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

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1
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1answer
11 views

What does it mean to attach a cell to a space by a map?

I am starting to study for my algebraic toplogy exam and a lot of the problems sound like: let $Y_p$ be the space obtained by attaching an $(n+1)$-cell to $S_n$ by a map of degree $p$, where $p$ ...
0
votes
2answers
31 views

about a sequence of isometries' convergency.

Let $M$ be a compact metric space, let $(i_n)$ be a sequence of isometries: $M \rightarrow M$. I've already showed that there exists a subsequence $(i_{n_k})$ that converges to $i$ which is also a ...
3
votes
3answers
50 views

Is such a map always null-homotopic?

Let $X,Y$ be CW-complexes with $X$ finite dimensional and $X = \bigcup_{n \in \Bbb N} X_n$ where the $X_n\subset X_{n+1}$ are finite sub-complexes of $X$. If $f: X \rightarrow Y$, with $f|_{X_n}$ ...
1
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0answers
54 views

Exercise 3.4 in Rotman's An Introduction to Algebraic Topology

I am self-learning algebraic topology by reading Rotman's An Introduction to Algebraic Topology. I am stuck on Exercise 3.4 on page 41. I'd be grateful for any hints or solution. Exercise 3.4: Let ...
4
votes
3answers
72 views

Is reduced homology a full functor on connected spaces?

Let $X$ and $Y$ be connected topological spaces. Can we then realize each map between their homologies as coming from a continuous map between their spaces? For any arrow $φ\colon \tilde H_•(X) → ...
0
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1answer
36 views

Attaching maps in a product cell complex

This is a follow-up to those two questions: Cartesian product of two CW-complexes, and Product of CW complexes question. Consider two cell complexes $A$ (with cells $e^m_\alpha$ and attaching maps ...
0
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0answers
60 views

A question about CW complex

Given a space $X$ and a collection of subspaces $X_\alpha$ whose union is $X$, these subspaces generate a possibly finer topology on $X$ by defining a set $A\subset X$ to be open iff $A\cap X_\alpha$ ...
4
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1answer
115 views

long exact sequence in $K$-theory

I am studying the basics of $K$-theory and given a CW pair $(X,A)$ I understand how to construct an long exact exact sequence $\cdots \rightarrow K(SX) \rightarrow K(SA) \rightarrow K(X/A) ...
0
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0answers
19 views

Why use class multiplication in Homotopy groups?

This is a related to a physics question Why use class multiplication to describe topological entangling and merging?. In physics, the homotopy theory is used to describing topological defects in order ...
33
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0answers
918 views

Is there a homology theory that counts connected components of a space?

It is well-known that the generators of the zeroth singular homology group $H_0(X)$ of a space $X$ correspond to the path components of $X$. I have recently learned that for Čech homology the ...
2
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0answers
20 views

Computing $\pi_1(\text{Pr}(S),\mathbb{P}_0)$

Let $(S,d)$ be a complete separable metric space, and consider the space $\text{Pr}(S)$ of probability measures on $S$ that are defined on Borel sets arising from the metric $d$. Now endow ...
12
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4answers
212 views

Fundamental group of the product of 3-tori minus the diagonal

I have a past qual question here: let $T^3 = S^1 \times S^1 \times S^1$ be the 3-torus, and let $\Delta = \{ (x,x) \in T^3 \times T^3 \colon x \in T^3 \}$ be the diagonal subspace. Compute $\pi_1(T^3 ...
0
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1answer
29 views

Deforming $\text{id}: S^1 \to S^1$ to the symmetry $S^1 \to S^1$ such that $x \mapsto -x$

I am trying to find a deformation retraction of $\text{id}: S^1 \to S^1$ to the symmetry $S^1 \to S^1$ such that $x \mapsto -x$. I guess this deformation of maps has to respect all homotopy rules, ...
0
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0answers
14 views

Reference Request: James reduced product

I would like to quickly learn the basics of James reduced product (also called James construction). Anyone know some suitable material for beginners?
1
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0answers
7 views

Citation: earliest incidence of the Borel localization theorem

The Borel localization theorem in (Borel) equivariant cohomology states that if $T$ is a torus and $M$ a smooth $T$-manifold, with fixed point set $M^T$, then upon localizing the coefficient ring ...
5
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1answer
39 views

Module structure in the Serre spectral sequence of the Borel construction

Let $G$ be a finite group, $M$ a reasonable (e.g. a closed manifold) $G$-space. Then there is a fibration $X \to EG \times_G X \to BG$, where $BG$ is the classifying space of $G$ and $EG$ is its ...
5
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0answers
73 views
+50

Quotient Groups and Covering Spaces in Painting Hanging

Consider the $1$-out-of-$n$ painting hanging problem: Given $n$ nails in a wall, how can we hang a painting such that upon removal of any nail, it falls. This has a nice interpretation as a problem in ...
1
vote
1answer
25 views

Can we deform continuously $\text{id}:\mathbb{R}^2 \to \mathbb{R}^2$ into the constant map $\mathbb{R}^2 \to \mathbb{R}^2?$

Can we deform continuously $\text{id}:\mathbb{R}^2 \to \mathbb{R}^2$ where $(x,y) \mapsto (x,y) $, into the constant map $\mathbb{R}^2 \to \mathbb{R}^2$ (where I guess $(x,y)\mapsto (0,0)\,\,$)?$ I ...
2
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2answers
61 views

Product of CW complexes question

I am having trouble understanding the product of CW complexes. I know how to actually do the computations and all, I just don't understand how exactly it works. So here's my questions specifically: ...
0
votes
2answers
15 views

minimal genus of surface representing a homology class

Can you give an example of a 4-manifold with embedded surfaces of different genera representing the same homology class? (If $i:\Sigma\to X$ is an embedding of a closed surface in a closed smooth ...
0
votes
0answers
44 views

how to prove that two sets have the same homotopy type

I want to prove that two bounded sets $A$ and $B$ of $R^2$ have the same Euler characteristic (number of connected components minus number of connected components of the complementary) but I think ...
0
votes
2answers
25 views

Converse of the path lifting lemma

We know that covering spaces have the path lifting property i.e. if $p:E \rightarrow B$ is a covering map and $u:I \rightarrow B$ is a path wish intial point $a$, then for each $w \in p^{-1}(a)$, ...
3
votes
0answers
34 views

Doubt about proof of factorization $f=pi$, where $i$ is acyclic cofibration and $p$ is fibration

I try to understand a proof in More Concise Algebraic Topology: Localization, completions and model categories by May & Ponto (pdf). The proof is on page 262, and it is for the statement Any ...
4
votes
1answer
131 views

Products of CW-complexes

I am currently reading through May's "Algebraic Topology" and in the chapter on CW-complexes he shows that a product of CW-complexes is again a CW-complex, because one can define product cells using ...
1
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0answers
29 views

Inflating open sets up to homotopy through CW skeletons

In preparation for an upcoming exam, I’m working through old exercises. Let $X$ be a CW complex with skeletons $X^0 ⊂ X^1 ⊂ … ⊂ X$. Show that for any open $U ⊂ X^{n-1}$ there is an open $V ⊂ ...
2
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0answers
43 views

Tensoring two short exact sequences

Let $R$ be a commutative ring with $1$ and consider the following short exact sequences of $R$-modules \begin{align} &0 \to M' \to M \stackrel{f}{\to} M'' {\to} 0 \qquad \text{and } \\ &0 \to ...
1
vote
1answer
28 views

Universal Cover of the Punctured Torus

I am currently trying to compute the universal cover of the punctured torus. I originally thought it would be a lattice, but while computing the fundamental group I saw that the punctured torus is ...
0
votes
0answers
25 views

Homology of Subspace vs. Homology of Ambient Space.

Let $M$ be a manifold embedded in $\mathbb R^n$ , so that the manifold has non-trivial $k-th$ homology for some $n \geq k\geq 0$ . How do we identify the fact that while there is a non-trivial cycle ...
0
votes
2answers
51 views

The quotient map and isomorphism of cohomology groups

Let $X$ be a closed $n$-manifold, $B$ an open $n$-disc in $M$. Suppose $p:X\rightarrow X/(X-B)$ is a quotient map. Notice that $X/(X-B)$ is homeomorphic to the sphere $\mathbb{S}^n$. My question is ...
2
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1answer
20 views

If the fibers of a quotient map are all discrete, is this map a covering map?

If $p:\tilde{X}\rightarrow X$ is a covering projection then I know that for every point $x \in X$ the fibre above $x$, i.e $p^{-1}(x)$, has the discrete topology. Here $p$ being a covering map means ...
1
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0answers
11 views

co H-space on CW-complex [on hold]

Does someone have any idea how to prove these? If X is (n-1)-connected CW-complex of dimension <2n then there exists homotopy co-multiplication on X. If X is (n-1)-connected CW-complex and ...
0
votes
1answer
34 views

Is the stable homotopy group of sphere a commutative ring? If not, are there easy examples?

Is the stable homotopy group of spheres a commutative ring? If not, are there easy examples? In the Adams spectral sequence converging to the stable homotopy group of spheres, it seems that any page ...
2
votes
2answers
24 views

Homology groups of orientable surfaces.

I am trying to show that the second (simplicial) homology group or an orientable surface is ismormophic to $\mathbb Z$. I can show that this group is non-trivial by triangulating the surface, and ...
2
votes
1answer
15 views

Homotopically equivalent to Čech nerve?

I see a theorem without proof on Gelfand & Manin: Suppose $\mathfrak U=\{U_\alpha\}_\alpha$ is a locally finite open covering of the topological space $X$ such that each finite intersection ...
6
votes
2answers
56 views

Counter example to Mostow's rigidity theorem for 2-manifolds.

I am trying to understand a counter-example to Mostow's rigidity theorem. Here is the counter example I want to understand. Take two non-isometric octagons with the sum of interior angles equal to ...
8
votes
2answers
203 views

What's the difference between cohomology theories of varieties and topological spaces

There is defined several cohomology theories for algebraic varieties, but in the situation is very different for topological spaces (up to homotopy) for which there is only one cohomology theory for ...
6
votes
1answer
42 views

Books or texts on singularity theory

So a friend is doing his PhD in maths (algebraic topology) and his advisor wants him to publish something on singularities (of which, as fas as I understand, he knows next to nothing). I want to give ...
2
votes
2answers
90 views

Computing $\pi_4(S^3)$ using Serre spectral sequence

I'm following Davis & Kirk's computation that $\pi_4(S^3)=\mathbb Z/2$ using the Serre spectral sequence but I'm having problems at the very end. We consider a homotopy fibration $X\to S^3 \to ...
1
vote
1answer
36 views

Fundamental group of two circles joined by an arc

What is the fundamental group of two circles joined by an arc? In other words, let $S_1$ and $S_2$ be two standard circles. Let $p_1$ and $p_2$ be two points in $S_1$ and $S_2$ respectively. Join ...
20
votes
10answers
924 views

Surprising applications of topology [on hold]

Today in class we got to see how to use the Brouwer Fixed Point theorem for $D^2$ to prove that a $3 \times 3$ matrix $M$ with positive real entries has an eigenvector with a positive eigenvalue. The ...
4
votes
1answer
60 views

A seemingly wrong definition of convergence of spectral sequences in Bott & Tu?

After introducing exact couples, Bott & Tu defines spectral sequences as follows: A sequence of differential groups $\{E_r,d_r\}$ in which each $E_r$ is the homology of its predecessor ...
2
votes
1answer
19 views

Boundary maps of the projective plane as a $\Delta$-complex (homology)

Hi, very simple question here. In Hatcher's 'Algebraic Topology' the diagram above is used to describe the projective plane as a $\Delta$-complex(see p.102). Later the 2-boundary maps are given by ...
4
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1answer
39 views

Poincare duality isomorphism problem in the book “characteristic classes”

This is the problem from the book, "characteristic classes" written by J.W. Milnor. [Problem 11-C] Let $M = M^n$ and $A = A^p$ be compact oriented manifolds with smooth embedding $i : M \rightarrow ...
5
votes
1answer
136 views

Why is SO(3) not $S^1 \times S^2$? (Where is the mistake?)

I was trying to calculate the fundamental group of SO(3). In order to represent the group I reasoned the following way: In order to build the 3X3 orthogonal matrix I need an orthonormal positive ...
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0answers
42 views

Simplicial homology [closed]

Let $\Delta$ be the simplicial complex on vertex set [5] whose Stanley-Reisner ideal is $I_{\Delta}=(x_{1}x_{4},x_{1}x_{5},x_{2}x_{5},x_{1}x_{2}x_{3},x_{3}x_{4}x_{5})$. Write the augmented oriented ...
3
votes
1answer
51 views

the top chern class of the holomorphic tangent bundle is the euler class

Is the following true? Let X be a complex manifold of complex dimension d and let V denote its holomorphic tangent bundle (ie it's $T^{1,0} \subset T \otimes_R C$, where T is the tangent bundle of ...
18
votes
4answers
1k views

Motivating Cohomology

Question: Are there intuitive ways to introduce cohomology? Pretend you're talking to a high school student; how could we use pictures and easy (even trivial!) examples to illustrate cohomology? Why ...
2
votes
1answer
26 views

The Oriented Universal Bundle in Characteristic classes by J.W. Milnor.

I have a problem to understand the section "The Oriented Universal Bundle" in the page 145 of "Characteristic classes" written by J.W. Milnor. The content in that page is like below, ...
2
votes
3answers
52 views

Universal Cover of $\mathbb{R}P^{2}$ minus a point

I've already calculated that the fundamental group of $\mathbb{R}P^{2}$ minus a point as $\mathbb{Z}$ since we can think of real projected space as an oriented unit square, and puncturing it we can ...
2
votes
1answer
585 views

Group action and covering spaces

Let $X$ be a path-connected and locally path-connected topological space. The action of a topolgical group $G$ on $X$ is a covering space action. For any subgroup $H < G$, we have a composition ...