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

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2
votes
2answers
117 views

Equivalent definition for a collection of simplices to be a simplicial complex

I am reading the following lemma from Munkres' Elements of Algebraic Topology: Lemma 2.1 A collection $K$ of simplices is a simplicial complex if and only if the follow hold: ...
2
votes
0answers
109 views

Fundamental Groups of Complements of Knots Algorithm

Is there any clear algorithm to compute the fundamental group of complements of knots?
2
votes
1answer
578 views

Surface of genus g is not homotopy equivalent to wedge of cell complexes both with non-trival H_1

I came across this question while studying for a qualifying exam: Prove that a closed orientable surface of genus $g \ge 1$ is not homotopy equivalent to the wedge $X \vee Y$ of two finite cell ...
9
votes
1answer
1k views

Hatcher Problem 1.2.11: Cell decomposition of Mapping Torus $T_f$ of $S^1 \times S^1$

Suppose we have continuous function $f : X \to X$ that sends the basepoint of $X$ to itself, viz. $f(x_0) = x_0$ where $x_0$ is the basepoint of $X$. Recall the definition of the mapping torus $T_f$ ...
5
votes
3answers
548 views

Are these two spaces homotopy equivalent?

Let $X$ be the $2$-sphere with two pairs of points identified, say $(1,0,0) \sim (-1,0,0)$ and $(0,1,0) \sim (0,-1,0)$. Write $Y$ for the wedge sum of two circles with a $2$-sphere: if it matters, the ...
7
votes
1answer
284 views

Confusion on Cech cohomology

From Harvard math qualification exam, 1990. Let $X$ be a smooth manifold with an open cover $N<\infty$ sets $\{B_{n}\}^{N}_{1}$ which are contractible. Assume that $$\pi_{0}(B_{n}\cap B_{m})\le ...
4
votes
2answers
1k views

Hatcher problem 1.2.3 - technicality in proof of simply connectedness

I am trying to prove that $\Bbb{R}^n$ minus finitely many points $x_1,\ldots,x_m$ is simply connected, where $n \geq 3$. For days now I have tried many different arguments but I have found flaws in ...
2
votes
1answer
156 views

Graph Homology and Rank-Nullity Theorem

Let $G$ be a connected, directed graph with $v$ vertices and $e$ edges. According to Massey (Ch. VIII, Section 3), the euler characteristic satisfies \begin{align} v - e = \chi(G) = \text{rank} \, ...
1
vote
1answer
858 views

How to classify 3-sheeted covering space for $S_{1}\vee S_{1}$?

This might be a duplicate. This question also feels routine (it is also the execrise 10, page 88 in Hatcher). From Harvard qualification exam, 1990. Let $X$ be figure eight. 1) How many 3-sheeted, ...
6
votes
1answer
831 views

Intersection form on quotient manifold

I have a simple algebraic topology question. Let $M$ and $N$ be 2-dimensional oriented manifolds (say $H^{2}(M,\mathbb{Z})\cong \mathbb{Z}\alpha_{M}$ and $H^{2}(M,\mathbb{Z})\cong ...
2
votes
1answer
86 views

What is the relationship between $\pi_{2}\overline{X}$ and $\pi_{2}(X)$?

From Harvard qualification exam, 1990. Consider the space $X=\mathbb{S}^{1}\wedge \mathbb{S}^{2}$, alternatively viewing it as a sphere with north and south poles connected. I was asked to: 1): the ...
18
votes
3answers
3k views

Good exercises to do/examples to illustrate Seifert - Van Kampen Theorem

I have just learned about the Seifert-Van Kampen theorem and I find it hard to get my head around. The version of this theorem that I know is the following (given in Hatcher): If $X$ is the ...
3
votes
0answers
573 views

Universal cover of wedge product of circles

I want to ask a question about universal covering of wedge space of two circles. It is known that the universal covering space is the cayley graph. I have another thing in mind which I came up with ...
2
votes
0answers
284 views

A boundary version of Cauchy's Integral Theorem

In Complex Analysis by Kodaira, a more powerful version of Cauchy's Integral Theorem (and consequently formula) was proven. The result generalizes the theorem to the boundary of an open set as follows ...
1
vote
2answers
215 views

CW complex structure on a disk with 2 smaller disks removed

For some reason, I'm having trouble visualizing how to put a CW structure on a disk with 2 smaller disks removed. What I'd like to do is have three 0-cells, five 1-cells, and a single 2-cell. Three of ...
4
votes
2answers
703 views

Hatcher Algebraic Topology 0.24

This is my second question from Hatcher chapter 0 (and final I think). For $X$, $Y$ CW complexes, it asks one to show that $$X \ast Y = S(X \wedge Y)$$ by showing $$X \ast Y/(X \ast y_0 \cup x_0 \ast ...
3
votes
1answer
270 views

Hatcher Question 0.27

I am self studying hatchers book however I have been stuck on some questions. This is one of them. Let (X,A) satisfy homotopy extension property. Let $f: A \rightarrow B$ be a homotopy equivalance. ...
1
vote
0answers
64 views

Can one define “simplicial” EM spaces?

Let $\Delta$ be the well-known simplicial category, and denote with $\widehat\Delta_\bf C$ the category of simplicial objects in $\bf C$ ($\widehat\Delta$ for short). Given $\mathcal ...
2
votes
1answer
446 views

Prove that the cone over $S^1$ is homeomorphic to $D^2$

As the title, I am thinking $$f :\ S^1\times I / S^1\times \lbrace1\rbrace \rightarrow D^2$$ as $f(x,y,z)= (x,y)$ and $$g :\ D^2 \rightarrow S^1\times I / S^1\times\lbrace1\rbrace$$ as ...
5
votes
1answer
188 views

Massey products in the Adams Spectral Sequence

I've never quite 'got' Massey products - this question, I guess, is to work out a small example that might shed some light for me. So following Wikipedia, let $\Gamma$ be a differential graded ...
7
votes
1answer
397 views

Poincaré Duality Example

It is intuitively clear that the Poincaré dual of a ray $\{(r,0);r>0\}$ in $\mathbb{R}^2-\{0\}$ is the form $\frac{1}{2\pi} d\theta$. For some reason I do not understand, I failed to prove this ...
2
votes
2answers
149 views

Showing that $S^2 \times S^3$ is not homotopy equivalent to $S^2 \vee S^3 \vee S^5$

I'm trying to show that $S^2 \times S^3$ is not homotopy equivalent to $S^2 \vee S^3 \vee S^5$. My argument is that removing any point from the first space leaves a space which is connected (since it ...
3
votes
2answers
302 views

The orientation of quotient manifold

If $T$ is a torus and $\mathbb Z_2$ acts on it by $(z_1,z_2)\rightarrow(z^{-1}_1,-z_2)$, then is $T/\mathbb Z_2$ orientable?
3
votes
1answer
82 views

Different ways to wedge spaces

I am not a topologist, so please excuse me if the question is trivial. Suppose I am given three nice, path-connected spaces. Then I can think of two ways to wedge these spaces: join all three at a ...
2
votes
1answer
196 views

Is $K(G,1) = BG$?

Is an Eilenberg-MacLane space $K(G,1)$ the same as the classifying space $BG$ for a group $G$ ? PS: When I tried to submit the question (without this PS), I got the message: "Oops! Your question ...
3
votes
1answer
569 views

“The space of all lines on a plane is an open Möbius Band.”

I have come across this sentence when I was reading something about algebraic topology: The space of all lines on a plane is an open Möbius Band. I don't quite understand this, can anyone ...
8
votes
0answers
167 views

How to prove that my sum coincides with a sequence on the Online Encyclopedia of Integer Sequences?

I have a topological space $X$ whose reduced $\bmod 2$ Betti numbers (that is to say, the dimension of the $\bmod 2$ reduced homology) I computed to be $$\small \dim \tilde{H}_t(X; {\mathbb{Z}}_2) = ...
1
vote
0answers
158 views

Rational cohomology of quotient by group action

Let $X$ be a topological space with a continuous action by a finite group $G$. Hopefully under some assumptions on $X$ one can identify the rational cohomology of $X/G$ with the $G$-invariants of the ...
5
votes
2answers
258 views

How to prove every non-compact, connected 2 dimensional surface is homotopical to a bouquet of flowers?

This is one of my old unsolved questions when I reading Novikov's book on homology theory. I do not know how to prove it because standard triangulation, fundamental diagram, etc does not help and it ...
4
votes
1answer
160 views

A question about homology group

For a pair of spaces $X,Y$ we have $H_*(X)=H_*(Y)$. Can we necessarily find a continuous function $f$ from $X$ to $Y$ or from $Y$ to $X$, such that $f_*$ induces the isomorphism of homology group?
2
votes
0answers
222 views

Importance of well-pointedness (in particular for the pointed mapping cylinder construction)

In a recent question, I asked wether well-pointedness was indeed necessary to make the canonical inclusion $$X\hookrightarrow M_f^{\bullet}$$ a pointed cofibration. Here $X,Y$ are pointed topological ...
1
vote
1answer
226 views

Why this is a covering map between compact Riemann surfaces?

Let $F:X\rightarrow Y$ be a non constant holomorphic map between compact Riemann surfaces , If we delet the branch points (in $Y$) of $F$ and all their pre-images(in $X$), we obtain a map ...
0
votes
1answer
101 views

Algebraic topology involved in the 1/4-pinched sphere theorem?

Can anyone familiar with this theorem and its proof let me know how much algebraic topology is involved, and where specifically? I am familiar with a lot of differential geometry, but not many of the ...
5
votes
0answers
203 views

Homotopy equivalence in the category of arrows.

I'm reading Jeff Strom's book on Homotopy Theory and I am trying to make some sense of a certain exercise. On page 91, "Homotopy in Mapping Categories" we consider the category of arrows of ...
1
vote
0answers
99 views

Analog of a tubular neighborhood for an embedded wedge sum

If you have some embedding of a path connected topological space wedge of spheres $N$ into a compact simply connected smooth $n$ manifold $M$ (like a sphere for example), then is there some kind of ...
5
votes
2answers
176 views

Loopspace of Eilenberg Mac Lane space

Is the loop space of the Eilenberg-MacLane space $K(G,1)$ dependent only on the cardinality of $G$? For instance, is the loop space of $K(\mathbb{Z}_4, 1)$ homotopy equivalent to that of ...
3
votes
0answers
76 views

For which $g,p$ does $\Sigma_{g,p}$ cover $\Sigma_{3,2}$?

I am preparing for my qualifying exams. There is an algebraic topology problem I don't know how to do it. Thanks a lot for your help. Let $\Sigma_{g,p}$ denote the surface of genus $g$ with $p$ ...
7
votes
1answer
542 views

Universal Cover of projective plane glued to Möbius strip

Consider the usual cell structure on $\mathbb R P^2$, with one 1-cell and one 2-cell attached via a map of degree 2. Consider the space $X$ obtained by gluing a Möbius band along the 1-cell via a ...
1
vote
3answers
269 views

Algebraic Topology pamphlets?

I'm looking to self-learn some Algebraic Topology and have found the books I've looked at so far (ie. Hatcher) to be rather tome-like for my tastes. Does anyone know of a good slim lecture notes style ...
15
votes
1answer
886 views

A counterexample in topology

Semi-local simple connectedness is a property that arises in Algebraic Topology in the study of covering spaces, namely, it is a necessary condition for the existence of the universal cover of a ...
8
votes
1answer
706 views

Braid groups and the fundamental group of the configuration space of $n$ points

I am giving a lecture on Braid Groups this month at a seminar and I am confused about how to understand the fundamental group of the configuration space of $n$ points, so I will define some ...
2
votes
1answer
129 views

Homology of $\Sigma_{2}\times S^{1}$?

I'm quite at a loss with this...I want to use Mayer-Vietoris with open covers $A=\Sigma_{2}\times (S^{1}\setminus \{p\})$ and $B=\Sigma_{2}\times (S^{1}\setminus \{q\})$ so that $A$ and $B$ both ...
7
votes
1answer
422 views

Map Surjective on a Disk

I've got another question from a student that has stumped me: Let $D^{n+1}$ be the $n+1$-disk, with boundary sphere $S^n$. Suppose $f:D^{n+1}\longrightarrow \mathbb{R}^{n+1}$ is a map such that ...
0
votes
1answer
148 views

Prove Without Using Borsuk-Ulam

I am teaching a summer course and a student asked me a question he found online. It asks "Show that there is not a continuous mapping $f:S^n\longrightarrow S^1$ with $f(-x)=-f(x)$ for all $x$." The ...
8
votes
2answers
412 views

Some Good Algebraic Topology Exercises

I am teaching a topology prep course for first year graduate students taking their qualifying exams. I have been able to think of about ten days' worth of exercises, but am running out of ideas. ...
13
votes
3answers
1k views

Category Theory usage in Algebraic Topology

First my question: How much category theory should someone studying algebraic topology generally know? Motivation: I am taking my first graduate course in algebraic topology next semester, and, ...
9
votes
1answer
114 views

Showing that a CW space is contractible if it is endowed with a certain binary operation

I am having trouble with the following homework problem, and was hoping someone could provide me with a hint: I am given a connected CW space $X$ which has a continuous associative operation $(x,\ ...
2
votes
1answer
82 views

Questions about complex bordism.

I have some questionss about the construction of the complex bordism ring MU and would appreciate every answer: I have read that the multiplication in MU is given by the tensor product of vector ...
1
vote
1answer
85 views

How do I find the number of vertices on a planar diagram of a surface?

Cheers, I have a question which I just do not seem to see the answer for: I am proving the classification theorem for compact surfaces and use planar diagrams as representation of the surfaces. I ...
0
votes
1answer
126 views

equivalence between reduced and unreduced homology theory

I'm trying to prove that a reduced homology theory can be define from an unreduced one, but the problem is to define a border map for the reduced homology using the unreduced homology groups.