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

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4
votes
1answer
190 views

Identifying a $\Delta$ complex

I am doing some self study and am having trouble with the following. I want to say the answer is a cone, but I do not think that this is correct. Help will be apreciated. What familiar space is the ...
4
votes
1answer
144 views

Representation of (co)homology classes of $3$-manifolds by embedded surfaces

Let $M$ be a closed oriented $3$-manifold. Theorems in algebraic topology allow us to identify $$H_2(M) \ \cong \ H^1(M) \ \cong \ \langle M,S^1\rangle$$ where (co)homology is meant with integer ...
8
votes
1answer
311 views

Why is a path-connected topological space homotopy equivalent to the classifying space of its loop space?

Given a path-connected topological space $X$ (lets say compactly generated; this entire post will be working in the category of compactly generated topological spaces) with a designated point $x$, we ...
0
votes
1answer
45 views

Monotonic in sigma algebra

Please help me prove that if $A \subseteq B$, then $m(A) \leq m(B)$ (That $m$ is monotonic). How would you prove this? Can we say $m^*(A \cup B) \leq m^*(A) + m^*(B)$ where $m^*(A \cup B) + m^*(A) = ...
4
votes
3answers
200 views

Interesting theorems/facts about identification spaces

I am now studying algebraic topology (still at the beginning). I am now studying identification spaces, adjunction spaces,... As I still don't know how these concepts are going to be used, I think I ...
4
votes
1answer
92 views

Fundamentalgroup of $\mathbb{R}P^2$

i have to compute the fundamentalgroup of the projective plane $\Bbb{R}P^2$ with Van-Kampen Theorem. Therefore i use the fundamentalpolygon given in Projective Plane. I get for the presentation of the ...
1
vote
0answers
79 views

Orientation of the barycentric subdivision

Two orderings of the vertices of an $n$-simplex are said to be equivalent if they differ by an even permutation. An orientation of an $n$-simplex is a choice of one of the two equivalence classes of ...
2
votes
1answer
88 views

fundamental group of a graph

let $G$ be a connected graph and $\Omega$ its universal covering. Let $\gamma_1,\dots,\gamma_r$ be free generators of $\Gamma:=\pi_1(G)$, $v\in\Omega$ be a vertex and $s_i$ a path from $v$ to ...
5
votes
2answers
268 views

Algebraic Topology Homology Group

How would one go about calculating the first and second homology groups of a cube inscribed in a sphere?
0
votes
1answer
36 views

Paths between 0-cells in a classifying space. II

Let $\mathcal{C}$ be a small category and $X,Y$ objects within. If $X$ and $Y$ (as $0$-cells) lie within the same path-component in $B\mathcal{C}$, can one say anything in general on how $X$ and ...
1
vote
1answer
219 views

Covering space of figure 8 corresponding to $\mathbb{Z}$

Provided that certain conditions are satisfied, we know that there's a one to one correspondence between covers of a space and subgroups of the fundamental group of that space. Since $\mathbb{Z}$ is a ...
1
vote
1answer
40 views

Paths between 0-cells in a classifying space.

Let $\mathcal{C}$ be a small category. Giving a morphism $u: X \to Y$ in $\mathcal{C}$ is equivalent to giving a functor $f: \{0 < 1 \} \to \mathcal{C}$ (with $f(0) = X$, $f(1) = Y$, and $f(0 \to ...
2
votes
1answer
57 views

Fundamental group of the following disc

What is the fundamental group of the following space in $\mathbf C^n$? This is the topological space given by $$\{(x_1,\ldots,x_n)\in \mathbf C^n-\{0\} : \vert x_1\vert < 1, \ldots, \vert x_n\vert ...
2
votes
2answers
94 views

Algebraic topology question (Qual)

I am having trouble with a QR problem. I would appreciate some help. Construct a connected $CW$-complex $X$ with $H_0(X, \mathbb{Z}) = \mathbb{Z}, H_1(X, \mathbb{Z}) = \mathbb{Z}\times ...
5
votes
1answer
127 views

What exactly is the CW complex structure on a geometric realisation?

This is likely a silly question. Definitions: $\bullet$ $\Delta_n = \{ (t_0, \dots, t_n) \: | \: 0 \leq t_i \leq 1, \sum_i t_i = 1 \}$ $\bullet$ Given $f: \underline{m} \to \underline{n}$ in ...
5
votes
1answer
117 views

Is There a Formalization of Cauchy's $F - E+V = 2$ proof?

Can anyone provide, or direct me to a formalized version of Cauchy's proof that for any convex polyhedron with $F$ faces, $E$ edges and $V$ vertices that $F - E + V = 2$. I am willing to accept the ...
1
vote
0answers
49 views

Hamiltonian of one and two unknots

Recently I calculated the Ising Hamiltonian of a Hopf link. First, I colored the Hopf link in a checker board pattern and drew the Seifert surface from it. Considering the shaded regions as vertices ...
2
votes
0answers
65 views

Calculation of the fundamental group from triangulations

Is there - say, for a triangulable surface - a concrete algorithm how to calculate the fundamental group of the surface from a given triangulation, seen as a graph (of its 1-skeleton), given as an ...
7
votes
2answers
158 views

Simultaneous CW Approximation

Given a topological space $X$, we know that there is a CW complex $Z$ with a map $Z\rightarrow X$ inducing an isomorphism on homotopy groups. If we are given two spaces $X_{1}$ and $X_{2}$ with ...
1
vote
0answers
100 views

universal coefficient theorem for cohomology

We all know that we can compute homology and cohomology with arbitrary coefficient if we already know the homology groups with coefficient in $\mathbb{Z}$. I wonder if it is possible if we know the ...
7
votes
1answer
146 views

Set theoretic implications of constructions in Differential Geometry/ Topology

In subjects like Differential Geometry/ General Topology one often constructs for each $x$ in a space $X$ a set $U_x$ satisfying certain properties. Examples where one does constructions like this: ...
1
vote
1answer
205 views

Simple cellular homology computation

Here's a very simple cellular homology computation that I'm a little confused about. Put a CW structure on the closed disc $X=D^{2}$ with two zero-cells $v_{0},v_{1}$, two one-cells $e_{0},e_{1}$ ...
2
votes
3answers
98 views

Identifying all points of edge of disc

If we identify all points of the edge of a disc, do we get the Moebius strip? Why?
1
vote
1answer
362 views

Prove that the quotient space is homeomorphic to the torus.

Let $X = \{(x, y) \in R_2 \mid 1 \leq \left||(x, y)\right|| ≤ 2\}$. We define an equivalence relation on $X$ as follows: $(x, y) \sim (x', y')$ if and only if $(x, y) = (x', y')$ or $\left||(x, y ...
3
votes
1answer
146 views

Let $X = [0,2]$ and $A = \{0,1,2\}$. Prove that $X / A$ is homeomorphic to $C_{1}$ ∪ $C_{-1}$

Let $X = [0,2]$ and $A = \{0,1,2\}$. Prove that $X / A$ is homeomorphic to $C_{1}$ ∪ $C_{-1}$, where $C_1$ and $C_{-1}$ are the circles of radius $1$ centered at $(1, 0)$ and $(-1, 0)$ , respectively. ...
3
votes
2answers
224 views

Homeomorphism of Klein Bottle

Proof that the Klein bottle is homeomorphic to $T/S$ where $T$ is the torus of revolution and $S$ is the equivalence relation given by $(x, y, z) \sim (x', y', z')$ if and only if $(x, y, z) = \pm ...
2
votes
1answer
75 views

intersection form of $CP^2$

I am trying to understand why the intersection form of $CP^2$ is <1>. First we introduce {[x:y:z], x=0} as a generator of second homology and then we say that it has one intersection with {[x:y:z], ...
3
votes
1answer
178 views

Classification of fundamental groups of non-orientable surfaces

I want to compute the presentation of the fundamental group of the non orientable surfaces $N_h$, thus $\pi_1(N_h)$. I notated with $N_h$ the sphere with $h$ crosscaps. Herefore I first have to ...
2
votes
1answer
49 views

Van Kampen theorem fig-8

How to prove the Van Kampen theorem for the space of figure-eight by showing that $\pi_1(X)$ can be expressed uniquely as finite product $<\alpha>^{m_1} <\beta>^{m_2} ...
3
votes
0answers
56 views

If $S_1$ is orientable and $S_2$ it isn't,

Let $S_1$ and $S_2$ be two closed surfaces. Demonstrate that the following conditions are necessary for there to be a $k$-sheeted covering $p: S_1 \rightarrow S_2 $. a) $\chi(S_1)=k \chi (S_2)$. b) ...
3
votes
1answer
80 views

Special case of invariance of domain

Let $A=\{(x_1, \cdots, x_n)\in \mathbb{R}^n: x_1\ge 0,\|(x_1, \cdots, x_n)\|<1\}$. I want to show that this is not homeomorphic to any open set of $\mathbb{R}^n$. I can use the theorem of ...
3
votes
0answers
119 views

fundamental theorem of algebra proof

I have seen some proofs of the fundamental theorem of algebra using algebraic topology. But I have seen an exercise to prove the theorem by defining this map $f_t: S^1\longrightarrow S^1$ by $f_t(z) ...
3
votes
1answer
97 views

Properties of the Category of topological spaces with $n$ basepoints.

I've recently encounted a problem in my reading which would seem to be more naturally phrased if the category we work in shifted from the category $\textbf{Top}^*$ of pointed topological spaces, to ...
5
votes
3answers
188 views

Cylinder object in the model category of chain complexes

Let $\text{Ch}⁺(R)$ be the category of non-negative chain complexes of $R$-modules where $R$ is a commutative ring. What is a cylinder object, in the sense of model categories, for a given complex ...
1
vote
1answer
111 views

Simple Sphere Suspension Question

I have heard it said that the suspension of an $n$-sphere is an $n+1$-sphere. This is stated without proof in chapter 0 of Hatcher's book on algebraic topology. More generally, it seems that the ...
2
votes
1answer
81 views

How to apply the matrix of a boundary operator on a k-chain

It is said that the boundary operator $\partial_k$ maps a $k$-chain to a $(k-1)$-chain. I've also seen that this operator can be represented with a matrix of dimension $|K^{k-1}|\times|K^k|$. I can't ...
4
votes
1answer
124 views

Hatcher - simplicial and quotient representations of spheres

I am reading chapter 2 of Hatcher's algebraic topology text. In it, he makes the following two claims: 1) Quotienting $D^n$ by $\partial D^n$ produces a space homeomorphic to $S^n$. 2) We may create ...
3
votes
1answer
169 views

Existence of minimizing geodesic in each fixed-end-point homotopy class in a complete manifold?

This is intuitively clear, but I cannot solve this homework problem: 1) Let $(M,g)$ be a complete Riemannian manifold, let $c:[0,1]\to M$ be a continuous curve in $M$ such that $c(0)=p, c(1)=q$. Then ...
3
votes
1answer
54 views

Two loops are intersecting each other in convex set of R

In the figure bellow I was trying to prove that $ γ_1$ is intersecting $ γ_2$ at some point inside the convex and compact subset of $R$, then I thought to join $x, y$ and $r, z$ by line segments ...
4
votes
2answers
186 views

How does Thurston's geometrisation conjecture imply Poincaré's conjecture?

I ran into the geometrisation conjecture a few days ago, and I started wondering how to prove Poincaré's conjecture. Let $M$ be a compact, simply connected, $3$-manifold. Clearly it is irreducible ...
3
votes
1answer
112 views

Induced de Rham map is a ring map

The de Rham Theorem states that for a smooth manifold $M$ the cochain map $R: \Omega^*(M) \to C^*(M;\mathbb{R})$ from differential forms to singular real cochains defined by $R(\omega)(\sigma)= ...
1
vote
3answers
51 views

$a.b$ is path homotopic to $c.d$

Let $h: I \times I \rightarrow X$ be a continuous function, and let $a, b, c, d$ be the paths in $X$ defined as follows: $a(s)=h(s,0)$ $b(s)=h(1,s)$ $c(s)=h(0,s)$ $d(s)=h(s,1)$ Then I want to ...
0
votes
0answers
46 views

Do “multiples” of open dense sets of an algebraic group union to the whole group?

Let $G$ be an algebraic group. From algebraic geometry we know there exists an open dense subset $U$ of $G$ such that $U$ is nonsingular. Since left multiplication of $U$ by elements of $G$ is an ...
2
votes
1answer
102 views

Homotopy Type of a Riemann Surface with and without Points Removed

Suppose $\Sigma$ is a Riemann surface of genus $g$ and with $b$ points removed. Is there any restriction on the possible homotopy type that $\Sigma$ can possess? What about the case when $\Sigma$ has ...
2
votes
1answer
41 views

Covering space for figure $8$ space, such that $\chi(G) = 0$

The covering space for the figure $8$ can be thought of as a graph. Is there a covering such that $\chi(G) = 0$ (i.e the number of vertices equals the number of edges)?
1
vote
2answers
313 views

Help understanding manifolds and topological spaces

I'm used to think about linear algebra with matrices and vectors, I don't have particular problems with geometry either, I'm having hard times understanding what is the meaning of a manifold and a ...
2
votes
1answer
97 views

on the factorization of maps between connected CW complexes [duplicate]

I'm working on problem 16 in section 4.1 of Hatcher's Algebraic Topology book. I really have no ideas so far: Show that a map $f: X \to Y$ between connected CW complexes factors as a composition $X ...
21
votes
1answer
354 views

How much of an $n$-dimensional manifold can we embed into $\mathbb{R}^n$?

I observed some naive examples. Spheres, for example, when we cut out one point, can be embedded into $\mathbb{R}^n$. And if we cut out a measure zero set of a projective space, it can be embedded ...
0
votes
1answer
106 views

Homotopy equivalence of smash products

Suppose that $f:A\rightarrow B$ is a homotopy equivalence (both $A$ and $B$ are CW complexes), and $Y$ is a CW complex. Then is it true that the induced map $f\wedge Id:A\wedge Y\rightarrow B\wedge ...
1
vote
0answers
58 views

more examples on games

As in the example of the game of $Hex$, it needs some tools from algebraic topology to prove that there is exactly one winner in the game I wonder if there are more examples on games to be ...