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
0answers
28 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 ...
4
votes
0answers
85 views

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 ...
4
votes
1answer
146 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 ...
9
votes
1answer
143 views

Prove $\mathbb{R}^3$ is not the product of two identical topological spaces

I can only prove this for $\mathbb{R}$: If $\mathbb{R}\cong T\times T$, then $T$ embeds in $\mathbb{R}$ as a closed subspace (e.g. $T\times pt$). Since $\mathbb{R}$ is connected, so is $T$. So $T$ ...
7
votes
2answers
127 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
79 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 ...
3
votes
3answers
66 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 ...
-1
votes
1answer
45 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 ...
0
votes
2answers
105 views

Finite fundamental group and covering spaces

Show that if a path-connected, locally path-connected space X has a finite fundamental group , then every map $X$ to $S^1 \times S^1$ is nullhomotopic (i.e. homotopic to a constant map) . Is the ...
13
votes
1answer
305 views

Maps from $D^n$ to $D^n$ with a single inverse set are open.

Let $D^n$ denote the closed unit ball in $\Bbb R^n$. In multiple sources proving Brown's generalized Schoenflies theorem (including a version in the original paper), the following consequence of ...
2
votes
1answer
51 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 ...
2
votes
3answers
47 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 ...
2
votes
3answers
39 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
66 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 ...
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
27 views

Covering Space of Orthogonal Group [closed]

What is the covering space of Special Orthogonal Group SO(3)?
1
vote
0answers
56 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 ...
0
votes
2answers
39 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 ...
2
votes
0answers
34 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 ...
0
votes
1answer
44 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 ...
2
votes
1answer
73 views

Chern class of complex vector bundles

Let $\xi$ be an $n$-dimensional complex vector bundle. It is claimed that the Chern class of $\xi$ is $$ c(\xi)=(1+x_1)\cdots (1+x_n), $$ $|x_k|=2$, $c_j(\xi)$ is the $j$-th symmetric polynomial of ...
5
votes
2answers
81 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 ...
0
votes
1answer
93 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 ...
2
votes
0answers
39 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 $X\subset C$, ...
2
votes
2answers
68 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 ...
1
vote
1answer
34 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 ...
3
votes
1answer
62 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 ...
0
votes
2answers
44 views

Orbits of mapping class group on four-punctured sphere

Let $\mathcal{M}_{0,4}$ be the mapping class group of the four-punctured sphere $S_{0,4}$. Denote the simple closed curve around boundary components $b_1$ and $b_2$ by $x$, the one around $b_2$ and ...
1
vote
1answer
43 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 ...
2
votes
0answers
43 views

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

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.
1
vote
1answer
50 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
2answers
32 views

CW -complex structure of boundary of a manifold

Given a CW-complex structure of manifold with boundary .Is there any natural way to construct CW-complex structure of its boundary? Thanking you.
4
votes
3answers
207 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 ...
0
votes
0answers
16 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 ...
11
votes
1answer
206 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
61 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 ...
4
votes
1answer
61 views

definitions of various spectra: $E^X$ and $E \wedge \Sigma^\infty X$

Let $E=\{E_n\}$ be a spectrum given by a sequence of pointed CW complexes $E_n$ and inclusions $\Sigma E_n \to E_{n+1}$. Let $X$ a pointed CW complex. I had a few very naive questions I had while ...
0
votes
0answers
16 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 $Y^0$ ...
1
vote
1answer
118 views

fundamental group of the complement of a circle

This should be a quick question. I am reading Hatcher's Algebraic Topology book. At page 46, in the example of computing the fundamental group of the complement in $\mathbb{R^3}$ of a single circle, ...
0
votes
1answer
37 views

Explicit construction of retraction for Brouwer's fixed point theorem (disk)

So I'm trying to prove the Brouwer fixed-point theorem for the disk, arguing by contradiction and using the theorem that states that there is no retraction from the unit disk $D^2$ to the unit circle ...
4
votes
1answer
246 views

Finite dimensional Eilenberg-Maclane spaces

Given a positive integer $n\geq 2$ and an abelian group $G$, is it possible to find a finite dimensional $K(G,n)$? In case it does, which are some examples? Thanks...
2
votes
1answer
61 views

Constructing a simply connected covering space

"Construct a simply connected covering space of the space that is the union of the sphere S2 with two of its intersecting diameters." can anyone help me with this? i don't know how to think , all ...
0
votes
1answer
22 views

Description of topological spaces with lifting property

We say that topological space $L$ has lifting property if for any covering $\tilde{X} \xrightarrow{\pi} X$ and any map $L \xrightarrow{f} X$ we can lift $f$ to $L \xrightarrow{\tilde{f}} \tilde X,$ ...
1
vote
1answer
55 views

Is the tensor product of a complex line bundle with itself trivial?

Let $\xi$ be a complex line bundle over a manifold $M$. Then $\xi\otimes \xi$ is a trivial complex line bundle. Is my statement right?
1
vote
0answers
21 views

Convolution product in Borel-Moore homology

I have a question about Exampla 2.7.10 from the book "Representation theory and complex geometry" by N. Chriss and V. Ginzburg. It concerns the convolution product. In the example we have $M_1 = M_2 ...
-1
votes
0answers
38 views

Fundamental group of R^2-s^1

what is the fundamental group of R^2-S^1? Here is my thinking R^2-S^1 is subset of R^2-{(0,0)} therefore there is continuous map from R^2-S^1 to R^2-{(0,0)} which says that the fundamental group of ...
7
votes
1answer
614 views

Question on the symmetric product $\mathrm{Sym}^g\Sigma$

Let $\Sigma$ be a genus $g$ closed Riemann surface. Recall that the symmetric product $\mathrm{Sym}^g\Sigma$ is the quotient of the $g$-fold product $\Sigma \times \dots \times \Sigma$ under the ...
1
vote
0answers
41 views

Showing the Sum of $n-1$ Tori is a Double Cover of the Sum of $n$ Copies of $\mathbb{RP}^2$

I want to show that the non-orientable surface of genus $n$ has a 2-sheeted cover by an orientable surface of genus $n-1$. The base cases are easy: $S^2$ covers $\mathbb{RP}^2$ and I worked on a ...
1
vote
0answers
41 views

Can someone explain what it means for S1 to map to S1 under a function f?

If we are trying to get a circle to map onto itself through a continuous function f, as shown here: https://youtu.be/CSC5Q6ULWrg?list=PL6763F57A61FE6FE8&t=66 why is the image of S1 a squiggly ...