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
240 views

The retraction of genus g surface

The problem is in the picture. My question is How could $M_g$ retract onto $C'$?That seems impossible.Thank you.
2
votes
1answer
348 views

Compute Fundamental Group

This is an exercise in Allen Hatcher's Algebraic Topology book(page 53, 7) Let X be the quotient space of $S^2$ obtained by identifying the north and south poles to a single point.Put a cell complex ...
4
votes
0answers
127 views

Why is $\pi_2(S^1 \vee S^2)$ not finitely generated?

This is an exercise following a discussion of fibrations. preceding that there was a discussion of cofibrations and the long exact sequence of homotopy groups of a pair. Any hints would be greatly ...
8
votes
1answer
550 views

Proof that two spaces that are homotopic have the same de Rham cohomology

I know this is true, but how do I prove it? Specifically, I'm trying to calculate the de Rham cohomology of the 3-sphere by using the Mayer-Vietoris sequence and covering $S^3$ with two hemispherical ...
8
votes
2answers
939 views

Finding the homology group of $H_n (X,A)$ when $A$ is a finite set of points

It is one of the problems in Hatcher's book. I need to find the homology group of $H_n (X,A)$ when $A$ is a finite set of points and $X$ is $S^2$ or $T^2$. I figured out that for $n>1$, I could ...
5
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3answers
116 views

How much does $\operatorname{Aut}(H_1(S))$ determine a homeomorphism $S \to S$?

Let $S$ be an orientable compact surface. A homeomorphism $f: S \to S$ induces an isomorphism $f_{*}: H_1(S) \to H_1(S)$. How much can we say the converse? Namely, if we are given an element of ...
5
votes
1answer
2k views

Fundamental group of complement of $n$ lines through the origin in $\mathbb{R}^3$

Just a quick question to verify whether I'm right. Claim: The fundamental group of the complement of $n$ lines through the origin in $\mathbb{R}^3$ is $F_n$, the free group on $n$ generators. ...
1
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0answers
56 views

Disjointness of stars in a simplicial complex in $\ell_2$

Definitions Let's consider a full $n$-dimensional simplicial complex $C$ in $\ell_2(X)$. By that I mean a set of all functions $f:X \to[0,1]$ such that $\sum_{x\in X}f(x)=1$ and there are at most ...
2
votes
0answers
104 views

A Heegaard splitting of $S^2\times S^1 \# S^2\times S^1$.

For a Heegaard splitting of $S^2 \times S^1$, we can take two copies of genus 1 handlebodies and glue boundaries with the identity map. I want to generalize this a little bit. In the case of ...
2
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3answers
189 views

$\pi_7(S^4)$ has element of infinite order?

I need to prove that this has an element of infinite order. Do I use Frudenthal suspension theorem. Any hints on what to use to prove this?
0
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1answer
269 views

How do you do this example on a video lecture about algebraic topology?

In 48:15 of this video, I don't understand what the product have to do with the problem 4 and what that product is meaning. Link: ...
-1
votes
2answers
217 views

Freudenthal suspension theorem plus latex question.

I was looking at this paper - are there any easier references that go through the proof of Freudenthal suspension theorem? I need to use this to prove something about spheres. My lecturer was saying ...
4
votes
4answers
176 views

Fiber bundles with same total spaces, but different base spaces

(may be silly question but), Does there exist two fiber bundles (or in particular vector bundles) whose total spaces are the same but base spaces are different?
1
vote
2answers
761 views

Covering spaces need big help Hatcher

Is there any good guide on covering space for idiots? Like a really dumped down approach to it . As I have an exam on this, but don't understand it and it's like 1/6th of the exam. So I'm doing ...
4
votes
2answers
267 views

Heegaard splitting of a 3-manifold with boundary

A Heegaard splitting of a closed orientable 3-manifold $M$ is $M=H \cup H'$, where $H$ and $H'$ are handlebodies. Is there any similar concept for orientable 3-manifolds with boundaries?
5
votes
4answers
928 views

Homotopy composition Hatcher exercise

Show that composition of paths satisfies the following cancellation property: if $f_0 \cdot g_0 \simeq f_1 \cdot g_1 $ and $g_0 \simeq g_1$, then $f_0 \simeq f_1$. So I have two homotopies. So say ...
4
votes
1answer
176 views

What does 'factors through' mean in this context?

I'm trying to understand the proof of the Fundamental Theorem of Algebra in Theorem 3.7 here. I can't get my head around this sentence though If $p(z)$ has no roots at all, the map $p|_{S1(R)}$ ...
4
votes
1answer
472 views

Determining the “positivity” or “negativity” of Chern class (number?) of zero-sets of homogeneous polynomials

If $\Omega$ is the curvature 2-form on a $n-$manifold, then I would think that the Chern classes (forms), $c_k$ are defined as, $det(I + \frac{it\Omega}{2\pi}) = \sum c_k t^k$ I would like to ...
2
votes
1answer
184 views

Betti Numbers with coefficients in reals, rationals & integers.

One knows from the Universal Coefficient Theorem that Integral Homology can be used to derive homology with coefficients in any other groups like e.g. Reals, Rationals, Z/2Z etc. Suppose you have ...
3
votes
1answer
144 views

Why is the group of covering transformations relative to the quotient map isomorphic to a subgroup of the Fundamental Group?

I'm trying to prove the classification theorem for covering spaces. I've got to the stage where I need to show the following: If $H$ a subgroup of $\Pi_1(X,x_0)$ then $\exists Y$ covering space of ...
1
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0answers
55 views

How can we prove that $\mathbb R P^n \to \mathbb R P^n/\mathbb R P^{n-1}$ induces isomorphism on $H_n$ when n is odd?

It is said that $\mathbb R P^n \to \mathbb R P^n/\mathbb R P^{n-1}$ induces isomorphism on $H_n$ when $n$ is odd. How can we prove this?
1
vote
1answer
129 views

What is the the fundamental group of $ H_{\mathbb{R}}/H_{\mathbb{Z}}$

Consider $M = H_{\mathbb{R}}/H_{\mathbb{Z}}$, where $H_{\mathbb{R}} = \lbrace\begin{bmatrix} 1 & x & z \\ 0 & 1 & y \\ 0 & 0 & 1\\ \end{bmatrix}|x,y,z \in ...
5
votes
2answers
385 views

Does homotopy equivalence of pairs $f:(X,A)\to(Y,B)$ induce the homotopy equivalence of pairs $f:(X,\bar A)\to(Y,\bar B)$?

When we have a homotopy equivalence through a pair $f:(X,A)\to (Y, B) $, it is said that we can induce a homotopy equivalence through a pair $f:(X,\bar A)\to (Y,\bar B) $, where $\bar A$ stands for ...
3
votes
0answers
320 views

Covering a connected sum

I have the following problem as a part of my homework: Let $S$ be a closed surface (compact and connected). Show that for every $k$ exists a covering map of $k$ folds $p_k:S_k \rightarrow ...
8
votes
1answer
345 views

A counter example in obstruction theory

Let $K$ denote a simplicial complex and $Y$ some topological space. Let us also denote by $K^n$ the $n$-skeleton of $K$. I would like to have an example for the following situation: There is a map ...
6
votes
0answers
153 views

Finding the universal cover of a matrix group

Consider the group $G = \left\{\begin{pmatrix} a & b\\ 0 & 1\end{pmatrix}: a \in \mathbb{C}^{\times}, b \in \mathbb{C}\right\} \subset GL(2, \mathbb{C})$. How does one find the universal cover ...
2
votes
0answers
156 views

left inverse to trivial fibration is trivial cofibration

It is claimed that in the model category of simplicial sets (with usual model structure), a trivial fibration $X \to Y$ has a section, which is a trivial cofibration. Now, I see that there is a ...
5
votes
1answer
864 views

Homeomorphism vs. Homotopy (Equivalence)

Trying to brush up on some geometric and algebraic topology, I got a little confused about the following: Suppose we have the standard unit sphere $S^2$, but we remove its north and south poles. Is ...
8
votes
1answer
1k views

question about an exercise in hatcher's book (algebraic topology)

I am starting to read Hatcher's book on Algebraic Topology, and I am a little stuck with exercise 6(c) in Chapter $0$. Unfortunately a picture is involved so it doesn't quite make sense for me to ...
7
votes
2answers
248 views

Geometric interpretation of injective/projective resolutions?

I understand the geometric interpretation of derived functors, as well as their usefulness in giving a simple, purely algebraic description of cohomology. I also understand how resolutions are used ...
2
votes
1answer
182 views

Homotopy inverses need not induce inverse homomorphisms

Let $f:X \rightarrow Y$ and $g : Y \rightarrow X$ be homotopy inverses, ie. $f \circ g$ and $g\circ f$ are homotopic to the identities on $X$ and $Y$. We know that $f_*$ and $g_*$ are isomorphisms on ...
14
votes
2answers
325 views

Given a fiber bundle $F\to E\overset{\pi}{\to} B$ such that $F,B$ are compact, is $E$ necessarily compact?

Consider a (locally trivial) fiber bundle $F\to E\overset{\pi}{\to} B$, where $F$ is the fiber, $E$ the total space and $B$ the base space. If $F$ and $B$ are compact, must $E$ be compact? This ...
4
votes
0answers
149 views

Can the disk bundle associated to a vector bundle over a finite CW-complex be obtained by attaching cells?

Let $V$ be a vector bundle (with a chosen metric) over a finite CW-complex $X$ and $B(V), S(V)$ the associated (unit) disk resp. sphere bundles. In the paper Clifford-Modules the authors remark that ...
0
votes
2answers
272 views

Thinking of abelian groups as Z-modules, and allowing alternate ground rings of coefficients

In a paper I am reading involving simplicial homology, I have been told to think about certain Abelian groups(the boundary group and cycle group) as Z-modules so we can allow alternate ground rings of ...
7
votes
0answers
257 views

Weak Bott periodicity vs. strong Bott periodicity

Bruno Harris' proof (or I guess also Bott's original proof) of Bott periodicity (see here for instance) shows that there is a homotopy equivalence $h\colon\mathbb{Z}\times BU \rightarrow \Omega^2 ...
6
votes
1answer
200 views

Orientation in Connected Sum

When define the connected sum of two oriented manifods, one gluing along the reversed orientation of the boundary spheres. I am wondering what is the connected sum of $S^2$ connected sum with $S^2$ ...
12
votes
5answers
3k views

Fundamental group of the special orthogonal group SO(n)

Question: What is the fundamental group of the special orthogonal group $SO(n)$, $n>2$? Clarification: The answer usually given is: $\mathbb{Z}_2$. But I would like to see a proof of that and an ...
11
votes
5answers
364 views

$\varphi$ in $\operatorname{Hom}{(S^1, S^1)}$ are of the form $z^n$

I'd like to see a proof why $\varphi \in \operatorname{Hom}{(S^1, S^1)}$ looks like $z^n$ for an integer $n$. At first I thought I could argue that if I have a homomorphism that maps $e^{ix}$ to some ...
4
votes
1answer
266 views

Applications of the Dold-Thom theorem

Say $X$ has the homotopy type of a CW-complex. The Dold-Thom theorem states that $\pi_i SP(X) \cong \tilde{H}_i(X;\mathbb{Z})$, where $SP(X)$ denotes the infinite symmetric product of $X$. I am ...
3
votes
0answers
278 views

Computing a fundamental group and describing a universal cover

This is the second half of exercise 1.3.21 from Hatcher. Let $Y$ be the space obtained by attaching a Möbius band $M$ to $\mathbb{R}P^2$ via a homeomorphism from its boundary circle to a circle in ...
0
votes
0answers
123 views

Hatcher covering spaces question 8. [duplicate]

Possible Duplicate: The simply connected coverings of two homotopy equivalent spaces are homotopy equivalent Given we have a covering space $\tilde{X}$ of X and a covering space $\tilde{Y}$ ...
1
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1answer
66 views

Quasi-isomorphism between $d$ and $(-1)^{i}d$?

Given a complex $M^{\bullet}$ (say in an abelian category, in particular a module category) with differential $d^{i}: M^{i} \rightarrow M^{i+1}$, we can form another complex $\widetilde{M}^{\bullet}$ ...
8
votes
1answer
614 views

Covering spaces Hatcher question 6.

Let X be the shrinking wedge of circles. Which is the radius of circles $X \in R^2$ such that it's the union of $C_n$ circles centered at $(\frac{1}{n},0)$ with radius $\frac{1}{n}$ for n=1,2,3... ...
5
votes
1answer
285 views

Fundamental groups

I need to calculate the fundamental groups of the following spaces: $X_1 = \{ (x, y, z) \in \mathbb{R}^3 | x \neq 0\} $ $X_2 = \mathbb{R}^3 \backslash \{ (x, y, z) | x = 0, y = 0, 0 \leq z \leq 1 ...
3
votes
2answers
221 views

Van Kampen theorem?

So I'm trying to use Van Kampen theorem to prove that a space is null-homotopic. The thing is I got it down to this $\langle a\mid a=1\rangle$, however I'm confused what does this mean. For ...
1
vote
1answer
174 views

Stiefel manifolds are $n - p - q$ connected.

I'm supposed to show that $V_{n,p}$ is $n - p - 1$-connected. In this case, $V_{n,p}$ is the topological group $O_{n}/O_{n - p}$, where $O_{n}$ is the set of $n\times n$ orthonormal matrices. So this ...
5
votes
4answers
910 views

How should I visualize $S^n$ as the reduced suspension of $S^{n-1}$?

Or is there a particularly canonical homeomorphism between $S^n\subset \mathbb{R} ^{n+1}$ and $SS^{n-1}$ (the reduced suspension)? Right now the only thing that comes to mind is to think of the sets ...
3
votes
2answers
220 views

Maps between real projective spaces

Say we have a linear map, $f: \mathbb{R}^{m+1} \to \mathbb{R}^{n+1}$, and we define $\mathbb{RP}^{n}$ as $(\mathbb{R}^{n+1} - \{0\})/{\sim}$ with $\sim$ define by $x \sim y$ if $y = \lambda x$ for ...
0
votes
3answers
327 views

Is there a covering space proof of $\pi_1(S^1) \cong \mathbb{Z}$ .

I'm looking at 29 pages. When he does the does the calculation of the fundmental group of circle. Was wondering is there a easier way using covering spaces to prove this. As I'm trying to find a ...
0
votes
1answer
214 views

A Reduced (Simplicial) Homology Question

I am aware of tools for computing reduced homology when dealing with nonempty simplicial complexes, but: What would be an effective approach for computing the reduced homology of the empty simplicial ...