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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257 views

Intersection of simply connected sets II

I read the following statement in the old question "Intersection of Simply-Connected Sets" (Intersection of Simply-Connected Sets): If $U$ and $V$ are simply connected and $U \cap V$ is path ...
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149 views

Cohomology ring of $U(n)$

As you know $$H^\ast (U(n);{\bf Z})=\bigwedge_{\bf Z}[x_1,x_3,...,x_{2n-1}]$$ where $|x_i|=i$ To prove this we use Leray-Hirsch Theorem for $$\tag{*}\ U(n-1)\rightarrow U(n)\rightarrow S^{2n-1}$$ ...
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29 views

Let $S^3 = \{(z_1, z_2) \in \mathbb{C}^2 | ||z_1||^2 + ||z_2||^2 = 1\}$, $D^2 = \{z \in \mathbb{C} | ||z|| \leq 1\}$.

Let $S^3 = \{(z_1, z_2) \in \mathbb{C}^2 | ||z_1||^2 + ||z_2||^2 = 1\}$, $D^2 = \{z \in \mathbb{C} | ||z|| \leq 1\}$. Let $X$ be the quotient of the disjoint union of $D^2$ and $S^3$ by the smallest ...
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39 views

Relative homology of interlevel set

Let us consider a function $f\colon \mathbb{R}^3\to\mathbb{R}$, $f(x,y,z) = x^3+y^3+z^3 - 5yz$. Can anybody drop a hint how to compute relative homology of interlevel sets with coefficients in ...
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428 views

Find all the covering spaces of the Klein Bottle

I am studying for a qualifying exam and it asks for all the covering spaces of the Klein Bottle. Does anyone have any suggestions? I am not sure how to go about finding ALL of them.
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54 views

Definition of HEP correct?

In Algebraic Topology (Allen Hatcher, pg 14) I read a definition of HEP: ...Suppose one is given a map $f_{0}:X\rightarrow Y$, and on a subspace $A\subset X$ one is also given a homotopy ...
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29 views

Critical group and morse Lemma

I have this theorem with a part of it's prove I have two questions: 1) what is the spectral decomposition of A ? 2)How to see that $B_{\varepsilon}\cap f_0 = \lbrace x\in H,||x||\leq \varepsilon ...
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41 views

Fiber product and $G$-invariant maps

Let $S^n \times_{\mathbb{Z}_2} \mathbb{R}$ be the fiber product of unitary sphere $S^n$ and $\mathbb{R}$ over $\mathbb{Z}_2$, where $\mathbb{Z}_2$ acts on $S^n$ by antipodal relation and on ...
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46 views

Geometric interpretation of $R$-orientations

For an $n$-dimensional manifold $M$ and a ring $R$ one can define $R$-orientability, e.g. by choosing local orientations, i.e. generators of $H_n(M, M - \{x\}; R)$ for every point $x$ in the manifold, ...
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47 views

Intersection between cycles and dominant maps

i must say in advance that i'm not very familiar with algebraic cycles and intersection theory, so i hope my question is not too trivial. Let $X$ and $Y$ be two K3 surfaces. Let ...
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55 views

Vector field and a solution of an ODE

I have this: And my questions are : 1)what is :"The local theory of differential equations in a Banach space" 2)Why it's implies that each solution of (6) is equal to $\eta$ Please Thank you . ...
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109 views

Why is the pullback of a connected cover not necessarily connected?

In particular, I read somewhere that the fiber product of the maps $S^1\rightarrow S^1$ sending $z\mapsto z^m$ and $S^1\rightarrow S^1$ sending $z\mapsto z^n$ is disconnected with $\gcd(n,m)$ ...
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39 views

Question on critical groups

I have this theoreme with it's proof But i don't understand who is $f_0$ ? Can someone help me please ? Thank you .
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255 views

degree of a continuous map

Let $f(z)$ be a polynomial with complex coefficients. $f$ can be extended to a continuous map $\tilde f: S^2 \rightarrow S^2$ ($\mathbb C \cup \{ \infty \} \cong S^2$). Prove that the degree of ...
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47 views

Stiefel-Whitney classes as induced application in cohomology

Let $\xi=(E,p,X)$ a bundle over a paracompact space $X$. We can consider the exact sequence $$ 0 \rightarrow SO(n) \stackrel{i}{\rightarrow} O(n) \stackrel{det}{\rightarrow} \mathbb{Z}_2 \rightarrow ...
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59 views

Form a space $X$ by identifying the boundary of $M$ with $C$ by a homeomorphism. Compute all the homology groups of $X$.

Let $T$ denote the torus $S^1\times S^1$ and let $M$ denote the Möbius band. Let $C$ be a simple closed curve in $T$ which bounds a 2-disk. Form a space $X$ by identifying the boundary of $M$ with $C$ ...
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53 views

problem related to the Mayer-Vietoris Sequence

Let $D_{k}$ be the surface obtained by removing k small disjoint open 2-discs from the unit disc $E^{2}$. Show that $D_{k}\simeq G_{k}$, the k-leaved rose. Let $M_{k}$ be the surface obtained by ...
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32 views

continuous discrete open map and topological dimension

Is there anyone who can help me to answer this question : Let $\Omega$ be an open bounded and connected set of $\mathbb{R}^n$. Let $A\subset \Omega$ be a closed set of Lebesgue measure zero and whose ...
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107 views

Cohomology and 1-forms with compact support

I'm, having troubles with the following Let $U$ be a bounded open set in $\mathbb{R}^{2}$ such that $\mathbb{R}^{2}\setminus U$ has $n+1$ connected components. Prove that $\dim(H_c^{1}(U))=n$. I ...
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91 views

Canonical bundle and Möbius bundle

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

Homotopy type of surface of revolution

Let $X$ be a finite graph lying in a half-plane $P\subset\mathbb{R}^{3}$ and intersecting the edge of $P$ in a subset of the vertices of $X$. Describe the homotopy type of the surface of revolution ...
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51 views

Prove homotopy of maps

I am studying for a qualifying exam and I am having difficulty on the following problem and I don't really know how to begin. Any help would be great. Let $M, N$ be compact manifolds and for all $t ...
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173 views

Weak deformation retraction

This is part of a problem from Hatcher's book. Let $Y$ be the subspace of $\mathbb{R}^{2}$ shown in the picture. Let $Z$ be the zigzag space of $Y$ indicated by the heavier line. Show that there is a ...
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101 views

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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92 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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77 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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252 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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752 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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113 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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132 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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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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113 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 ...