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

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10
votes
2answers
312 views

Ehresmann Connection of the tangential bundle & Chern classes

I must have mistunderstood something, this is giving me quite a headache. Please, do stop me once you notice an error in my thinking. The Ehresmann Connection $v$ of some Bundle, $E\to M$, is the ...
4
votes
2answers
63 views

How do I show that $S^1$ is the suspension of $S^0$?

How do I show that $S^1$ is the suspension of $S^0$? I have all the definitions here, I'm just bad at applying them. The suspension of a topological space $X$ is the quotient $CX / (X × ${$1$}$)$, ...
4
votes
2answers
199 views

Cellular Boundary Formula

In Hatcher's book we find, when computing the boundary maps of cellular homology, Cellular Boundary Formula: $d_n(e^{n}_\alpha)=\sum_\beta d_{\alpha\beta}e^{n-1}_{\beta} $ where $d_{\alpha\beta}$ is ...
3
votes
2answers
198 views

Topological degree and homology

Let $f : \mathbb{S}^n \to \mathbb{S}^n$ be a continuous function. From homology, the degree $\deg_1(f)$ of $f$ can be defined as the integer $n$ such that $f_* : x \mapsto n \cdot x$, where $f$ ...
2
votes
2answers
36 views

boundary( geometric realization of the standard n-simplex) is not equal to the geometric realization of the boundary(standard n-simplex) in general

Consider $|\Delta^n|$ the geometric realization of the standard n-simplex. I know that the $|\delta \Delta^n|=\delta|\Delta^n|$ isn't true in general, whereby $\delta \Delta^n$ is the boundary of the ...
2
votes
2answers
41 views

Is composition of covering maps covering map?

In Munkres book, composition of covering maps is covering map when $r^{-1}(z)$ is finite for each $z$ in $Z$ where $q : X\to Y$ , $r:Y\to Z$ are the covering maps. I tried hard to find an example that ...
2
votes
2answers
50 views

intersection number of twocompact oriented manifolds

I have an oriented manifold M of n dimension and 2 oriented submanifolds, one of dimension k and the other of dimension n-k , I have to understand which is the intersection number of those manifolds. ...
2
votes
2answers
114 views

Morse Theory and critical groups

Please I have a question: What is the relation between Morse theory and critical point theory ? I studied the Morse inequalities and critical groups, but i can not not find or at least i do not ...
0
votes
1answer
68 views

Reduced suspension and unreduced suspension

In May's "A concise course in Algebraic Topology" Chap 14 section 1, the author says $\Sigma (X_+)$ is $\Sigma X\vee S^1$ where $X$ is an unbased space and $X_+$ is the union of a disjoint basepoint ...
0
votes
1answer
43 views

Empty set in a simplicial complex

Should the empty set be considered a simplex in a simplicial complex? Which justifications exist for the answer? I guess it is somewhat comparable to $1$ not being a prime number.
0
votes
1answer
42 views

a topological space is the union of its irreducible components

if we define irreducible component of a topological space $X$ as the maximal closed irreducible subset of $X$,prove that we can write $X$ as the union of its irreducible components. how to approach ...
0
votes
1answer
26 views

a region homeomorphic with klein bottle

prove that if we consider this shape in the picture below with the equivalency relation that : a & b are in one class if they are antipoles in inner or outer circles, then the induced quotient ...
0
votes
1answer
35 views

CW complex with no cells in dimension $n$

Hi need some help with the following problem: if $X$ is a CW complex with no cells of dimension $n$ then $\tilde{H}^n(X,G)=0$. where $G$ is any group. thanx.
0
votes
1answer
49 views

Lifting property of a covering space

A common example of a covering space is $p:\mathbb R \rightarrow S^1 $, $p(t)= e^{2 \pi it}$, which can be looked at as an infinitely long spiral placed over a circle. Consider the double loop ...
0
votes
1answer
25 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 ...
0
votes
1answer
21 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 ...
0
votes
1answer
83 views

Show that an inclusion is an isomorphism in homology

I'm struggling a bit with an exercise from a book, in a chapter about the Jordan-Brouwer separation theorem. It goes as follows: (note: $s_{n-1}$ is a topological space homeomorphic to ...
0
votes
1answer
29 views

Necessary and sufficient condition for existence of a deck transformation

I am considering the following problem: $\tilde X$ path connected, $X$ path connected, locally path connected, $P:\tilde X \to X$ covering map, $x_0 \in X, \tilde x_0, \tilde x_1 \in p^{-1}(x_0).$ I ...
0
votes
1answer
46 views

The exact sequence of a pair - something fishy going on here!

This question is related to my previous question. After knowing that $X \cong S^3$ and $A \cong S^1$, with $X/A \cong S^2$, I attempt to construct the long exact sequence of a pair. I need to use the ...
0
votes
1answer
74 views

Liftings in Covering Map Closed

Let $p:\tilde{X} \mapsto X$ be a covering map with $\tilde{X}$ path connected. Why are all liftings of a closed path $f$ in $X$ either closed or not closed? If $\omega$ is a path from $\tilde{x_0}$ ...
0
votes
1answer
16 views

Show that a finite graph product of finitely generated groups is finitely generated, and similarly for finitely presented groups.

Show that a finite graph product of finitely generated groups is finitely generated, and similarly for finitely presented groups. I think when we have a finitely generated groups,the graph product of ...
0
votes
1answer
52 views

Mayer-Vietoris sequence confusion

In the Mayer-Vietoris exact sequence $$... \rightarrow H_n(A) \oplus H_n(B) \rightarrow H_n(X) \rightarrow H_{n-1}(A \cap B) \rightarrow H_{n-1}(A) \oplus H_{n-1}(B) \rightarrow...$$ I am confused ...
0
votes
1answer
63 views

Question about degrees of maps from $S^1 \rightarrow S^1$

Note: Since the degree of a map is independent of the base-point I'll speak loosely and just say $\pi_1(S^1)$. One definition of the degree of a map $f$ from the circle to itself is the number $k$ ...
0
votes
1answer
29 views

Showing that a map can be deformed into the identity.

Suppose $F((a,b), k) = (ae^{\pi i k}, be^{\pi i k})$ where $0 \leq k \leq 1$. Now would $g(a,b)$ = $(-a,-b)$ if $g : S^{1} \rightarrow S^{1}$?
0
votes
1answer
69 views

$X \simeq Y \Rightarrow \pi_0(x) \cong \pi_0(y)$

Let $X$ and $Y$ be topological spaces. $X \simeq Y \Rightarrow \pi_0(X) \cong \pi_0(Y)$ I'm trying to prove this and I have no idea where to begin. Any hints would be helpful but I really don't want ...
0
votes
1answer
54 views

Homotopy equivalences

If $x\in X$, let $C(x)$ the path component of $x$ (the biggest path connected set containing $x$), and similarly if $y\in Y$. Let $C(X)$ and $C(Y)$ the family of all path components of $X$ and $Y$. ...
0
votes
1answer
27 views

The line between the centers of two k-simplices crosses through the centers of lower dimensional simplices

"The line between the centers of two k-simplices crosses through the centers of lower dimensional simplices." Is this always true? The barycentric centers are equidistant from the vertices. k=2 So ...
0
votes
1answer
94 views

Manifolds Resulting from Gluing Tori

I'm trying to show that if solid tori $T_1, T_2; T_i=S^1 \times D^2$ ,are glued by homeomorphisms between their respective boundaries, then the homeomorphism type of the identification space depends ...
0
votes
1answer
58 views

Isomorphic homology and cohomology groups

Let $X$ be a CW-complex of finite dimension and $F$ be a field. Do we have that $H^q(X;F)=H_q(X;F)$ for each $q\leq n$? I know that with filed coefficients the universal coefficient theorem simplifies ...
0
votes
1answer
30 views

inverse equivalence $\Pi(X) \to \pi_1(X,x)$ in proof of Van Kampen theorem

I would like to ask for help understanding May's (concise course of algebraic topology) proof of the Van Kampen theorem through colimits. Explicitly, I don't understand how to construct the inverse ...
0
votes
1answer
49 views

Sufficient conditions for smooth pushout

We restrict ourselves to the category of smooth manifolds and smooth maps. Suppose we have a pair of smooth maps $f:A \to X$ and $g:A \to Y$. A pushout is a pair of smooth maps $p:X \to Z$ and $q:Y ...
0
votes
1answer
68 views

Homology isomorphism of $H_n(S^d\times X)$ and $H_{n-1}(S^{d-1}\times X)$

$X$ is an arbitrary space, $d\geq 1$. The existence of such isomorphism in the title supposedly follows from the Mayer-Vietoris sequence of $(S^d\times X,S^d_{+}\times X,S^d_{-}\times X)$: ...
0
votes
1answer
81 views

Singular complex is a delta complex

If I understand correctly, $\Delta$-complex on a space $X$ is defined to be a collection $\Delta(X)$ of cont. funtions $\sigma:\Delta^n\to X$ such that: 1) restriction of $\sigma$ to any face is in ...
0
votes
1answer
41 views

a bout minimal basic open sets

If f is a continuous function from a finite space to another finite space does the image of minimal basic open set at a point x in the domain is equal to the minimal basic open set at f(x) in the ...
0
votes
1answer
86 views

Relationship between fundamental polygon and its side edges

Here is the fundamental polygon diagram for torus and the diagram for its edge of the square region: My question is why the direction of loops in both circles in the right diagram must be ...
0
votes
1answer
51 views

Need help on finding homotopy

Define a continuous map $\ell:(I,\partial I)\to (SO(3),1)$ by $\ell(t) = \left( \begin{array}{ccc} \cos 2\pi t & -\sin 2\pi t & 0 \\ \sin 2\pi t & \cos 2\pi t & 0 \\ 0 & 0 & 1 ...
0
votes
1answer
66 views

the induced homomorphism $i_\#:\pi_1(P,x_0) \to \pi_1 (X,x_0)$ is an isomorphism.

If $P$ is a path component of $X$ and $X_0\in P$, then the inclusion map $i:P\to X$ can be regarded as a map of pointed spaces $P(X.x_0) \to (X,x_0)$. Prove that the induced homomorphism ...
0
votes
1answer
45 views

Labeling edges of a cube with + and - so each face has an odd number of +s.

I am looking for a specific proof, using tools from cellular homology, of the following theorem. Let $I^n$ be the standard $n$-dimensional hypercube with its standard cellular structure. There ...
0
votes
1answer
78 views

Why call them cycles and boundaries?

I have a small question. Why we have this designation: $n$-cycles for $Z_n$ and $n$-boundaries for $B_n$ ? Why they are called cycles and boundaries ? ...
0
votes
1answer
60 views

The intersection of two or more open neighborhood deformation retracts onto the maximal tree.

At Hatcher Page 43, there's a statement next to this picture says The intersection of two or more $A_\alpha's$ deformation retracts onto $T$. Here $T$ is the maximal tree of the graph $X$ shown ...
0
votes
1answer
114 views

Virtual dimesion and index?

I was told that the virtual dimension (not sure if it is virtual cohomology dimension) is equal to the index of certain operators on a manifold. Does anyone know more about this possible relation? ...
0
votes
1answer
38 views

Describing the Tychonoff topology

my question is: Describe the Tychonoff topology on $Y^X$ in a manner similar to the description in below proposition of the compact-open topology. Proposition: If $X$ is locally compact ...
0
votes
1answer
92 views

On an open set $U ⊂ \mathbb{R}^n$ show that the exterior derivative d is the only operator $d : Ω^p(U) → Ω^{p+1}(U)$ satisfying

On an open set $U ⊂ \mathbb{R}^n$ show that the exterior derivative $d$ is the only operator $d : Ω^p(U) → Ω^{p+1}(U)$ satisfying: $d(ω + η) = dω + dη$; $ω ∈ Ω^p(U), η ∈ Ω^q(U) ⇒ d(ω ∧ η) = dω ∧ η + ...
0
votes
1answer
91 views

Is the closure of an open bounded convex set already a ball?

Does the "closure of an open bounded convex set in ${R}^n$ symmetric wrt. the origin" has to be already homeomorphic to a ball? (My motivation is this: one version of Borsuks theorem says that if ...
0
votes
1answer
45 views

The fundamental group of a band

I need some help with this please, my problem is to find the fundamental group of a "band".I have the circular region that we obtain from taking away an open disk from a closed disk in R^2 and then ...
0
votes
1answer
72 views

Can anyone check my proof that $H^1(\Sigma-\{p\})=0$ for a compact and orientable surface $\Sigma$?

I have the following problem: Let $\Sigma$ be a compact and orientable surface. Show that $H^1(\Sigma-\{p\})=0$ for every $p\in \Sigma$. Can anyone check my proof and give suggestions? Sketch of ...
0
votes
1answer
74 views

Why is this homotopy an isotopy?

I am trying to follow the proof of Willem's quantitative deformation lemma and I get everything except the justification for the (iv) property which states: $\eta_t(u)$ is a homeomorphism for ...
0
votes
1answer
37 views

Isomorphism canonical and Moebius bundle.

I have to prove that the canonical bundle over $\mathbb{R}P^1$ is isomorphic to Moebius bundle. We define the caninical bundle as $\xi=\{E^{\perp},p,S^1 \}$ where $E^\perp:=\{(l,v) \in \mathbb{R}P^{1} ...
0
votes
1answer
72 views

Understanding the proof that if two loops in $S^1$ are equivalent then their degrees are equal

I am trying to understand the proof of the following: Theorem: For loops $\alpha$, $\beta$ in $S^1$ with base point $1=(1,0)$, $[\alpha]=[\beta]$ if and only if ...
0
votes
1answer
96 views

Algebraic topology involved in the 1/4-pinched sphere theorem?

Can anyone familiar with this theorem and its proof let me know how much algebraic topology is involved, and where specifically? I am familiar with a lot of differential geometry, but not many of the ...