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

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0
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3answers
122 views

Function to uniquely map a set of rectangles in space to a number?

I am trying to build a new way of indexing spatial data. Is there a function that takes as parameter a number of rectangles in euclidean space, and outputs an unique number?Can such a function be ...
11
votes
2answers
381 views

A comparison between the fundamental groupoid and the fundamental group

Are there two path connected topological spaces $X,Y$ such that the fundamental groupoid of $X$ is not isomorphic to the fundamental groupoid of $Y$ but the fundamental group of $X$ is isomorphic to ...
1
vote
0answers
46 views

What is the closed orientable surface of genus 2?

I just have a very simple question. Could someone please explain to me what the closed orientable surface of genus 2 is? Thank you so very much
9
votes
2answers
178 views

Are there any simply connected parallelizable 4-manifolds?

On pp. 166 of Scorpan's "The Wild World of 4--manifolds", he gives an example of a parallelizable 4-manifold ($S^1\times S^3$) and then asserts: "there are no simply connected examples". It's ...
1
vote
0answers
26 views

Critical group and morse Lemma

I have this theorem with a part of it's prove I have two questions: 1) what is the spectral decomposition of A ? 2)How to see that $B_{\varepsilon}\cap f_0 = \lbrace x\in H,||x||\leq \varepsilon ...
1
vote
0answers
40 views

Fiber product and $G$-invariant maps

Let $S^n \times_{\mathbb{Z}_2} \mathbb{R}$ be the fiber product of unitary sphere $S^n$ and $\mathbb{R}$ over $\mathbb{Z}_2$, where $\mathbb{Z}_2$ acts on $S^n$ by antipodal relation and on ...
3
votes
1answer
251 views

On the Hopf invariant

It was an important problem of topology to determine for which dimensions the Hopf invariant was one. There are several clear expositions giving the definition of the Hopf invariant including the ...
2
votes
2answers
115 views

Morse Theory and critical groups

Please I have a question: What is the relation between Morse theory and critical point theory ? I studied the Morse inequalities and critical groups, but i can not not find or at least i do not ...
2
votes
1answer
255 views

Is there a compact contractible manifold?

Does there exist a compact connected manifold (without boundary), that has a trivial homotopy type?
6
votes
1answer
132 views

Surgery on manifold

In this article on surgery on manifolds it is explained that from an $n$-manifold $M$ an $n$-manifold $M'$ can be constructed by cutting out $S^p \times D^q$ and gluing in $D^{p+1}\times S^{q-1}$. ...
1
vote
1answer
119 views

Definition of cohomology with compact support

We can define a cohomology on open manifold: Define a simplicial cochain group $$ \Delta^i_c(X;G)$$ consisting of cochains that are compactly supported in the sense that they take nonzero values on ...
5
votes
2answers
569 views

Contractible spaces has trivial fundamental group.

I have to prove the following: Show that if $X$ is contractible (the def. I have is that $I:X\rightarrow X$ the identity function is homotopic to the constant function $p$ for some $p\in X$), then its ...
2
votes
1answer
172 views

Is the product of covering maps a covering map?

I have a question about covering maps. If $\phi_1: X_1 \rightarrow Y_1$ is a covering map, and $\phi_2: X_2 \rightarrow Y_2$ is a covering map, then is it true that $\phi_1 \times \phi_2: X_1 \times ...
3
votes
1answer
54 views

Universal covering space of connected open subset of $\mathbb R^n$

Is the universal covering of an open connected subset $U$ of $\mathbb{R}^n$ homeomorphic to $\mathbb{R}^n$?
4
votes
2answers
195 views

'homotopy' between morphisms of a 'topological' or 'algebraic' category (Stanley-Reisner ring)

In what follows, a homotopy is a congruence $\simeq$ on a given category. Given such a homotopy, objects $X$ and $Y$ of the given category are homotopy equivalent when there exist morphisms ...
5
votes
2answers
94 views

Two defintions of simply connected

I'm showing that the first definiton here implies the second (the other implication is obvious). My thoughts: Let $p,q$ be two paths in the space $X$. Then since $X$ is path connected there are two ...
2
votes
0answers
53 views

Quotients of infinite dimensional sphere

Recall $$S^\infty = \cup S^n,\ {\bf RP}^\infty = \cup {\bf RP}^n,\ {\bf CP}^\infty = \cup {\bf CP}^n$$ Hence $S^\infty / {\bf Z}_2 ={\bf RP}^\infty$. And I think that the following is possible : ...
1
vote
0answers
41 views

Geometric interpretation of $R$-orientations

For an $n$-dimensional manifold $M$ and a ring $R$ one can define $R$-orientability, e.g. by choosing local orientations, i.e. generators of $H_n(M, M - \{x\}; R)$ for every point $x$ in the manifold, ...
3
votes
1answer
91 views

Visualizing the group operation in higher homotopy groups

I'm having trouble picturing the homotopy group operation of concatenation between two pointed spaces. For $n$-spheres, we have for $f,g: S^n \to X$ $$(f * g)(s_1,\ldots, s_n) = \begin{cases} ...
4
votes
1answer
96 views

Different point-set level definitions of spectra

I've been trying to understand the Adams spectral sequence and one of the more accessible sources is the (unfinished) book on spectral sequences by Hatcher. The usual (only?) construction of the ...
28
votes
3answers
654 views

How did we know to invent homological algebra?

Update: Qiaochu Yuan points out in the comments that the title of the question is misleading, as homological algebra did not begin with long exact sequences as I'd thought. (Original question ...
2
votes
1answer
59 views

What would be an example of a nontrivial, i.e. multi-sheeted, covering of the torus?

What would be an example of a nontrivial, i.e. multi-sheeted, covering of the torus? I would greatly appreciate any help that you could give me.
2
votes
0answers
114 views

Product of two Kähler manifolds

Let $M$ and $N$ be Kähler manifolds. I know that the cartesian product $M\times N$ is a kahler manifold. I was wondering which form has the Kähler form on $M\times N$. Here's what I thought: Let ...
3
votes
2answers
325 views

covering space of a particular CW complex

I am trying to find all connected covers of the following space $X$ (up to isomorphisms) $X$ has one $0$-cell, two $1$-cells labeled $a$ and $b$, and three $2$-cells attached via $a^2$, $b^2$ and ...
1
vote
0answers
43 views

Intersection between cycles and dominant maps

i must say in advance that i'm not very familiar with algebraic cycles and intersection theory, so i hope my question is not too trivial. Let $X$ and $Y$ be two K3 surfaces. Let ...
8
votes
2answers
442 views

Motivation of stable homotopy theory

A stable homotopy category can be obtained by modifying the category of pointed CW-complexes: objects are pointed CW-complexes, and for two CW-complexes $X$ and $Y$, we take $$\lbrace X,Y \rbrace = ...
1
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0answers
53 views

Vector field and a solution of an ODE

I have this: And my questions are : 1)what is :"The local theory of differential equations in a Banach space" 2)Why it's implies that each solution of (6) is equal to $\eta$ Please Thank you . ...
10
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0answers
496 views

When is a fibration a fiber bundle?

In this question I am using Wiki's definitions for fibration and fiber bundle. I want to be general in asking my question, but I am mostly interested in smooth compact manifolds and smooth fibrations ...
5
votes
1answer
123 views

Explanation of example 3F.7 in Hatcher

The section I am refering to is the following example on page 314 of Hatcher's Algebraic Topology: I'm a bit confused by his statement about relations and can't quite see what he is trying to say. ...
5
votes
3answers
285 views

Homomorphism/map in both direction implies isomorphism/homeomorphism or not?

I was working on a homework, and my first attempt get me to a deadend, but I was eventually able to solve it using a different method. But the fail attempt make me curious, and I wonder if it could ...
0
votes
1answer
77 views

Properly discontinuous action on $\mathbb{R}^2$

Given a group $H$ with two generators $a$ and $b$, and one relation $abab^{-1}=e,$ I want to show that $H$ acts properly discontinuously on $\mathbb{R}^2$. I think that I need to show that for every ...
6
votes
2answers
315 views

Cup product on torus

I want to calculate the cup product on torus (cf. Hatcher's book) If $\pi_1(T^2) = ([a]) + ([b])$, then by universal coefficient theorem we have a cocycle $\alpha$ (resp. $\beta$) which have a ...
2
votes
0answers
56 views

Homomorphism and boundary operator

I have this : I dont know how to verify 1) and 2)? Please help me Thank you.
5
votes
1answer
134 views

Sheafification of singular cochains

Let $S^k$ be the presheaf on a space $X$ that assigns to every open set $U$ the abelian group $S^k(U)$ of singular k- cochains on $U$. This is clearly not a sheaf. Consider the sheafification $F^k$ of ...
0
votes
1answer
56 views

Isomorphism of Covers

On page 26 of Peter May's A Concise Course on Algebraic Topology, it is claimed that given any two covers of a space $X$, $(E, p)$ and $(E', p')$ are isomorphic iff for any points $e \in E, e' \in E'$ ...
12
votes
1answer
153 views

The fundamental group functor is not full. Counterexample? Subcategories with full restriction?

Anyone aware of a nice counterexample to "The fundamental group functor is full?" (Which is...false, right?) and are there a nontrivial subcategories on which its restriction is full? I.e. Can you ...
1
vote
2answers
239 views

CW-pairs are good pairs

Hatcher uses in a proof that every subcomplex of a CW-complex is a deformation retract of some neighborhood. In what way can I see this in the infinite dimensional case?
7
votes
2answers
547 views

Two spaces homotopy equivalent to eachother, attaching maps, Algebraic Topology.

I have a question regarding algebraic topology with which I was hoping someone could help me with. I've managed to show the following: If $f,g:S^{n-1} \to X$ are homotopic maps, then $X\sqcup_fD^n$ ...
0
votes
1answer
67 views

Prove $SO(3)$, the group of rotations of $\mathbb{R}^3$, is not homotopically equivalent to $S^1\times S^2$

Prove $SO(3)$, the group of rotations of $\mathbb{R}^3$, is not homotopically equivalent to $S^1\times S^2$. I know that $\pi_1(SO(n))\cong \mathbb{Z}_2$ and I think that $P:\mathbb{R}\times S^2 \to ...
0
votes
0answers
80 views

Question about an isomorphism result in homotopy theory

I have another question regarding homotopy theory and winding numbers (or degrees). In Manton and Sutcliffe they state the following theorem: $\pi_2(G/H)=\pi_1(H)$ provided $G$ is a compact, ...
1
vote
0answers
81 views

Why is the pullback of a connected cover not necessarily connected?

In particular, I read somewhere that the fiber product of the maps $S^1\rightarrow S^1$ sending $z\mapsto z^m$ and $S^1\rightarrow S^1$ sending $z\mapsto z^n$ is disconnected with $\gcd(n,m)$ ...
1
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0answers
38 views

Question on critical groups

I have this theoreme with it's proof But i don't understand who is $f_0$ ? Can someone help me please ? Thank you .
1
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1answer
61 views

Finite group acting freely on Haussdorf space- Topology problem

How to prove the following problem: It is given Hausdorff space $X$ and finite group $G$ (with neutral $e$) that is acting freely on $X$. For $g\in G$, $\overline{g}:X\rightarrow X$ is ...
1
vote
2answers
84 views

horn of a simplex

I'm reading one book of simplicial homotopy. It's just amazing. But I am stuck at the very beginning of the book. He let a simplicial set be a contravariant functor between the category $ \Delta $ of ...
0
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2answers
60 views

Question about “THE MORSE INEQUALITIES”in Milnor's book

in this paragraph what is $H_{*}$ ? Please help me Thank you .
2
votes
1answer
99 views

A Borsuk theorem

Let $M$ and $L$ be two subspaces of Banach space $X$ such that $\dim L<\dim M<\infty$. Let $S=\{m\in M : \|m\|=1\}$ and let $g$ be a continuous function from $S$ to $L$ such that $g(-m)=-g(m)$ ...
11
votes
4answers
702 views

What are two continuous maps from $S^1$ to $S^1$ which are not homotopic?

This is an exam question I encountered while studying for my exam for our topology course: Give two continuous maps from $S^1$ to $S^1$ which are not homotopic. (Of course, provide a proof as ...
19
votes
2answers
297 views

Curious remark of D. Ravenel

In his beautiful (but difficult) book "Complex cobordism and stable homotopy groups of spheres", concerned mostly with methods of computing homotopy groups of spheres, D. Ravenel describes a general ...
6
votes
1answer
138 views

Perverse sheaves (or D-modules) on vector spaces, constructible with respect to a hyperplane arrangement

Let $V$ be a finite dimensional complex vector space and let $\mathcal{A}$ be a finite collection of hyperplanes in $V$. Stratify $V$ by the intersections of elements of $\mathcal{A}$, and consider ...
2
votes
2answers
135 views

Quotient space and equivalence relation

Define an equivalence relation $\sim$ on $X={\bf C}^2\setminus \{(0,0)\}$ by $(x_1,y_1)\sim(x_2,y_2)$ if and only if there exists $t \in C\setminus\{0\}$ such that $(x_1,y_1)=(tx_2,ty_2)$ show that ...