let $A$ be strong deformation retract of $X$ and $A=\alpha^{-1}(\{0\}) $ for some continuous $\alpha:X\to I$. If $H:X\times I\to X$ is homotopy between $i\circ r $ and $1_X$ (rel $A$) , where ...
0
votes
0answers
35 views

Compactness of covering space

If we have space $X$ with and $n$ sheeted covering space $Y$ is $Y$ compact iff $X$ is? Torus or sphere, make me believe the answer is yes.
5
votes
1answer
56 views

Triangulation of a manifold adapted to a submanifold

I am not extremely proficient in topology, and am concerned with the following question : Given a compact manifold $X$ and a submanifold $Y \subset X$, is it always possible to find a triangulation ...
1
vote
1answer
45 views

Covering infinite sheeted covering of torus

Suppose I have subgroup $H=\operatorname{span}\langle (a,b)\rangle\subset \pi_1(\mathbb{T}^2)=\mathbb{Z}^2$, where $a,b$ are integers where $(a,b)\neq(0,0)$. I know the covering space is $S^1\times ...
1
vote
1answer
186 views

Is the product of two contractible spaces contractible?

If $X$ and $Y$ are contractible spaces, is $X\times Y$ a contractible space?
1
vote
1answer
56 views

induced map homology example

I am having trouble understanding how to compute the induced map for the second homology. For example say I have $\varphi:\mathbb{T}^2\rightarrow \mathbb{T}^2$ that is a self homeomorphism, then what ...
1
vote
1answer
39 views

All the compact covering spaces of torus.

I know the covering spaces of the of a torus $T^2$ are homeomorphic to $T^2,S^1\times\mathbb{R},\mathbb{R}^2$. I am interested in finding all of the covers with covering space $T^2$. The subgroups of ...
2
votes
1answer
31 views

Show that if each $X_k$ is open in $X,$ then $X$ is simply connected.

I am having difficulty with the following Qualifying exam problem. I am not sure what to use. Was wondering if there was an algebraic topology approach, or if there was something that I could use ...
1
vote
2answers
47 views

Homotopy vs homeomorphism.

If we have a topolological space $X$ and 2 auto-homeomorphisms $\phi_1,\phi_2:X\rightarrow X$ then are $\phi_1,\phi_2$ homotopic as well? If so why?
1
vote
1answer
36 views

Are homotopy equivalent path-connected spaces homotopy equivalent as pointed spaces?

Let $(X,x)$ and $(Y,y)$ be path-connected pointed topological spaces. Is it true that the statement ''$X$ and $Y$ are homotopy equivalent'' implies ''$(X,x)$ and $(Y,y)$ are homotopy equivalent as ...
1
vote
1answer
20 views

Analogous notion of knot complements for braids

Knots/links seem to be studied quite a lot for their topological connection to 3-manifolds by considering knot complements in $S^{3}$. Is there an analogous topological entity for braids? They appear ...
7
votes
1answer
218 views

Contractibility of convex set

Suppose that $\Omega$ is a convex open subset of an infinite dimensional vector space $E$ such that $\Omega$ is not contained in any finite dimensional subspace of $E$. Let $Q_m\subset \Omega$ denote ...
0
votes
0answers
37 views

Question on homotopies

I am working through algebraic topology questions and I am having trouble with the following question. Any help is appreciated. Let $X$ be a finite simplicial complex of dimension 1. Prove that ...
0
votes
1answer
93 views

Qualifying Exam Question on Manifolds

I am practicing qualifying exam problems and I am having trouble with the following question. Any help is greatly appreciated. Let $P$ be a polygon with an even number of sides. Suppose that the ...
1
vote
2answers
65 views

Does there exist a retraction for these spaces?

I have some qualifying exam questions using retractions that I do not know how to solve. The only tool that I know for solving retraction problems is by using the fundamental group trick. Here are ...
1
vote
1answer
47 views

Giving $Top(X,Y)$ an appropriate topology

Let $Top$ be the category of topological spaces. Let $X,Y$ be objects in $Top$. Let $F:\mathbb{I}\rightarrow Top(X,Y)$ be a function (I will denote the image of $t$ by $F_t$). Let $F_{*}:X\times ...
2
votes
1answer
23 views

Homotopy equivalence of torus and $\mathbb{C^2}$ without coordinate cross.

I have difficulties with one intermediate result. I know it is right, but it is not obviously for me. I'm trying to prove homotopy equivalence of $\mathbb{C^2}\setminus\{(a,b)\in\mathbb{C^2}\mid ...
2
votes
2answers
47 views

Show the wedge product of two tori is not homotopy equivalent to a 2-manifold.

I was doing a practice qualifying exam test and I was having trouble with the following question. Show the wedge sum of two tori is not homotopy equivalent to a 2-manifold. It seems obvious why it ...

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