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
493 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
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1answer
47 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
67 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
122 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
205 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
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1answer
129 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 ...
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0answers
51 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
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0answers
129 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
175 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 ...
3
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0answers
139 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
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1answer
171 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: ...
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2answers
443 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
121 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
663 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 ...
4
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1answer
165 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. ...
4
votes
2answers
397 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 ...
3
votes
1answer
136 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
2answers
390 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
66 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
63 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
113 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
160 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
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1answer
124 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
364 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
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1answer
171 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
132 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
162 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
304 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
63 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
339 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
135 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
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3answers
55 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
50 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
139 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
47 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
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2answers
643 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
128 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
425 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
159 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 ...
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0answers
65 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 ...
5
votes
1answer
625 views

what is the universal cover for $S_g$?

Denote $S_g$ to be orientable, compact closed surface with genus $g$; especially, $g=1$, $S_1$ is just the torus, it has the universal cover $\mathbb{R}^2$. I heard the universal cover for $S_2$ is ...
1
vote
1answer
212 views

real Grassmann manifolds, Schubert cells, and boundary maps for computing homology

The problem: Compute $H_\ast (G(n,k); \mathbb{Z}/m\mathbb{Z})$. Define $G(n,k)$ to be the space of $k$-dimensional vector subspaces of $\mathbb{R}^n$. Define the Schubert cell $e(m_1,...,m_s)$, so ...
3
votes
0answers
124 views

Extension of Brouwer's degree to continuous functions.

I am studying the first chapter of this book: Topological Degree Theory and Applications At page 13 of this document, Definition 1.2.5, it is essentially said that to define the degree of a ...
0
votes
0answers
46 views

An variation of “Lk” in simplicial complexes

Let $\mathcal S$ be a simplicial complex. For a simplex $S \in \mathcal S$, we have the closure, the star and the link $\operatorname{Cl} S := \{ F \in \mathcal S : F \leq S \}$ $\operatorname{St} S ...
4
votes
1answer
607 views

Fundamental group of the wedge sum of two spaces

Let $X,Y$ be two path-connected topological spaces and $\langle A\mid R\rangle,\langle B\mid S\rangle$ respectively presentations for their fundamental groups. I think that a presentation for the ...
5
votes
0answers
124 views

Homological definition of orientation at a boundary point?

For a topological manifold $M^m$, an orientation at a point $x \in M$ can be defined as a choice of generator for $H_m (M, M-x)$. For a topological manifold with boundary this definition still makes ...
6
votes
2answers
255 views

This set is a manifold

let $S$ be the set of pairs $(x,y)$ where x,y are orthogonal unit vectors in $\mathbb R^3$. i am trying to show this is a topological manifold. for starters one needs to define a suitable topology on ...
4
votes
1answer
262 views

Defining multiplication on a Koszul complex

Let $R$ be a Noetherian commutative ring and $x$ and $y$ two elements in $R$. We construct the Koszul complex on $x$ and $y$. We start by the following two chain complexes: $$ C_2=0\to ...
1
vote
1answer
52 views

Orientation of closed combinatorical surfaces

I have some problems with the equivalence of definitions of orientation. I know two definitions of orientation, namely: A surface is orientable if it contains no 1-sided curves (a 1-sided curve in a ...
3
votes
1answer
133 views

Topology of manifolds

Where can I find a stricter presentation of topology of manifolds, then in section 0.4 in Griffiths-Harris? For example, they define the map $H_k \times H_{n-k}$ by presenting a cycle by a submanifold ...