Questions about algebraic methods and invariants to study and classify topological spaces: homotopy groups, (co)-homology groups, fundamental groups, covering spaces, and beyond.

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4
votes
1answer
843 views

deformation-retracts into a point / contractible : what is the difference?

Okay, I keep reading the definitions over and over, but I don't see the difference between the two ; apparently spaces that deformation retract onto a point are contractible, but the opposite is not ...
2
votes
1answer
184 views

What is the fundamental group of $X=S^2\cup\text{a disc parallel to the plane of equator}$?

How I imagine it: I think a cell structure of $X$ consists of 1 0-cell, 1 1-cell (the equator)-call it $a$, 3 2-cells (north-south hemispheres and the disc). But I'm not sure about the 2-cells, ...
21
votes
2answers
825 views

How do you define the “boundary” of a topological space?

As described here (and as I always thought was the most general definition of boundary), a possible definition of the boundary of a subset $S$ of a topological space $X$ is $\partial S = \overline S \...
2
votes
0answers
1k views

Two definition of simply connected region in complex analysis.

I have some question on the definition of simply connected region in complex analysis. In the textbook of complex analysis I have, the author defined the definition of simply connected region as ...
2
votes
1answer
209 views

The definition of a contractible space.

I wonder if my approach is completely wrong. If so, may I request for some hints for heading to the right direction? Thank you! Show that a retract of a contractible space is contractible. The ...
3
votes
0answers
479 views

Hatcher's Algebraic Topology Problem 0.6bc - is this proof legit?

I am tackling Hatcher's Algebraic Topology Problem 6b. But I am wonder if my proof is okay, or that is too descriptive and I need to carry out explicit computation in coordinates? Also, I am not sure ...
2
votes
0answers
301 views

Hatcher's Algebraic Topology 0.6(a) - Is this proof legit?

I am primarily not sure about my two-step homotopy construction: Let $X$ be the subspace of $\mathbb{R}^2$ consisting of the horizontal segment $[0, 1] \times \{0\}$ together with all the ...
0
votes
1answer
234 views

Hatcher's Algebraic Topology 0.6(a)

Let $X$ be the subspace of $\mathbb{R}^2$ consisting of the horizontal segment $[0, 1] \times \{0\}$ together with all the vertical segments $\{r\}\times[0, 1 - r]$ for $r$ a rational number in [0,...
0
votes
0answers
76 views

Is a uniquely geodesic space contractible? II

Is a uniquely geodesic space, contractible ? With the extra assumption that closed metric balls are compact, there is an answer here. We expect here an answer beyond this extra assumption (...
1
vote
1answer
100 views

fundamental group of punctuated plane

Let $X$ be the plane punctuated at the origin. Let $C$ be the unit circle, with each point being identified by an angle between $0$ and $2\pi$. $f$ is a function $[0,1] \rightarrow C$ so that $f(t)=2\...
2
votes
0answers
388 views

The inclusion map $V\subset U$ is nullhomotopic. - Is this proof legit?

There are many discussions has done on Hatcher 0.5, some here, and some there. But I am primarily uncertain about the invoke of tube lemma. Wonder if someone would kind help me take a look at my proof ...
1
vote
1answer
526 views

Cohomology of complex projective space

Hello : I would like to know how to compute the cohomology of complexe projective space : $ H^p ( \mathbb{P}^n ( \mathbb{C} ) , \mathbb{Z} ) $. Thanks a lot.
2
votes
1answer
332 views

Construct an explicit deformation retraction of $\mathbb{R}^n - \{0\}$ onto $S^{n−1}$.

Just getting started with Hatcher, wondering if I get the right idea at the beginning? Problem 0.2, Page 18: Construct an explicit deformation retraction of $\mathbb{R}^n - \{0\}$ onto $S^{n−1}$. ...
7
votes
1answer
602 views

Turning higher spheres inside out

I know that one can turn a sphere $S^2$ in $\mathbb{R}^3$ inside-out having at each time an immersion of $S^2$ into $\mathbb{R}^3$. It is called Smale paradox. There is beautiful animation about that. ...
8
votes
1answer
883 views

constructing a genus 2 surface from 8-gon

I am requesting some help or reference for visualization? I am having a hard time constructing a genus 2 surface from 8-gon. May I request for some reference? Here's the construction I used from ...
0
votes
0answers
154 views

What's the classification of CW complexes formed by gluing a 2-cell to a circle?

After this answer, the following question comes : What's the classification (up to homeo.) of CW complexes formed by gluing a 2-cell to a circle ?
7
votes
2answers
653 views

Is a CW complex, homeomorphic to a regular CW complex?

Is a CW complex, homeomorphic to a regular CW complex ? Regular means that the attaching maps are homeomorphisms (1-1). In particular, an open (resp. closed) $n$-cell, is homeomorphic to an open (...
2
votes
1answer
159 views

Is a uniquely geodesic space contractible? I

Is a uniquely geodesic space contractible ? We assume in addition that closed metric balls are compact. A post without this extra assumption is here.
0
votes
1answer
105 views

Is the closure of an open bounded convex set already a ball?

Does the "closure of an open bounded convex set in ${R}^n$ symmetric wrt. the origin" has to be already homeomorphic to a ball? (My motivation is this: one version of Borsuks theorem says that if $f:...
5
votes
1answer
340 views

Homework: The Klein bottle retracts onto a loop.

From the UCLA topology qualifying exam, Fall 2011: Let $K$ be the Klein bottle. Show that there exist homotopically nontrivial simple closed curves $\gamma_1,\gamma_2$ such that $K$ retracts to $\...
3
votes
1answer
205 views

Homotopy problem for infinite dimensional topological space III

This post here is a specification of this post. Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of complete geodesic metric spaces verifying : $X_{n}$ is a $n$-dimensional regular CW complex. $...
2
votes
1answer
76 views

An issue with computing the attachment map for $S^2$

I am watching a video here: http://youtu.be/KVjRILkbILA?t=1m42s one can see that the author explains that: $$\rho_{\alpha}:S^1 \rightarrow e^0$$ The problem that I am having is how he got there, ...
4
votes
1answer
77 views

When is the functor $\otimes:\mathcal{C}^\omega\times\mathcal{C}^\omega$ is monoidal functor?

Admittedly, the question in the title is not a yet a precise question, so I must make it a precise question. Let $\mathcal{C}$ be a monoidal category, with monoidal product, $\otimes$. We will let $\...
1
vote
1answer
115 views

Group Extension and Classifying Space

If $$ 0 \to H \to G \to G/H \to 0\ $$ is a group extension, under what conditions do we have a fibration of the form $$ BH \to BG \to B(G/H), $$ where $BG$ is a classifying space of $G$? Suppose ...
0
votes
1answer
44 views

Deformation retract needs to be smooth?

So I am not quite sure that why none of these three is a deformation retract - is that because of the corners? But I don't remember deformation retract rely on smooth criteria, instead, on continuous ...
1
vote
1answer
106 views

Hatcher's Algebraic Topology Chapter 0 at Page 3

I have a question from Hatcher's Algebraic Topology Chapter 0 at Page 3: http://www.math.cornell.edu/~hatcher/AT/ATch0.pdf One could equally well regard a retraction as a map $X\to A$ restricting ...
0
votes
1answer
108 views

Why does the letter $X$ retract onto a point? (Hatcher's Algebraic Topology, Chapter 0, pg 2)

I am stumble at a statement from Hatcher's Algebraic Topology Chapter 0 at Page 2 which can be found here. The thick $\mathbf{X}$ deformation retracts to the thin $X$, which in turn deformation ...
2
votes
1answer
257 views

Find the Fundamental group of this space

Let X be the space obtained from $S^2$ by identifying (x; y; 0) with (-x; -y; 0), for all (x; y; 0)$\in$ S2. Compute $\pi_(X).$. I know to choose the open sets so that they each deformation retract ...
3
votes
3answers
328 views

Uniquely geodesic spaces

The purpose of this list issue is to better understand the class of uniquely geodesic spaces. I'm looking for two different things : Overclass : for example geodesic space or contractible space. ...
1
vote
0answers
63 views

Dimension of the homology group with coefficients in $\mathbb{Z}/2\mathbb{Z}$.

Charles Weibel writes in his survey of homological algebra Riemann de fined a surface $S$ to be $(n + 1)$-fold connected if there exists a family $C$ of $n$ closed curves $C_j$ on $S$ such that ...
0
votes
1answer
313 views

Equivalence of path-connected CW-complexes and CW-complexes with one 0-cell

Proposition Any path-connected CW-complex is homotopy equivalent to a CW-complex with precisely one 0-cell. Proof (Sketch) Let $X$ be a path-connected CW-complex, so $sk_1(X)$ is a connected graph. ...
6
votes
1answer
218 views

Retractions and fundamental groups

On page 39 of Allen Hatcher's Algebraic topology (page 19 in the pdf document available on Allen Hatcher's website), exercise 16 c), I have the following proof : Let $\varphi : S^1 \to S^1 \times D^...
1
vote
1answer
193 views

Stiefel classes and generic sections

One can say that the Stiefel-Whitney classes is dual classes to the locus of linearly dependence of generic sections. What means "generic"? It would be great, if you show me some relation in local ...
3
votes
2answers
1k views

Characteristic map of a n-cell in a CW complex

I have a problem in understanding the purpose of the definition of a CW complex. What really would help me is to understand the following: Let $\sigma$ be a n-cell and $\Phi_\sigma:\mathbb D^n \to X$ ...
0
votes
1answer
39 views

SvKT application to CW-complexes

There are a few things I didn't understand about the proof, could you help me? Proposition Let $X$ be a topological space and denote the homotopy class of $\phi: ∂D^2\to X$ by $[\phi]$. Let $Y$ be ...
0
votes
1answer
61 views

The fundamental group of a band

I need some help with this please, my problem is to find the fundamental group of a "band".I have the circular region that we obtain from taking away an open disk from a closed disk in R^2 and then ...
8
votes
1answer
381 views

Calculating homology of a Klein bottle (Using only axioms)

so let K be the Klein Bottle. I am perfectly aware how to calculate it for singular homology, using some properties of chains, and an explicit description of the differential. This is not my problem. ...
3
votes
1answer
218 views

Prove that this covering map is a homeomorphism

Let $p \colon E \to X$ be a covering map. Let $s \colon X \to E$ be continuous. If $p \circ s = \operatorname{id}_{X}$, show that $p$ is a homeomorphism. We know that $p$ is a continuous surjection. ...
2
votes
2answers
205 views

Algebraic Topology: CW complexes and their associated $n$-skeletons questions.

The following is from Hatcher's book on Algebraic Topology: we first start with a discrete set $X^0$ whose points are $0$-cells. Then "Inductively, form the n skeleton $X^n$ from $X^{n−1}$ by ...
2
votes
1answer
88 views

Excisive triads and weak equivalences

Let us suppose that $f(X',A',B',x_0') \rightarrow (X;A,B,x_0)$ is a map of triads such that $$f_\ast:\pi_\ast (A' \cap B',x_0') \rightarrow \pi_\ast(A \cap B, x_0)$$ $$f_\ast:\pi_\ast(A',x_0')\...
4
votes
1answer
161 views

4-manifold: $0$-handle $\cup$ $2$-handles along a framed link in $S^3$ (intersection form = linking matrix = presentation matrix of $H_1(\partial M)$)

Let $L$ be a framed link in $S^3$, consisting of framed knots $L_1,\ldots,L_m$. Let $A=[a_{ij}]\in\mathbb{Z}^{m\times m}$ be its linking matrix, with $a_{ii}=$ framing of $L_i$ and $a_{ij}=$ linking ...
3
votes
1answer
115 views

A specific example of a CW complex and a few questions concerning it.

The question I am facing is this one: Construct a CW complex X with a 0-cell x(n) for each natural number $n \geq 0$ and a 1-cell $D_{n}^1, n \geq 1$ which is glued to $x(0)$ at one end and $x(n)$ at ...
4
votes
1answer
197 views

Doubt from Allen Hatcher's AT about CW Complex

I am trying to learn Cellular Homology from Allen Hatcher's AT book, but stuck in first Lemma (2.34) itself. While introducing Cellular Homology Hatcher in his AT, Section 2.2 Lemma 2.34 says $X^n/X^{...
5
votes
2answers
432 views

Mapping cylinder of $z\rightarrow z^2$

A question was asked in my topology course the other day (not an assignment for credit). Let $f:S^1\rightarrow S^1$ by $f(z)=z^2$ ($S^1$ is considered to be in the complex plane). What is the mapping ...
3
votes
0answers
97 views

Understanding the proof of the Seifert-van Kampen theorem

Here is the bit of the proof I didn't really understand: Let $X=X_1\cup X_2$, where $X_1$, $X_2$, $X_1\cap X_2$ and $X$ are path-connected and $X_1, X_2$ are open subsets of $X$. Moreover, assume ...
1
vote
1answer
111 views

Why is an $E_\infty$-operad a kind of ''strictification'' for a non-commutative operation?

A topological space $X$ which is an algebra over an $E_\infty$-operad $E$ consists of a sequence of maps $$ \mu_n':E(n)\times X^n\to X $$ with compatibility conditions. The spaces $E(n)$ are ...
3
votes
2answers
103 views

Example of an oriented manifold with cohomology not isomorphic to a homogeneus space

The question as in the title: Is there a simple example of a compact orientable smooth finite-dimensional manifold whose singular cohomology groups with integer coefficients are not isomorphic to ...
9
votes
0answers
319 views

De Rham Cohomology of $M \times \mathbb{S}^1$

Let $M$ be a closed (compact, without boundary) $m$-dimensional manifold. I want to prove that $H^{k+1}(M \times \mathbb{S}^1) = H^k(M) \oplus H^{k+1}(M)$. ($H^k$ is the $k$-th De Rham cohomology ...
4
votes
1answer
96 views

Local triviality condition on line bundles

We recall that a complex line bundle consists of a triple $(\pi,E,B)$ where $E,B$ are topological spaces, $\pi : E \to B$ a continuous map satisfying the following local triviality condition: ...
2
votes
0answers
131 views

When do equivariant quasi-isomorphisms of chain complexes induce a quasi isomorphism on the tensor product

Suppose I have chain complexes $A,B,C,D$ where $A$ and $C$ have right $R$-module structures and $B$ and $D$ have left $R$-module structures, and that I have maps $f:A\to C$ and $g:B\to D$ which ...