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
1answer
28 views

The cone minus its apex deformation retracts onto its basis

Let $X$ be a topological space and $$C(X)=X\times [0,1]/X\times \{0\}$$ be the cone on $X$. Call $P$ the apex of the cone. I want to show that $C(X)-P$ deformation retracts onto $X\times \{1\}$. My ...
2
votes
2answers
56 views

Isomorphism of modules arising from algebraic topology

While studying for a course in algebraic topology, the following question popped out: Let $S,R$ be two commutative rings with unit, $A,B$ two $S$-modules, and assume that $R$ is also an ...
-1
votes
0answers
19 views

AI / Machine Learning related to high/modern/front mathematics [on hold]

I major math and cs. and i'm interested in ai/machine learning/data mining. so i want to know what math subjects are used in frontier of these technology. especially, high mathematical tool, like ...
1
vote
0answers
29 views

Relative cohomology of a vector space module non-zero vectors

I am trying to explicitly compute the relative cohomology groups $H^m(\mathbb R^n,\mathbb R^n_0;\mathbb Z)$, where $\mathbb R^n_0$ is all the non-zero vectors in $\mathbb R^n$. I think that the answer ...
0
votes
0answers
31 views

Why is $(X\times EG)/G\to X/G$ a fibration if $G$ acts freely on $X$?

Suppose that $G$ acts freely on $X$, and let $EG$ be a contractible space on which $G$ acts freely. According to many references, the projection $(X\times EG)/G\to X/G$ is a fibration. However, I ...
3
votes
1answer
38 views

Extending cellular maps between aspherical complexes

In a paper I read, the author seemed to use a property similar to: Let $X, Y$ be two aspherical CW-complexes and $f : X^{(2)} \to Y^{(2)}$ be a cellular map between their 2-skeletons. Then $f$ ...
3
votes
1answer
52 views

vector bundles on $X \times S^2$ built via clutching functions

Let $X$ be a compact Hausdorff space and $E \to X$ a (complex) vector bundle over $X$. We can build a vector bundle on $X \times S^2$ by clutching functions, i.e an automorphism $f:E \times S^1 \to E ...
2
votes
4answers
273 views

Should I read about Manifolds or Algebraic Topology?

I really enjoy doing maths and it fills quite a lot of my spare time. I'm starting my first year in the university on october and I probably won't have that much time for independent reading once ...
0
votes
1answer
33 views

What are the deck transformations of this threefold cover of the figure 8?

Hatcher lists some examples of covers of a figure 8 (page 58). One of them corresponds with the group with two generators $a$ and $b$ and the relations $a^2, b^2, aba^-1, bab^-1$. I thought ...
2
votes
0answers
39 views

Role of the Thom space in the Pontryagin-Thom construction

I am trying to understand the Pontryagin-Thom theorem; especially how the Thom space comes into play. Just to bring everyone on the same page: I am specifically talking about the construction of an ...
3
votes
0answers
42 views

Surgery and Euler Characteristic

I am trying to find out how a $(p,n-p)$ surgery affects the Euler Characteristic of an orientable, $n-$ dimensional, compact manifold. Call the initial manifold $M$ and the post-op manifold $M'$. This ...
11
votes
3answers
139 views

Fundamental group of the product of 3-tori minus the diagonal

I have a past qual question here: let $T^3 = S^1 \times S^1 \times S^1$ be the 3-torus, and let $\Delta = \{ (x,x) \in T^3 \times T^3 \colon x \in T^3 \}$ be the diagonal subspace. Compute $\pi_1(T^3 ...
0
votes
2answers
33 views

Why is $H_k(X^n) = H_k(X)$, $k < n$, where $X^n$ is the $n$-skeleton of the CW-complex $X$?

Why is $H_k(X^n) = H_k(X)$, $k < n$, where $X^n$ is the $n$-skeleton of the CW-complex $X$? I am probably overlooking something trivial. I tried using the fact that $H_k(X^n,\emptyset)\cong ...
3
votes
1answer
38 views

Question about the proof of the universal coefficient theorem

When deriving the universal coefficient theorem, in class we proceeded as follows: We have the SES: $$0\to ...
1
vote
0answers
33 views

Homology group of the join

Prove that $$\tilde{H_n}(X*Y) \cong H_{n-1}(X \times Y/X \vee Y)$$ Firstly, I set $$A=X \times Y \times [0,1) / (x,y_0,0) \sim (x,y_1,0) $$ $$B=X \times Y \times (0,1] / (x_0,y,1) \sim (x_1,y,1) $$ ...
3
votes
1answer
49 views

Eilenberg-Maclane space $K(G\rtimes H, 1)$ for a semi-direct product.

We know that $K(G\times H, 1)=K(G,1)\times K(H,1)$. Do we know something like this for a semi-direct product, where $K(G,1)$ denotes the Eilenberg-Maclane space.
2
votes
2answers
48 views

Find a CW complex with prescribed homology groups

A past qual question asks to construct a CW-complex $X$ with $H_0(X) = \mathbb{Z}$, $H_5(X) = \mathbb{Z} \oplus \mathbb{Z}_6$, and $H_n(X) = 0$ for $n\not= 0, 5$. One can build a CW-complex $Y$ by ...
1
vote
1answer
33 views

Show this Map $f:M\rightarrow S^n$ is a Diffeomorphism

Let $M$ be a compact, connected smooth manifold of dimension $n>1$, and $f:M\rightarrow S^n$ a smooth map such that the differential $df_x:T_xM\rightarrow T_{f(x)}S^{n}$ has full rank $= n$ for all ...
2
votes
1answer
28 views

Homology groups of a simplicial complex

I have a question from a qualifying exam: let $X$ be the simplicial complex that consists of the 3-simplices $(v_1,v_2,v_3,v_4)$, $(v_3,v_4,v_5,v_6)$, $(v_1,v_2,v_5,v_6)$, where the $v_i$'s are all ...
0
votes
1answer
24 views

Connected components of the complement of a closed geodesic on a hyperbolic surface.

Let $M$ be homeomorphic to a 2-sphere with a finite number $\geq 3$ of points removed. This implies that $M$ can be equipped with a complete, finite area hyperbolic metric. I imagine $M$ as an ideal ...
1
vote
1answer
29 views

Number of connected components of the complement of a closed curve.

Let $\gamma:[0,1] \rightarrow \mathbb{R}^2$ be a continuous, closed curve (i.e. $\gamma(0) = \gamma(1)$). My question is about the number of connected components of the complement $\mathbb{R}^2 ...
0
votes
1answer
56 views

A detail in the proof of Jordan's theorem

The (usual) proof with homology first inductively shows that complements of embedded disks are acyclic. In doing so, Mayer-Vietoris is applied, and this assumes that their complements in the sphere ...
3
votes
1answer
65 views

Algebraic question (In Hatcher's book, exercise: 1.1.16-c:)

In Hatcher's book, exercise: 1.1.16-c: Show that there are no retractions $r :X \rightarrow A$ in the following cases: (c) $X = S_1 × D_2$ and A the circle shown in the figure. Page 39 By ...
3
votes
1answer
28 views

Two definitions of framed manifolds/cobordism

One notion of framed cobordism is that $\Omega_{fr}^n(X)$ is a certain equivalence class of $n$-manifolds $M^n$ in $X$ with a trivialization of the normal bundle $N M^n\subseteq TX$. For compact ...
1
vote
2answers
63 views

How to call two subsets that can be deformed into each other?

Given a topological space $X$, is there a canonical name for the equivalence relation generated by the following relation on the subsets of $X$? $A \sim B :\Leftrightarrow \exists \text{ continuous } ...
6
votes
2answers
99 views

Is this Space Homotopy Equivalent to $S^2$

Let $X$ be the space $S^1$ with two $2$-cells attached via maps of relatively prime degrees. This space is simply connected and has the homology of $S^2$, but is it homotopy equivalent to $S^2$?
2
votes
0answers
33 views

Cellular structure on a manifold [duplicate]

Is it always possible to put a cell structure on a manifold? in other words is it possible to decompose a manifold as a CW complex? I know that by Morse theory we always have a handle decomposition of ...
5
votes
1answer
74 views

Tensor product of real line bundles is trivial as a map $\mathbb{R}P^\infty\to\mathbb{R}P^\infty\times\mathbb{R}P^\infty\to\mathbb{R}P^\infty$

The tensor product of a real line bundle with itself is trivial, as is easily seen by looking at the transition functions or checking the Stiefel-Whitney class. Real line bundles are classified by the ...
2
votes
1answer
56 views

Proof of an algebraic topological lemma

I have been given the following result without proof, so I would like to show it is true: Let $I=[0,1]$, then: $$H^\bullet(I,\partial I;R)\cong H^\bullet(I/\partial I,*;R)\cong ...
1
vote
2answers
102 views

Representation of nullhomologous loop on compact surface as a product of commutators.

Why this sentence is true?: Assume that $M$ is compact surface and $f: S^1 \to M$ is nullhomologous and without selfintersections. Letting $g$ be the genus and $b$ the number of boundary components ...
0
votes
0answers
29 views

How closely is algebraic topology related to mathematical physics? [closed]

I have heard that research in string theory uses mostly pure math . Sounds fascinating....
1
vote
0answers
39 views

Homotopy groups of unitary groups

in this paper I found some explicit generators of homotopy groups of unitary groups, for example $\pi_3[SU_2]$: $\begin{bmatrix}z_1\\z_2\end{bmatrix}$$\rightarrow $$\begin{bmatrix}z_1 ...
1
vote
1answer
21 views

A further question on reparametrization.

Hatcher contains the following paragraph: Define a reparametrization of a path $f$ to be composition $f\psi$ where $\psi:I\to I$ is any continuous map such that $\psi(0)=0$ and $\psi(1)=1$. ...
5
votes
2answers
117 views

Is there a self-homeomorphism of the 2-sphere with exactly 3 fixed points?

I don't believe so, but I'm not sure how to prove it. The Lefschetz-Hopf theorem says in this case that the sum of the fixed point indices is 0 or 2 (since our map is a self-homeomorphism). My ...
7
votes
2answers
102 views

Long exact sequence of a fibration, center

Let $p:E \rightarrow B$ be a fibration with fiber $F$ . Associated to this we have a long exact sequence $$\cdots \rightarrow \pi_n(F) \rightarrow \pi_n(E) \rightarrow \pi_n(B) \rightarrow ...
1
vote
2answers
49 views

$H_n(\mathbb{R}P^4 \times S^1)$

I have been trying to compute the homology of $\mathbb{R}P^4 \times S^1$ by using cellular homology. Nevertheless, I cannot see what the attaching maps are.
1
vote
0answers
24 views

How to show that the local homology at a point on the boundary of the half-plane is zero?

Let $\mathbb{H}^n = \{x = (x_1,\dots,x_n) \in \mathbb{R}^n \mid x_n \geq 0 \}$. i.e. the half plane. Let $y \in \mathbb{R}^{n-1} \times \{0\}$ be a point on the boundary. How do I show that ...
1
vote
2answers
143 views

A doubt in Hatcher's explanation of reparametrization.

Hatcher contains the following paragraph: Define a reparametrization of a path $f$ to be composition $f\psi$ where $\psi:I\to I$ is any continuous map such that $\psi(0)=0$ and $\psi(1)=1$. ...
1
vote
1answer
38 views

Covering $S^{n-1}$ with $n+1$ closed sets containing no antipodal points

For proving the equivalence of the Theorems of Borsuk-Ulam and Lusternik-Schnirelmann, we need to cover $S^{n-1}$ with closed sets $F_1,\dots,F_{n+1}$ such that none of the $F_i$ contains a pair of ...
3
votes
1answer
37 views

The long exact sequence in reduced K-theory. How to glue together the short ones?

I'm trying to filling the details about the construction of the long exact sequence in K-theory. I'm using the notation of Hatcher's book (pages 52-53). here is a related question, even thought it's ...
3
votes
1answer
90 views

Complex (topological) K-theory of $S \Sigma$ for a surface $\Sigma$.

In the following question here: What's the K-group of a surface? The $K$-theory of a compact orientable surface is computed. I was curious if it was also possible to compute the "higher" ...
2
votes
1answer
30 views

Relative homology of disk and any of its subspace is isomorphic to reduced homology of the subspace?

Consider the pair of topological space $(\mathbb{D}^n,X)$, where $X \subset \mathbb{D}^n$ is a subspace. We know that there is a long exact sequence of reduced homology groups, $$\cdots \to ...
2
votes
1answer
43 views

“Visual” interpretation of the Bott Periodicity for complex vector bundles

I've just read the Bott Periodicity proof (complex case) in Hatcher book about K-theory. At first it seems that using this theorem I could conclude that every complex vector bundles over $S^{2n+1}$ ...
0
votes
0answers
25 views

Prove that reduced homology is relative homology to a point.

I am trying to follow the proof here that reduced homology $\tilde{H}_n(X)$ is the same as $H(X,x_0)$, the relative homology of $X$ to a point $x_0 \in X$. What I don't get is the very last step, ...
2
votes
1answer
33 views

Attaching space, help on visualization

Let $X$ be a topological space, $A\subset X$ a closed subspace. $CA$ means the cone of $A$, and by $SA$ I'll denote the suspension of $A$. I need to prove that $$ \left( \left( (X \cup CA) \cup ...
1
vote
0answers
26 views

Two sheeted disconnected cover of a connected topological space has exactly two components

Let $\tilde{X}$ be a two-sheeted cover of a connected topological space X. If $\tilde{X}$ is disconnected then this has exactly two components. Further each component is homeomorphic to X by the ...
2
votes
2answers
68 views

Homology groups equal when $\cdots \rightarrow 0 \rightarrow H_n(\mathbb{S}^m) \rightarrow H_{n-1}(\mathbb{S}^m) \rightarrow 0 \rightarrow \cdots$

I'm reading a set of notes but I don't understand the following concept. We have a long exact sequence $\cdots \rightarrow 0 \rightarrow H_n(\mathbb{S}^m) \rightarrow H_{n-1}(\mathbb{S}^m) ...
1
vote
0answers
33 views

$\pi_0$ of $M(2) \wedge M(2)$

My motivation is trying to understand Tom Goodwillie's argument here: http://mathoverflow.net/questions/87919/difficulties-with-the-mod-2-moore-spectrum and the only thing I don't get is why ...
3
votes
1answer
29 views

Equivalence of relative and (reduced) homology for arbitrary pairs

I could not find my mistake in the following argument, though I know it is wrong. This is more like a "Q&A", since there is nothing to "prove" in the positive sense. Here it goes: For an ...
0
votes
0answers
35 views

How to describe a homomorphism from a fundamental group to a finite group?

Let $S$ be a connected locally noetherian scheme with $s$ a geometric point of $S$. I read something like this: to give a surjective continuous homomorphism from $\pi_1(S, \bar{s})$ to a finite group ...