-2
votes
1answer
65 views

Cw complex $\Sigma_g$

Consider the oriented connected compact surface $\Sigma_g$ of genus $g$ with its standard CW structure. How do I write down the attaching map for the single $2$-cell and how can it be proven that it ...
1
vote
1answer
18 views

Topological Tensor Product is a Topological Ring Independent of the Choice of Basis

Let $A, B$ be commutative rings containing a field $k$, with $B$ a finite dimensional $k$-module, $w_1, ... , w_N$ a basis. If $w_iw_j = \sum\limits_{n=1}^N c_{ijn}w_n$, then we can define $C$ to be ...
2
votes
0answers
23 views

Good pair vs. cofibration

It can be shown that $i:A\hookrightarrow X$ is a closed cofibration if and only if there is a map $\varphi:X\to I=[0,1]$ and a homotopy $H:U\times I\to X$ on some neighborhood $U$ of $A$ such that ...
1
vote
1answer
42 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
59 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
1answer
46 views

Universal cover of the pinched sphere?

Consider the sphere $S^2$ and identify its north and south poles to get a "pinched" sphere. What is the universal cover of this space?
2
votes
1answer
83 views

$\mathbb{R}^2$ is a Retract of $\mathbb{R}^5$

Looking through some old qualifying exams, I noticed the question: Let $X\subseteq \mathbb{R}^5$ be homeomorphic to $\mathbb{R}^2$. Prove that $X$ is a retract of $\mathbb{R}^5$. I'm not even sure ...
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 ...
3
votes
2answers
71 views

every continuous map $S^1 \rightarrow S^1$ can be extended to continuous map $B^2 \rightarrow B^2$

Let $S^1$ denote the unit circle, and $B^2$ denote the closed unit disk. I came across this question and got stuck: Q:) Every continuous map $f :S^1 \rightarrow S^1$ can be extended to continuous map ...
0
votes
1answer
22 views

a region homeomorphic with klein bottle

prove that if we consider this shape in the picture below with the equivalency relation that : a & b are in one class if they are antipoles in inner or outer circles, then the induced quotient ...
-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 ...
1
vote
0answers
36 views

constructing the klein bottle by gluing the sides of a triangle

can any one tell me why gluing the sides as in this picture would make a klein bottle for us? thank you
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 ...
1
vote
1answer
34 views

Is $\textrm{im}(f)$ homeomorphic to the torus less the inner equator?

Consider the map $id_{S^1}\times f:S^1\times [0, 1]\longrightarrow S^1\times S^1$ where $f:[0, 1]\longrightarrow S^1$ is given by $$f(t)=(\cos(\pi t), \sin(\pi t)).$$ Is it true that $\textrm{im}(f)$ ...
2
votes
2answers
97 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 ...
5
votes
2answers
81 views

Explicit expression for homeomorphism and homotopy equivalence

In many cases of topology when one needs to show two spaces $X$ and $Y$ are homeomorphic or homotopy equivalent, one uses some description instead of constructing an explicit homeomorphism or homotopy ...
4
votes
2answers
76 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 ...
1
vote
1answer
30 views

Closed regular neighborhood

I would like to understand the following sentences. Let $L$ be a framed link in the three dimensional sphere $S^3$. Suppose $L$ has $m$ components $L_1, \cdots, L_m$. Let $U$ be a closed regular ...
0
votes
0answers
52 views

Finite covering space with compact spce.

Prove that if $p: \ Y \rightarrow X$ is finite covering, then if $Y$ is compact so it is X. Can someone check my attempt? :) Let $\mathcal{U}$ be any open cover of $X$. For every $x \in X$ let us ...
0
votes
0answers
25 views

every path in $X$ is homotopic with endpoints fixed to a path passing through $b$

$X$ is path connected and b$\in$X, show every path in $X$ is homotopic with endpoints fixed to a path passing through $b$ This is the hint in the book: Let $\gamma$ be a path from $x$ to $y$. If ...
2
votes
1answer
25 views

Degree of a restriction of a continuous map?

I have a map $f:D^2 \rightarrow S^2$ and $f(-x)=-f(x)$ for $x \in S^1$. Does this mean that $\deg(f|_{S^1})=0$? if so, why? We defined this degree on $S^1$ as $f(\exp(t))=\exp(F(t))$ then ...
1
vote
2answers
70 views

Fundamental groups and some properties

I have some basic questions about fundamental groups that came up when I tried to prove a few things: I am sorry that they are kind of informal questions, but I could not find any answers to them in ...
0
votes
2answers
52 views

Fundamental group of torus by van Kampens theorem

So I am currently going through some lecture notes where the fundamental group of a torus is calculated by van Kampen's theorem: The torus is decomposed into its characteristic fundamental polygon ...
0
votes
1answer
45 views

Questions about van Kampen's theorem.

I just read some things about van Kampen's theorem that threat this one from a different perspective than we discussed in class and this brought up a few questions: It was said that the images of the ...
0
votes
1answer
51 views

Determine if these spaces are connected, Hausdorff, or compact.

Let $X = [0,1]/(0,1)$ and let $\pi: [0,1] \rightarrow X$ be the quotient map. Answer the following questions, proving your assertions: a) Is $X$ contractible? We need $s:X \rightarrow ...
1
vote
2answers
57 views

boundary of $M \times I$ where $M$ is the Möbius band

Let $M$ be the Möbius band and $I$ be the closed interval $[0,1]$. What is the boundary of $M \times I$? Is it orientable? What can I do when I want to know the boundary of such space? Please give an ...
3
votes
1answer
42 views

Local topological properties

Could we define the terms locally connected/compact/contractible/simply-connected/whatever to mean that there is a basis (for the topology on our space) of ...
1
vote
1answer
43 views

Determining if certian properties of a topological space pass to its image under a quotient map.

A property $P$ of topological spaces is said to "pass to quotients" if whenever $p : X \rightarrow Y$ is a quotient map and $X$ has property $P$ then $Y$ has property $P$. For the following ...
3
votes
1answer
59 views

Question about Hatcher's book CW complex

I am currently reading in Hatcher's book at page 522 about the construction of open sets in a CW complex. They start with an arbitrary set $A \subset X$ and want to construct an open neighborhood ...
1
vote
1answer
34 views

Winding number from complex analysis and differential geometry

I showed that for a differentiable function $f:S^1 \rightarrow S^1$, the winding number is given by $\frac{1}{2 \pi i } \int_{S^1} \frac{f'(z)}{f(z)} dz$. Now I want to show that the winding number ...
10
votes
1answer
231 views

What is the Atiyah-Singer index theorem about?

I was just a little bit curious about the general statement of this theorem. Honestly, I am not at all interested in fully understanding this, so it is not that I am too lazy to read plenty of books ...
1
vote
1answer
64 views

Number of connected components of this complement

Let $X$ be a locally finite simplicial complex and let $K$ be a finite subcomplex of $X$. Why is the number of connected components of the complement $X-K$ finite?
4
votes
0answers
47 views

Does $\pi_0$ commute with sequential colimits?

I'm looking for an example of a sequence of topological spaces $Y_1 \rightarrow Y_2 \rightarrow \cdots$ such that the induced map $$ \text{colim}\, \pi_0(Y_i) \rightarrow \pi_0 (\text{colim}\, Y_i) $$ ...
2
votes
0answers
28 views

example of a regular Y such that F(X,Y) with open-compact topology is not regular? [closed]

this question from (elementary topology by s.willard ) page 288 give an example of a regular Y such that F(X,Y) (space of function not space of continuous function) with open-compact topology is not ...
1
vote
1answer
39 views

Prove $j_{*}:\pi_{1}(X,b) \rightarrow \pi_{1}(Y,b)$ is surjective for certain X,Y.

Let $P=(2,0)$ and $O=(0,0)$. Let $Y=\mathbb{R}^2\backslash\{O\}$ and $X = \mathbb{R}^{2}\backslash \{O,P\}$ and let $j:X \rightarrow Y$ be an inclusion. Prove $j_{*}:\pi_{1}(X,b) \rightarrow ...
0
votes
1answer
34 views

Lifting property of a covering space

A common example of a covering space is $p:\mathbb R \rightarrow S^1 $, $p(t)= e^{2 \pi it}$, which can be looked at as an infinitely long spiral placed over a circle. Consider the double loop ...
0
votes
1answer
29 views

A question about abstract simplicial complexes and discs.

I find the following definitions in a book about algebraic topology: Definition: Let $K$ be an abstract simplicial complex. $(1)$ If $K$ is finite, simply connected and with nonempty ...
0
votes
1answer
51 views

How does one give topological structure to an abstract simplicial complex?

Given an abstract simplicial complex $K,$ I'd like to know how I can endow it with a topology.
8
votes
2answers
157 views

What is the way to see $(S^1\times S^1)/(S^1\vee S^1)\simeq S^2$?

What is the way to see $S^1\times S^1/(S^1\vee S^1)\simeq S^2$? Even just an intuitive walkthrough. I can't visualize this quotient in my head.
1
vote
1answer
42 views

Gluing $2$-cells when viewing $S^2$ with antipodal points on the equator $S^1$ identified.

Consider the space $X$ which is $S^2$ with $x\sim -x$ for $x$ on the equator $S^1$. I was reading this answer here. When putting a cell structure on $X$, it says the two hemispheres are wound twice ...
0
votes
1answer
25 views

Euler class is odd under orientation, thus its integral over a manifold will be even.

I learned a statement from others: "Euler class is odd under orientation, thus its integral over a manifold $M$ is even." I cannot fully appreciate it, can someone show this explicitly? The ...
3
votes
1answer
77 views

Topological degree of a complex valued map defined over a circle

Given a continuous map $f \colon S^n \to S^n$, it induces a map $f_{*} \colon \tilde{H}_n(S^n) \to \tilde{H}_n(S^n)$ of the form $f_{*}(z)=k*z$, where $k$ is an integer. Define the degree of $f$ as ...
2
votes
1answer
81 views

Open covers by simply connected sets and fundamental group

I have a set $X$ which is path connected and it have an open cover by sets $U$ and $V$ which are simply connected, I am looking for a reference that shows that $\pi_1(X)$ is the free group with number ...
5
votes
2answers
193 views

Does the Euler characteristic of a manifold depend upon the field of coefficients?

Define the Euler characteristic of a space $X$ to be $$\chi(X)= \sum_i \dim H_i(X, \mathbb Q)$$ This is obviously not necessarily well-defined for an arbitrary space $X$, so let $X$ be a manifold ...
0
votes
0answers
47 views

Video lectures about algebraic topology

I am looking for a video lectures about algebraic topology in graduate level course. If someone know some site, please let me know
0
votes
0answers
38 views

Classification of closed surfaces

I am doing a course in topology and is currently working on the classification theorem for closed surfaces. After realizing that every closed surface is either homeomorphic to the sphere or the sphere ...
3
votes
1answer
77 views

Are two spaces obtained from homeomorphic spaces by removing a ball still homeomorphic?

I have a specific example in mind. Consider $S_1,S_2$ two surfaces. Remove two discs to obtain surfaces with boundary $S_1',S_2'.$ If $S_1 \cong S_2,$ does it necessarily follow that $S_1' \cong ...
6
votes
1answer
58 views

Baumslag–Solitar $B(1,2)$ is not hyperbolic

I have a question which asks me to show that the Baumslag–Solitar $B(1,2)$ is not hyperbolic by considering its Cayley graph and showing that triangles can be arbitrarily fat. The Cayley graph can be ...
1
vote
0answers
30 views

Can a factor map be a Serre fibration?

Let $D_n$ be an $n$-disc. Is the factor map $p: D_n\to D_n/S^{n-1}\simeq S^n$ a Serre fibration, in other words, can any homotopy $F: [0,1]\times X\to S^n$ be lifted to $\tilde{F}: [0,1]\times X\to ...
3
votes
2answers
51 views

How to show $\frac{X\sqcup_{S^1}D^2}{D^2}\simeq \frac{X}{S^1}$

Let $X$ be a $CW$-complex which contains $S^1$. How to show $X\sqcup_{S^1} D^2/D^2$ is homeomorphic to $X/S^1$? Here $D^2$ is two dimensional closed unit disc in $\mathbb R^2$. My Atempt: Let ...