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

learn more… | top users | synonyms (1)

1
vote
3answers
20 views

How many elements does the free product $\mathbb{Z}/2\mathbb{Z}\ast\mathbb{Z}/2\mathbb{Z}$ have?

Taken from Hatcher, I think the free product $G =\mathbb{Z}/2\mathbb{Z}\ast\mathbb{Z}/2\mathbb{Z}$ should have infinite elements taking the form of words containing alternate elements, i.e. $a, b, ab, ...
-10
votes
0answers
44 views

i want to know about cube root of unity [on hold]

Blockquotecomlex number cube root of unity origin of comlex numbers origin of cube root ofunity
0
votes
0answers
14 views

Questions about strata of a variety: the nilpotent cone of $\mathfrak{sl}_2$.

I am reading the lecture notes. In the end of page 3, let $X = \{(a, b; c, -a): a, b, c \in \mathbb{C}^3, a^2 + bc = 0\}$. It is said that there are two strata: the regular orbit $U$ and $0$. What is ...
0
votes
3answers
31 views

Relative homology $H_n(S^2,S^0)$, or other examples

I've been reading Hatcher and think I understand the idea of relative homology, but he only provides two (fairly trivial) examples, homology relative to a point computing $H(S^n)$ using $D^n$s. My ...
5
votes
1answer
55 views

fiber bundle in topological category and smooth category.

Let $M$ be a smooth manifold and $G$ be a Lie group. Denote by $Bun(M,G)$ the set of all equivalent smooth Principal bundle on $M$ with structural group $G$ in smooth category. And denote by ...
3
votes
3answers
52 views

The significance of filtered colimits in homotopy theory

I have been allowed to attend some preparatory lectures for a seminar on the Goodwillie Calculus of Functors. I found in my notes from one of the lectures two statements which I would like to ask ...
3
votes
1answer
28 views

A question about the degree of a map

Consider $f,g: S^{n}\rightarrow S^{n}$ to be continuous maps. Seeing $S^{n}\subset\mathbb{R}^{n+1}$, let's say that $f$ and $g$ are orthogonal at $x\in S^{n}$ whenever $\langle f(x),g(x)\rangle = 0$ ...
1
vote
0answers
13 views

Conjugate group homomorphisms induce homotopic maps on classifying spaces

Let $G$ be a group and let $\phi: G \to G$ be the inner automorphism given by conjugation by an element $g' \in G$, i.e., $\phi(g) = g'^{-1} g g'$. I want to show that the induced map on classifying ...
1
vote
1answer
20 views

Is a covering space of a topological group a fiber bundle with the structure group the fundamental group of the topological space?

Question. Is a covering space of a topological group a fiber bundle with the structure group the fundamental group of the topological space? Let $p:E\rightarrow X$ be a covering space of X. I ...
1
vote
1answer
35 views

Fiber bundle beginner question.

I'm reading some notes on fiber bundles. Let $f:X \rightarrow Y$ be a continuous map of topological spaces. The author states: We say $f$ makes $Y$ a fiber space over $X$ if $f$ is locally trivial ...
4
votes
1answer
112 views

A question on Hawaiian earring

Consider the Hawaiian earring. Suppose $f_n$'s are the loops representing the circles of radius $1/n$ centered at $(1/n, 0)$ for $n=1,2\cdots$. Suppose the following holds in the fundamental group ...
1
vote
1answer
50 views

On the associative property of a binary operation of the fundamental group.

I was reading about the proof of associativity property of the operation on the fundamental group here. The book gives the following diagram then it says the reader should supply the elementary ...
-1
votes
0answers
61 views

How to show $S^n$ is not contractible without using Homology..

I know the prove $S^n$ is not contractible using homology.But I don't know how to prove it from definition of contractibility.Can anyone help me in this direction? Thanks.
7
votes
2answers
104 views

Fundamental group of a quotient on a solid torus.

It is easy to compute the fundamental group of a solid torus. You easily get $\mathbb{Z}$ just because the torus is the cartesian product of a circumference and a closed disk. The next step is ...
2
votes
1answer
68 views

Shrinking wedge of circles

I'm spending too much time thinking about this problem : I need to show that the shrinking wedge of circles which is path connected, locally path connected ,doesn't have a simply connected covering ...
-1
votes
0answers
21 views

Finding whole number coordinates on continuous curves [on hold]

Imagine two surfaces in 3D space defined by known equations intersect and form a line in 3D. How could you find out if that curve formed by the intersection goes through any points where all 3 ...
2
votes
3answers
39 views

Difference between Wedge of countable infinite circle and Hawaiian ear ring?

Hawaiian ear ring is the union of countable circles at points (0,1/n) with radius 1/n.It seems to me that wedge sum of countable infinite circle is same as Hawaiian ring.But I found that this not ...
4
votes
1answer
59 views

Do join and suspension commute?

Do join and suspension of topological spaces always commute, i.e. is it true that $\sum(A\star B)=A\star(\sum B)$? I suppose that it is not true in general (but, for example, everything works in the ...
1
vote
1answer
61 views

Is there a more general obstruction to the existence of moduli spaces than the existence of automorphisms?

We are taught that, in general: A type of objects that has nontrivial automorphisms cannot have a fine moduli space. The proof generally goes along the lines of: Take an object $X$ with a ...
2
votes
3answers
37 views

locally path connectedness

While studying covering spaces , hatcher mentioned the "shrinking wedge of circles" this space is locally path connected as I was told , but I wasn't able to prove it nor to see it, it looks like comb ...
2
votes
1answer
47 views

Calculating $H_1(\mathbb{R})$

Given the space $X=\mathbb{R}$, how can we calculate its first homology group $H_1(\mathbb{R})$? Intuitively, the object of first homology describes 1-dimensioal holes in the set which here doesn't ...
-1
votes
1answer
44 views

classification theorem in a subset of R^2

I need some very simple results of algebraic topology but I am not sure where I can find them without having to swallow the whole theory. What I want: -An open bounded subset $A$ of $R^2$ is ...
3
votes
3answers
57 views

The biggest degree of a map between fixed surfaces

Let $S_1$ and $S_2$ be compact surfaces (real manifolds of dimension 2). None of them is a sphere. Question: What is the biggest degree of a smooth map from $S_1$ to $S_2$? Comment 1. I have a ...
2
votes
1answer
50 views

Universal covering of the complement of a circle in $\mathbb{R}^3$

What is the universal covering of $X=\mathbb{R}^3\setminus(S^1\times\{0\})$? I've been trying to build a covering map from $\mathbb{R}^3$ onto $X$ via composition of $p:\mathbb{R}^3\to Y$ and $q:Y\to ...
-1
votes
1answer
26 views

Covering Space of Orthogonal Group [on hold]

What is the covering space of Special Orthogonal Group SO(3)?
2
votes
1answer
50 views

Proving that $H_0(X)=\tilde{H_0}(X)\oplus\mathbb{Z}$

In the article about the Reduced Homology it's stated that $$H_0(X)=\tilde{H_0}(X)\oplus\mathbb{Z}$$ but I don't know how to prove that. I know $$H_0(X)=\bigoplus_{\alpha\in ...
-1
votes
1answer
49 views

Induced homomorphism - Homology [on hold]

I have some difficulty understanding what is a homomorphism induced by a certain function f. For example, let f: X->Y a function such that f(x)=y. Now, suppose H⊂X et H'⊂Y, hence f induces a ...
1
vote
0answers
55 views

Finest good cover of a topological space

Let $X$ be a topological space. Does there exists a good open cover $\left\{ U_{a}\right\}_{a\in I}$ finer than any other open cover of $X$? A good cover $\{U_\alpha\}_\alpha$ of $X$ is a ...
2
votes
0answers
30 views

How to compute perverse sheaves?

In the video, from 49:00 to the end of the video, there is an example of computing $IC(S, L)$ and equivariant local systems. I don't understand some parts of the computations. Let $X$ be the variety ...
-1
votes
1answer
37 views

$\mathbb{R}^3 \setminus A$ deformation retracts onto $S^1 \vee S^2$

Hatcher says the following: The complement $\mathbb{R}^3 \setminus A$ of a single circle $A$ deformation retracts onto a wedge sum $S^1 \vee S^2$. He then goes on to explain it, but I do not ...
0
votes
2answers
34 views

Show the two fundamental groups representation of the Klein bottle are the same

Show the two different representation of the fundamental groups of the Klein bottle $G=<a,b:aba^{-1} b>$ and $H=<c,d:c^2 d^{-2}>$. As we calculate in different and gett one result as ...
4
votes
0answers
25 views

How do we see the rank of the braid group?

The only presentation of the braid group that most people ever see is the standard Artin presentation $$B_n = \langle \sigma_1, \cdots, \sigma_{n-1} | \ \ \sigma_i\sigma_j = \sigma_j\sigma_i\ \ (|i - ...
2
votes
0answers
37 views

Fundamental group of infinitely many glued copies of a space

Let $C=\{p\in \mathbb{R^3}: \left\|p\right\|_{\infty}\le 1\}$ be a cube in $\mathbb{R}^3$. Let $X$ be a compact, path-connected topological space in $\mathbb{R}^3$ such that $X\subset C$, ...
1
vote
1answer
31 views

References request on characteristic class

I am planing to learn something about characteristic classes on my own. I am wondering if anyone could recommend me something on such materials like constructions of vector bundles, Thom isomorphism ...
1
vote
1answer
33 views

Let $S$ be a regular surface and let $F$ be a diffeomorphism. How can I prove that the image $F(S)$ is a regular surface?

Let $S$ be a regular surface and let $F$ be a diffeomorphism. How can I prove that the image $F(S)$ is a regular surface ? Consider a regular surface $S \subset \mathbb R^3$ and a diffeomorphism ...
5
votes
2answers
75 views

What are monomorphisms in the category of real vector bundles over a fixed base space $X$?

I have a (perhaps stupid) question dealing with vector bundles. In most textbooks, a morphism of real vector bundles $f:\xi\to \eta$ over a fixed base space $X$ is said to be a ...
2
votes
0answers
36 views

Find cohomology ring of $H^*(S^n \times X)$

I know that $H^* (S^1 \times X) \equiv H^*(S^1) \otimes H^*(X)$ . Now how to generalize it for $S^n$. Please state in brief and without using category and Kanneth formula.
0
votes
0answers
34 views

Is there a spectral sequence to estimate the connectivity of a homotopy limit of a cosimplicial space?

Given a diagram $$ D\colon \Delta\to sSet_* $$ i.e. a cosimplicial pointed simplicial set by $[k]\mapsto S^n\wedge D'([k])$ for some other diagram $D'\colon \Delta\to sSet_*$ and a fixed integer ...
0
votes
1answer
87 views

Is it necessary to read point set topology to read differential geometry?

I want a quick insight in differential geometry but it is hard to start directly although i have done courses in calculus and basic algebra .is it necessary to get through point set topology and ...
1
vote
1answer
44 views

Simply connected covering space

"Find an example of a path connected, locally path-connected space which does not have a simply connected covering space". I was reading hatcher and he gives an example of shrinking wedge of circles, ...
0
votes
0answers
29 views

Fundamental group of the complement homogeneous variety in $\mathbb{C}P^{2}$

Let $f,g:\mathbb{C}^3\to \mathbb{C}$ are two irreducible homogeneous polynomials. If there is a homeomorphism $h:\mathbb{C}^3\to \mathbb{C}^3$ such that $h(X)=Y$ and $h(0)=0$, where $X=f^{-1}(0)$ and ...
3
votes
0answers
54 views
+50

Textbook on infinite loop spaces

I'm looking for a good update reference covering the material in first three chapters of "Adams, Infinite loop spaces" (specially construction of delooping functors and group completion) with exact ...
3
votes
3answers
196 views

Question about closed sets

Let $A$ and $B$ be subsets of $\mathbb{R}^n$ (where $\mathbb{R}^n$ is Euclidean n-space). Define $A + B = \{ x + y : x \in A , y \in B \}.$ Now If $A$ and $B$ are closed sets, is $A+B$ also a closed ...
2
votes
2answers
65 views

Clarification about the Computation of the Homology of the Connected Sum in degree $n-1$.

There are plenty of questions about the homology of the connected sum of two $n$-manifolds, but I didn't find an explicit explanation of the computation done in degree $n-1$. Let's show some examples ...
0
votes
0answers
15 views

How to show that the boundary of an antipodally symmetric 1-chain contains an even number of antipodal pairs?

This is an exercise in Jiri Matousek's book 'Using the Borsuk-Ulam Theorem' which I'm going through. A 1-chain is of course a collection of 1 dimensional simplices (edges). A chain is antipodally ...
3
votes
1answer
55 views

The Poincaré dual of a space-time curve

We have a smooth space-time curve defined by $f:C{\mapsto}M$, where $M$ is a typical curved space-time manifold. ${\eta}^{(4)}$ is the volume 4-form defined on $M$ and ${\varepsilon}^{(1)}$ is the ...
10
votes
1answer
187 views

Is there a map from the torus to the genus 2 surface which is injective on homology?

Let $T$ denote the torus and $M_2$ the genus 2 surface. Specifically, I am wondering if there is a map $f\colon T\to M_2$ such that $f_*\colon H_1(T)\to H_1(M_2)$ is injective. By thinking about the ...
4
votes
0answers
55 views

Proof of the isomorphism $[X, \mathbb{R} P^\infty] \cong H^1(X; \mathbb{Z}/2).$

I want to prove the following: $[X, \mathbb{R} P^\infty] \cong H^1(X; \mathbb{Z}/2)$ via the map $[f] \mapsto f^* w_1(\gamma^1)$ where $\gamma^1$ denotes the tautological line bundle over ...
0
votes
1answer
34 views

Hatcher's Algebraic topology, section 2.2 exerise 26, page 157

In part (a) of this question, we show that $X$ is a retract of $X \cup CA$ if and only if $A$ is contractible in $X$. Then in part (b), the question is to show that if $A$ is contractible in $X$ then ...
0
votes
0answers
15 views

Structure of simply-connected cw-complex

Let $X$ be connected CW-complex. I want to prove that $\pi_1(X) = 0 \implies X \simeq Y$ where $Y^0$ consist only one point, $X^1$ is empty. I have theorem which says that $X \simeq Y$ where $Y^0$ ...