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

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7
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0answers
124 views

There does not exist a map $S^2\times S^2\to \mathbb{CP}^2$ with odd degree.

The following is a problem from a topology qualifying exam I am studying for: Show there does not exist a map $S^2\times S^2\to \mathbb{CP}^2$ with odd degree. I think I am doing something wrong, ...
1
vote
0answers
32 views

Composition of covering maps is a covering map if the inverse image is finite. [duplicate]

Let $q: X \to Y$ and $r: Y \to Z$ be covering maps. Let $p=r \circ q$. Show that if $r^{-1}(z)$ is finite $\forall z$, then $p$ is a covering map. $\textbf{My Attempt:}$ Let $U$ be an arbitrary open ...
0
votes
1answer
93 views

Why is this induced homomorphism not surjective?

Let $p: Y \to X$ be a covering with fix base point. I have already shown that the induced homomorphism $p_{*}=\pi_1(Y,y_0) \to \pi_1(X,x_0)$ is injective. However, since we are not calling it an ...
3
votes
1answer
84 views

Relative Simplicial Approximation

While I was studying the cellular approximation theorem on May's "A Concise Course in Algebraic Topology" I found something a bit unclear. I agree with the fact that, given two CW-complexes $X,Y$, and ...
2
votes
1answer
82 views

Calculating fundamental group of adjunction space with linear transformation.

$X = D^{2} \times S^{1} \cup_{f} S^{1} \times D^{2}$, where $f : S^{1} \times S^{1} \to S^{1} \times S^{1}$ is a map induced by the linear map on $\mathbf{R}^{2}$ given by the matrix $$\left( \begin{...
3
votes
0answers
149 views

Comparing the torus and a small wedge product, and their universal coverings

I have the following problem: Let $A = S^1 \times S^1$ and $B = S^1 \vee S^1 \vee S^2$. Compute their universal coverings. Prove that $A$ and $B$ have isomorphic homology groups for any $n \in \...
0
votes
1answer
38 views

problem with $g^{-1}g$ homotopic to $e_a$

I found the following problem in my lecture notes. We need to prove $g^{-1}g$ is homotopic to $e_a$ where $g$ is a path from $a$ to $b$ and $e_a$ is the identity path at $a$. My lecture notes say ...
0
votes
1answer
32 views

Connectedness of combinatorial complexes with no free faces

I'm currently reading the paper "$\mathcal{VH}$ complexes, towers and subgroups of $F \times F$" by Bridson & Wise. There they define combinatorial complexes as follows: A continous map between ...
8
votes
1answer
186 views

What is the intended solution to exercise 0.16 in Hatcher? (Contractibility of $S^\infty$)

Hatcher, Algebraic Topology, Exercise 0.16 reads: Show that $S^\infty$ is contractible. Let's look at the definition Hatcher gives for $S^\infty$: There are natural inclusions $S^0 \subset S^...
1
vote
0answers
42 views

the Pontryagin number for 4-dim surface bundle

Corollary 1.8 in arxiv.org/pdf/1103.0218 implies that the Pontryagin number for a 4-dim surface bundle is non-zero only when the surface has a genus $g>2$. I would like to ask what is the minimal ...
1
vote
1answer
48 views

What is meant by the “collar of a disk”?

I've seen people use the term "collar of a disk" within the context of algebraic topology. What is meant by this term? Is it the "boundary" of a disk?
1
vote
1answer
39 views

Queries about why $p: S^1 \to S^1$ such that $p(z)=z^n$ is a covering map.

To understand why this is true I have been told to imagine the covering space as ..."drawing a circle wrapping around a cylinder $n$ times with $(n-1)$ intersections". The covering space $p: \mathbb{...
0
votes
1answer
171 views

A differential topology lemma

Consider the following lemma (1) How come he talks about degrees here, after all he doesn't assume $X$ to be oriented? (2) Why is $\bar{v}|\partial X$ homotopic to $g$? (NOTE: we consider them as ...
0
votes
1answer
62 views

Finding lifted paths, homotopy lifting

I am given a covering map $p: \mathbb{R}^+ \times \mathbb{R} \to \mathbb{R}^2 \setminus \{0,0\}$ defined by $p(r, \theta)=(r \cos 2 \theta,r \sin 2 \theta)$ Let $\alpha: [0,1] \to \mathbb{R}^2 \...
2
votes
0answers
34 views

How do I prove that $U(r) \to S(r,n) \to G(r,n)$ is a fibration?

$U(r)$ here is unitary group of $\mathbb{C}^n$, $S(r,n)$ is the Stiefel manifold of $r$-frames in $\mathbb{C}^n$ and $G(r,n)$ is the Grassmannian manifold of $r$-planes in $\mathbb{C}^n$. I've tried a ...
2
votes
0answers
79 views

Stereographic projection for a link/knot

I've been trying to understand the topological "link" between algebraic varieties and their associated knots/links and to this end I've been reading F. Kirwan's book, "Complex algebraic Curves". The ...
1
vote
2answers
124 views

Homotopy equivalence between $X/A$ and $X$?

Consider the following definition: Definition: Let $(X, A)$ be a topological pair. We say $A$ has the homotopy extension property with respect to a space $Y$ if given any continuous map $f:X\...
3
votes
1answer
152 views

Clarifying the definition of regular simplicial action

I'm currently studying Chapter 3 of the book "Introduction to compact transformation groups" by Bredon. The main matter that is discussed in this chapter are simplicial actions of a finite group $G$ ...
1
vote
2answers
148 views

Homology of manifolds with boundary

If $M$ is a compact topological manifold WITH boundary does it follow that its homology groups are finitely generated and zero almost all of them? I know it is true in case it has no boundary (i.e. is ...
1
vote
1answer
187 views

Determing if the fundamental group of the following is isomorphic to either the trivial, infinite cyclic, figure eight fundamental groups

Hello there i am having trouble to determine isomorphisms of the following fundamental groups: 1) the torus $T$ with a removed point. 2) $\mathbb{R}^3$ with nonnegative axes 3) $S^1 \cup (\mathbb{R}...
1
vote
0answers
53 views

Question about the Fundamental group of circle.

We were taught in class that $\pi(S^1)=\mathbb{Z}$ I am a bit confused as to why this is true. The motivation behind this was that if we have $\alpha_1$ moving around the circle once in a clockwise ...
3
votes
1answer
83 views

Homology of the pair $(S^2, \{\pm x\})$?

I'm trying to compute the homology of the pair $(X, A)$ where $X=S^2$ and $A=\{\pm x\}$ is a pair of antipode points. Can anyone check if I'm doing it right? What I tried: 1. $A$ has two path ...
1
vote
0answers
27 views

Prove that if $p: Y \to X$ is a covering space and $X$ is path connected, then the cardinality of $p^{-1}(X)$ is constant. [duplicate]

Let $p: Y \to X$ be a covering space and $X$ is connected. I want to show that $\forall x \in X$ the cardinality of $p^{-x}$ is the same. $\textbf{My Attempt:}$ Let us first fix a point $x_0 \in X$ ...
3
votes
0answers
112 views

The vanishing (?) cohomology of the Milnor fiber

Setup. Say we have a germ of a holomorphic function $f:(\mathbb C^{n+1},0)\to (\mathbb C,0)$ with a critical point at the origin. There is an $\epsilon>0$ small enough so that $f$ becomes a ...
1
vote
1answer
60 views

How to find the braids that when closed make the $6_1$ knot.

I have the $6_1$ knot and my question is how can I easily find the braids that when closed make this knot, what's the easiest way in general for any knot.
2
votes
0answers
150 views

group cohomology for SO(3) and SO(3,1)

I am studying relativistic quantum mechanics and I have encountered the concept of projective representation for a group. I have read in http://groupprops.subwiki.org/wiki/Projective_representation ...
8
votes
1answer
162 views

Cohomology of $S^2\times S^2/\mathbb{Z}_2$

The product of two spheres admits a diagonal $\mathbb{Z}_2$-action, $(x,y)\mapsto (-x,-y)$. I'm trying to compute the integral singular cohomology ring of the orbit space $X$ of this action. $X$ is ...
0
votes
0answers
120 views

lifting a closed curve

Is it always true (because of covering spaces has homotopy lifting property)? loop $f$ lifts to a closed curve if and only if any curve freely homotopic to $f$ lifts to a closed curve. or we have to ...
0
votes
1answer
37 views

How do we represent relations in a free group by loops in a manifold?

Hypothesis: Let $F$ be a finitely presented group s.t. $$ F = \left\langle S \mid R \right\rangle $$ Let $X$ be a $4$-manifold. Question: I've seen it asserted that we can represent each relation ...
2
votes
1answer
159 views

Geometric picture of Stiefel-Whitney class of a tangent bundle?

The first Stiefel-Whitney class $w_1$ of a tangent bundle of a manifold $M$ has a simple geometric picture: if there is a loop that the orientation of the tangent space reverses, then $w_1\neq 0$ and ...
3
votes
1answer
268 views

Number of Fixed Points in a Map from the Torus to itself using Lefschetz Trace

Let $f: X \to X$ be a continuous map. For any fixed point $f(x) = x$ with $x \in X$, we can find the index of that fixed point $i(f,x)$. The Lefschetz-Hopf formula says: $$ \sum_{x \in \mathrm{Fix}(...
11
votes
1answer
371 views

Showing that every finitely presented group has a $4$-manifold with it as its fundamental group

Wikipedia: For any finitely presented group it is easy to construct a (smooth) compact 4-manifold with it as its fundamental group. Question: How do we do this? EDIT: Below is a proof sketch ...
5
votes
3answers
238 views

Showing $\pi_1(M) = F$ ($F$ a finitely generated free group and $M$ an $n$-manifold of dimension greater than $2$)

Problem: Let $F$ be a finitely generated free group. Prove that there is an $n$ manifold, $M$, $n > 2$ with $\pi(M) = F$. Let $F = F_S$ s.t. $|S| \in \mathbb{N}$. If I could show that there ...
2
votes
0answers
256 views

Homeomorphic or Homotopic

Q1: Are the Fig (a) and (b), the equivalence "=" is Homeomorphic or Homotopic? ps. for details of the figures see Ref here. I learned that "The characterization of a homeomorphism often leads ...
10
votes
1answer
259 views

Showing that $\pi(M \# N) = \pi(M) \ast \pi(N)$ for $n$-dimensional manifolds $M$,$N$

Problem: Let $M$ and $N$ be $n$-dimensional manifolds, where $n > 2$. Let $M \# N$ be their connected sum. Show that $\pi(M \# N) = \pi(M) \ast \pi(N)$. RE-EDITED Attempt: Let $U_2$ and ...
8
votes
1answer
310 views

Relation between Stiefel-Whitney class and Chern class

A complex vector bundle of rank n can be viewed as a real vector bundle of rank 2n. From nLab, we have that the second Stiefel-Whitney class of the real vector bundle is given by the first Chern class ...
4
votes
0answers
49 views

Are 4-dimensional mapping tori always spin?

We know that all compact orientable manifolds of dimension 3 are spin. In 4 dimensions, $CP^2$ is not spin. I would like to ask if all 4-dimensional compact orientable mapping tori are spin?
45
votes
7answers
9k views

Hanging a picture on the wall using two nails in such a way that removing any nail makes the picture fall down

A friend of mine told me that it's possible to hang a picture on the wall from a string using two nails in such a way that removing either of the two nails will make both the string and picture fall ...
5
votes
1answer
105 views

Why are the total spaces of two Serre fibrations equivalent when the bases and the fibers are equivalent?

Suppose $B$ is a pointed space and suppose $f\colon E\to B$ and $f\colon E'\to B$ are two Serre fibrations. Let moreover a map $g\colon E\to E'$ be given such that $f=f'\circ g$ which is a weak ...
3
votes
0answers
71 views

The Morse complex of a manifold with boundary

For a smooth manifold with boundary $M$ and $\partial M = V_+ \cup V_-$ two disjoint sets of boundary components, one usually defines the Morse complex of $M$ using a Morse-Smale pair $(f,X)$ such ...
0
votes
1answer
49 views

Why must $\phi(1) \in H$ if $\phi$ is a deck transformation of $G$, with normal subgroup $H$

Hypothesis: Let $H$ be a normal subgroup of $G$. Let $p: G \rightarrow G/H$ form the universal covering over $G/H$. Let $\phi$ be an arbitrary deck transformation of $G$. Question: Why is it that $\...
0
votes
1answer
94 views

Two Basic Questions about Deck Transformations

Let $p: \widetilde{X} \rightarrow X$ form a covering space over $X$. Let $$ D = \{\sigma : \sigma \text{ is an $\widetilde{X}$ isomorphism}\} $$ form the set of deck transformations of $\widetilde{...
1
vote
1answer
37 views

Is it true that $\pi(G/H, x_0) \cong \pi(G, x_o)/ \pi(H, x_0)$?

Is it true that $$ \pi(G/H, x_0) \cong \pi(G, x_o)/ \pi(H, x_0) \text{?} $$ if $H$ is a subgroup of $G$?
3
votes
2answers
104 views

Showing that $\pi(G/H, 1) = H$ under a condition

Problem: Let $G$ be a simply connected (i.e., $\pi(G)=1$) topological group, and let $H$ be a discrete normal subgroup. Prove that $\pi(G/H,1) = H$. I know that since $H$ is a discrete subgroup of $...
0
votes
1answer
149 views

Product of weak Hausdorff space is weak Hausdorff

I have read on May's Algebraic Topology such that the category of weak Hausdorff space $\mathcal{wTop}$ has same limit as $\mathcal{Top}$, which means Product of weak Hausdorff space is weak ...
9
votes
1answer
598 views

Easier proof about suspension of a manifold

For what manifolds $M$ is the suspension $\Sigma M$ also a manifold? By the suspension of a topological space $X$ (not necessarily a manifold), I mean the space $$\Sigma X = (X \times [0,1])/{\sim}$$ ...
6
votes
1answer
206 views

Examples of de Rham cohomology being easier to compute that singular cohomology

De Rham's theorem states that for any smooth manifold $M$ the singular cohomology and de Rham cohomology of $M$ are isomorphic. Are there any examples of manifolds for which it is easier to compute ...
7
votes
1answer
112 views

Describe all the quotient groups of $\mathbb{Z} \oplus \mathbb{Z}$

I'm thinking about a problem in algebraic topology of how to determine all the $k$-fold covers of $\mathbb{T}^2$, where $k$ is a certain integer. In thinking about the problem, I came upon the ...
5
votes
0answers
63 views

Prove that if $f:D^2\to D^2$ is a homeomorphism, then $f(S^1)=S^1$

I've already proved that set of points $z\in D^2$ such that $D^2-z$ is simply connected is precisely $S^1$. Now from this, I'm supposed to conclude that if $f:D^2\to D^2$ is a homeomorphism, then $f(S^...
0
votes
1answer
29 views

Prove that $h$ and $p*q$ are homotopic relative to {$0,1$}

Let $0<s<1$. Given paths $p$ and $q$ with $p(1)=q(0)$, define $h$ by the formula $$h(t) = \begin{cases} p(t/s),& \text{if} \quad 0 \leq t \leq s \\ q((t-s)/(1-s)), &\text{if} \quad s\...