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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Topolologics $(2n+1)$-dimensional Manifold with ball removed orientable?

Suppose you have a compact, orientable $(2n+1)$-manifold $M$. You take a neighbourhood about a point homeomorphic to $\mathbb{R}^{2n+1}$, and remove a small ball $B$. So the the boundary of ...
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216 views

Computing singular homology

could you help me solving this questions,please Let $D^2$ be a 2-dimensional disc and $M$ be the Möbius strip. Note that the boundary of both $D^2$ and of $M$ is homeomorphic to the circle ...
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27 views

Covering space problem from an old Qual

Suppose that S$^1 \times P^2$ covers some space, and let $h$ be a covering translation. Show that the induced isomorphism $h$ of $H_1(S^1, P^2)$ must be the identity
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90 views

Reduced homology exact sequences

Given $H$ a homology theory, let $f: X \to P$ the unique map from $X$ to a one point space $P$. This induces a map $f_{*}: H_i(X) \to H_{i}(P)$. We define the reduced homology to be $\tilde{H}_i(X, ...
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39 views

Connected sum of $n$ $\mathbb{R}P^n$

Does anyone know a way of computing the fundamental group of the connected sum of $n$ copies of $\mathbb{R}P^n$? Any help will be appreciated. Thank you!
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62 views

$f^\ast (a \smile b) = f^\ast(a) \smile f^\ast(b)$ using simplicial chains to define cochains

Let $f \colon X \to Y$ be a continuous map between topological spaces $X$ and $Y$, $f_\ast$ be the induced homomorphism of singular chains $C_k^s(X;G)$, $C_k^s(Y;G)$ and $f^\ast$ be the induced ...
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76 views

Cohomology of a chain complex

I know that one can define a chain complex for a CW complex X by taking the chain groups $C_n(X)$ as the free group generated by the $n$-cells, $C_n(X;\mathbb{Z}) = \mathbb{Z}\langle ...
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249 views

Proving that $\deg(fg) = \deg (f) \deg (g)$ for $f:S^1 \to S^1$

I'm trying to prove that $\deg(fg) = \deg (f) \deg (g)$ for $f:S^1 \to S^1$. The intermediate step is proving that: if $a$ is a lift of $f \circ \exp$ and $b$ is a lift of $g \circ \exp$ then $a+b$ ...
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90 views

applications of uncountability of $π_1(\mathbb{R}^2−\mathbb{Q}^2)$

Is there any application on the fact that $π_1(\mathbb{R}^2−\mathbb{Q}^2)$ is uncountable?
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739 views

Homology of the mapping torus

Let $f,g:X\rightarrow Y$ be two continuous maps. Consider the space $Z$ obtained from the disjoint union $Y\sqcup (X\times[0,1])$ by identifying $(x,0)\sim f(x)$ and $(x,1)\sim g(x)$ for all $x\in X$. ...
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112 views

Question on homotopy lifting

I'm studying covering maps and homotopy lifting and I would like to clarify a few things which my lecture notes doesn't seem to make clear. A lemma in my lecture notes says: Let $p: \tilde Y \to ...
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106 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 ...
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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 ...
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69 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 ...
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106 views

Reference for Topology and Geometry

While trying to read the book "Three-Dimensional Geometry and Topology" by William P. Thurston and Silvio Levy I just realized that my knowledge of Topology is still very scarce...Is there any other ...
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70 views

Extending simplicial complex

Let $X$ be a simplex and $Y\subseteq |X|$ a simplicial complex. Can I construct a simplicial complex $X'\supseteq Y$ s.t. $|X'|=|X|$? Can I do it without introducing new vertices apart from ...
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114 views

Subcomplexes of the infinitely-dimensional sphere

I'm trying to understand better what Hatcher does in the beginning chapter on cell complexes and so in this sense I would really like someone to elucidate for me what are the subcomplexes of ...
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145 views

2-sphere union with unit 2-cell is homotopy equivalent to one point union of two 2-spheres.

This is problem 2 in Bredon I.14, on homotopy. I need to prove that $X$ = union of the 2-sphere with the unit 2-cell going through the origin is homotopy equivalent to $Y$ = one-point union of two ...
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131 views

De Rham cohomology

I have some question on De Rham cohomology: the first one is general. If we calculate De Rham cohomology of a manifold with Mayer-Vietoris sequences we discover that the cohomology is the difect of ...
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74 views

Integration equivariant form

We consider the action of $S^{1}$ on $S^{2}$ by rotation respect the vertical axes. We want to integrate the $2-$ equivariant form $$\alpha(X)= -X\cos(\phi) +\sin(\phi)\,d\phi\,d\theta.$$ We have ...
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103 views

Equivariant localization and integration equivariant forms

I have two problems: Let it $\Omega^{*}_{G}:=(\mathbb{C}[\mathfrak{g}]\otimes\Omega^{*}(M))^{G}$ be the complex of equivariant differential forms on a differential manifold $M$ (in which acts a Lie ...
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59 views

Classification theorem of the coverings of a given space

I'm trying a lot to find easy examples of classification theorems of covering spaces of a given space. I've already read some examples here at Mathexchange such as Classification of covering spaces ...
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63 views

Direct limits and germs of continuous functions

Consider the germs of continuous functions about some real number; say 0 for simplicity. Is there a nice way of quantifying the germs, in the sense of putting them into a bijection with a simpler set? ...
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108 views

Which space this space drawn in this picture is homeomorphic?

Based in this question Why this space is homeomorphic to the plane? I would like to know which space this space is homeomorphic, any help or an intuitive idea are welcome. [Context of Image: ...
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61 views

Why is a loop going once around $S^1$ a generator of its fundamental group?

Referencing Munkres' Topology - in many proofs the authors use the fact that a loop going once around a circle ($S^{1}$) generates that circle's fundamental group. Why is this so?
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110 views

Postnikov invariants & Cohomology of EM spaces

I'm in trouble in understanding these two statements in Morita's Geometry of Characteristic Classes book: First: what's up with the "twisted" product $K(\pi_2(X),2)\times_{k^4} ...
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208 views

homotopy groups of mapping space

I got this homework problem: $X,Y$ finite CW-complexes with $\dim X=m$ and $Y$ is $n$-connected. Prove that $\pi_k(map(X,Y))=0$ for all $k \le n-m$. Thanks for the help!
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76 views

How do I prove that a map is not a covering map?

I'm thinking how to prove that a map is not a covering map. For example let $p:\mathbb R_+\to S^1$ be a map defined by $p(\theta)=(\cos(2\pi\theta),\sin(2\pi\theta))$. I'm trying to find a point which ...
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199 views

Winding number of algebraic curves

Given a closed curve $C$ in the affine plane, we define its winding number around a point (that does not meet $C$) as the total number of times $C$ travels counter-clockwise around the point. -2 -1 ...
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186 views

Any reflection is homotopic to the antipodal map on $ \ S^{2n} $.

By the degree argument we see that any reflection on $\ S^{2n}$ is homotopic to the antipodal map. But that seems a big theorem. I am looking for a straightforward argument.
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53 views

Generalizations of Sperner's Lemma?

Consider colorings of n-dimensional complexes. Given a complex $C$, and a coloring of $C$, a simplex $\alpha \in C$ is said (n-x)-complete with respect to this coloring if all its vertices receive at ...
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149 views

How do I prove this surface is homeomorphic to the sphere

Let $S$ a compact and connected surface. If $S=U_1\cup U_2$, where $U_1,U_2$ are of finite character and the boundary of $U_1$ is in $U_2$. How can I prove that S is homeomorphic to the sphere? ...
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77 views

Special case for Alexander's Duality

Let $C_1$ and $C_2$ be homeomorphic closed sets in $S^n$.Prove that: $$H^{r}_{dR}(S^n\setminus C_1 )\cong H^{r}_{dR}(S^n\setminus C_2 )$$ Here's what I tried: Let $ p \in S^n\setminus C_1 $ , $ ...
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68 views

2-morphisms from spans of spans

I have a question about the construction of 2-morphisms from spans of spans in the paper "2-vector spaces and groupoid" by Jeffrey Morton . Suppose we have a span of span of groupoids as follows and ...
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191 views

How do mathematicians formally prove the connected sum of two disks is homeomorphic to annulus and similar kinds?

I was tyring to read this fact that the connected sum of two disks is homeomorphic to annulus. By intutitive picture, it is obvious, but I wanted to it in a formalistic way. So I was reading many ...
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377 views

triangulation of the closed disc

I'm starting to study triangulations of topological spaces by myself. I find really difficult since I've never seen any formal example of a triangulation in any book! So, I began with the one of the ...
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585 views

$\Delta$ complex structure for $S^n$.

For n=1. We consider two copies of $\Delta^1$, $1$ simplicies and identifying their boundaries we get a loop, that is $S^1$. For n=2, identifying boundaries of two copies of $\Delta^2$ via identify ...
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First examples in triangulations

I am starting to study about triangulations in my algebraic topology course. We have seen the triangulation of the sphere, the closed disc and so on. Intuitively it's ok, however I couldn't find any ...
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135 views

Universal cover as a principal $\pi_1$ bundle.

Let $M$ be a connected manifold with universal cover $\tilde M$ and fix $x_0 \in M$. Then it is well-known that $\tilde M \to M$ is a principal $\pi_1(M,x_0)$ bundle. I'm a bit confused about the ...
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65 views

Can one define “simplicial” EM spaces?

Let $\Delta$ be the well-known simplicial category, and denote with $\widehat\Delta_\bf C$ the category of simplicial objects in $\bf C$ ($\widehat\Delta$ for short). Given $\mathcal ...
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186 views

Rational cohomology of quotient by group action

Let $X$ be a topological space with a continuous action by a finite group $G$. Hopefully under some assumptions on $X$ one can identify the rational cohomology of $X/G$ with the $G$-invariants of the ...
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102 views

Analog of a tubular neighborhood for an embedded wedge sum

If you have some embedding of a path connected topological space wedge of spheres $N$ into a compact simply connected smooth $n$ manifold $M$ (like a sphere for example), then is there some kind of ...
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264 views

Homotopy versus Isotopy

First of all, I apologize for the crudeness of my question. Consider the construction of the homotopy groups. We mod out the space of "loops" at point by the equivalence relation generated by homotopy ...
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112 views

About manifolds after attaching handles.

I'm reading R.E. Gompf and A.I. Stipsicz, 4-Manifolds and Kirby Calculus. I don't understand Remarks 4.4.1 on page 116-117 Google books here. At first I can't understand why we take immersed disk $D ...
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125 views

Question on essential map

Suppose $M$ is a closed smooth n-manifold. a)Does there always exist a smooth map $f:M\to S^n$ from $M$ into the n-sphere, such that $f$ is essential(i.e. $f$ is not homotopic to a ...
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193 views

Definition of topologically onto map and Covering Spaces

In the chapter on Covering Spaces in his book A Basic Course in Algebraic Topology Massey uses the term topologically onto when defining covering spaces (see below for Massey's definition). What does ...
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154 views

Terminology in an Exercise of Hatcher

I am trying to solve an exercise in Hatcher "Algebraic Topology" but am a little confused by the terminology he is using (just so you know, it is exercise 5 in chapter 1.1). He writes that in the ...
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58 views

Disjointness of stars in a simplicial complex in $\ell_2$

Definitions Let's consider a full $n$-dimensional simplicial complex $C$ in $\ell_2(X)$. By that I mean a set of all functions $f:X \to[0,1]$ such that $\sum_{x\in X}f(x)=1$ and there are at most ...
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55 views

How can we prove that $\mathbb R P^n \to \mathbb R P^n/\mathbb R P^{n-1}$ induces isomorphism on $H_n$ when n is odd?

It is said that $\mathbb R P^n \to \mathbb R P^n/\mathbb R P^{n-1}$ induces isomorphism on $H_n$ when $n$ is odd. How can we prove this?
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26 views

Isomorphims on homologies induced by cylindrical structure

This question is related to my previous question Cylindrical structure and homology. Let $S$ be a oriented compact (topological) 2-manifold. We consider a cylinder $M=S\times I$ over $S$, here $I=[0, ...