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

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0
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0answers
26 views

Extend vector fields from several $S^1$ to $D^2$

Let's take a disk $D^2 \subset \mathbb{R}^2$ with $n$ holes ($n = 0, 1, ...$). In case $n = 0, 1$ it's clear how to extend any non-zero (i.e. with no singular points) vector field from $S^1$ to disk ...
2
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1answer
25 views

Thom space of unit circle

Say we embed $S^1$ into $\mathbb{R}^2$ as the unit circle. What is the Thom space $Th(i)$ associated to this embedding $i:S^1 \to \mathbb{R}^2$? By definition, the Thom space is the one point ...
5
votes
2answers
52 views

Definition of covering (deck) transformation for smooth manifolds: Are they diffeomorphisms?

In John Lee's book Riemannian Manifolds, a covering transformation (or deck transformation) of a smooth covering map $\pi:\tilde{M}\to M$ (of connected smooth manifolds) is defined to be a smooth map ...
0
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0answers
24 views

Associativity of Operation * on Path-homotopy Classes Proof (Supposedly Trivial Question)

In Munkres' Topology Book where the Proof of Associativity of Operation * on Path-homotopy Classes, there is a statement which I don't quite understand. Background Info: Munkres defines a path ...
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0answers
24 views

Complex Projective Space as a Quotient of a Disc

I am reading Hatcher's book and I have a problem understading how the complex projective space $\mathbb CP^n$ can be realised as a quotient of $D^{2n}$ (page 7) Let me briefly outline his arguments ...
0
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1answer
35 views

Degree of a map over a different ring in homology

The degree of a map $f: S^n \to S^n$ is definied as the unique integer $H_n(f;\mathbb{Z} ): H_n(S^n;\mathbb{Z}) \to H_n(S^n;\mathbb{Z})$ since $H_n(S^n;\mathbb{Z}) \cong \mathbb{Z}$. Now my question ...
2
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1answer
44 views

Why is a simply connected 3-manifold a homotopy 3-sphere?

I recently looked at the statement of the Poincare conjecture, and realized I didn't know why the fact that a 3-manifold is simply connected implies that it is homotopic to a 3-sphere. Could someone ...
1
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0answers
15 views

Module structure of the homology of spaces endowed with a group action

If $X$ is a topological space endowed with the action of a group $G$, is it true that $$ H_n(X,\mathbb{C}) \cong H_n(X) \otimes_{\mathbb{Z}[G]} \mathbb{C}[G]$$ as $\mathbb{C}[G]$-modules? Edit: ...
0
votes
1answer
21 views

Diffeomorphism between covering spaces

Let $\pi_1: M \rightarrow M_1$ and $\pi_2: N \rightarrow M_2$ be two smooth covering maps. Now $\phi: M \rightarrow N$ is a smooth diffeomorphism. Does this induce a smooth diffeomorphism $f: M_1 ...
7
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1answer
77 views

General relationship between braid groups and mapping class groups

I just finished correcting my answer on visualizing braid groups as fundamental groups of configuration spaces, and in the process became interested in the other pictorial definition of the braid ...
1
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0answers
38 views

Problem of Arnold's book and covering spaces

I am currently reading Arnold's book "Mathematical Methods of classical mechanics" on page 278 and I don't see through his arguments there at a point. Especially, I am talking about the part that ...
1
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1answer
31 views

Homeomorphism of the closed unit ball not preserving the sphere?

Exercise 2.9.12 in Ronnie Brown's Categories and Groupoids asks the reader to show that if $f:\mathbb{R}^n \to \mathbb{R}^n$ is continuous such that $f$ restricts to a homeomorphism from the open ...
4
votes
0answers
55 views

Properties of $\mathbb{C}P^n$

I'm currently working on a somewhat deformed version of $\mathbb{C}P^2$ and want to check some properties from a geometrical and/or topological point of view. Of course, $\mathbb{C}P^2$ is Kähler ...
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0answers
25 views

finite graphs with homeomorphic covering space that do not cover the same graph.

I came across this exercise from section 1.3 in Hatcher's "Algebric topology". Construct finite graphs $X_1$ and $X_2$ having a common finite-sheeted covering space $X_1 \cong X_2$ , but such ...
3
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0answers
51 views
+50

Cantor Set in Alexander Horned Sphere Construction

I have seen it said in several different places that in the standard construction of the Alexander horned sphere, given by successive embeddings of a sphere with $2^n$ handles, either limited or ...
0
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0answers
54 views

How do I visualize this quotient space?

If $V = [0,1] \times [0,1] \subset \mathbb{R}^2$. We define the equivalence relation $\sim$ on $V$ as follows: every element $(x,y) \in V$ is equivalent with itself and besides that the three ...
1
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2answers
56 views

Book recommendation: Homology and Cohomology

I need to learn some concepts about Homology and Cohomology theory to apply in riemannian geometry basically, but really I have not time to read about that. I know just two books of W. S. Massey, ...
0
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0answers
37 views

Can we recover homology from cohomology [duplicate]

The universal coefficient theorem allows one to calculate cohomology by homology. Can we recover singular homology by cohomology for a complex manifold? Can a complex manifold (algebraic manifold) ...
0
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0answers
44 views

Does the restriction of the Thom class of a submanifold to the cohomology of the submanifold give the Chern class of the normal bundle?

Let $X$ be a compact complex manifold of (complex) dimension $d$, let $i\colon Y\hookrightarrow X$ be a regular complex submanifold of (complex) codimension $k$, let $\mathcal{N}_{Y/X}$ be the normal ...
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0answers
38 views

Triangulation of triangle [on hold]

Why it's not triangulation(named in Hatcher as $\Delta$-complex structure)?(edges are glueing)
3
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1answer
30 views

Homology and Neighborhood

Let $X$ a connected manifold, $x \in X$ and $V$ a neighborhood of $x$. Assume $i:V \to X$ induce isomorphism between all homology groups. Does $X-p$ and $V-p$ still have the same homology groups ? ...
2
votes
1answer
45 views

Covering spaces of $S^1 \vee S^1$

The question is: Let $x_0$ be the common point of two circles in $X = S^1 \vee S^1$. Let $a$ and $b$ be the standard generators of $\pi_1(X, x_0) = \langle a, b\rangle$ corresponding to the two ...
0
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5answers
59 views

Finite set of points of $R^n$ is compact

In order to show that a finite set of points of $R^n$ is compact, I just need to show that the set is closed and bounded. First of all, since it's a finite set, I can Always pick the greatest ...
6
votes
1answer
54 views

A Ham Sandwich type problem

If $A_1,...,A_n$ are measurable subsets of $S^n$, then there is a great $S^{n-1}$ cutting each $A_i$ exactly in half. The tools I have at my disposal are the Borsuk Ulam theorem and the Ham ...
2
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0answers
28 views

Analyzing the following space:

I recently encountered the following space: the underlying set is $C = C_1 \cup C_2$, where $C_i$ is the circle of radius i and centre 0 in the complex plane. Basic open sets are: • {z} for every z ...
1
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1answer
26 views

van Kampen theorem for fundamental groupoid of $X$ relative to $A$

Let $X$ be a manifold with submanifold $A \subseteq X$. Let $\Pi_{1}(X,A)$ denote the homotopy classes of paths with endpoints lying in $A$. This is a Lie groupoid with set of objects $A$. For ...
0
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0answers
29 views

Derivative group action [duplicate]

Let $\phi : G \times M \rightarrow M$ be a group action on a smooth manifold $M$ and Lie group $G$. Then we define $$f(t):=\phi(g(t),d(t)).$$ Now I'd say: $$f'(t) = D\phi(g(t),d(t))(g'(t),d'(t)).$$ ...
2
votes
0answers
54 views

Compute the fundamental and homology groups of $S^3 \setminus K$, where $K$ is two linked copies of $S^1$ in $\mathbb R^3$

Compute the homology groups of $S^3 \setminus K$, where $K$ is two linked copies of circles in $\mathbb R^3$. How about the homology group of $S^3 \setminus K'$ where $K'$ is just one copies ...
2
votes
1answer
27 views

Find all surfaces that can be obtained from an octagon by identifying edges in pairs.

Find all surfaces that can be obtained from an octagon by identifying edges in pairs. I think there are many many surfaces. Can anyone give some hints for the question?Thanks.
0
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0answers
22 views

recover (pontrjagin) ring structure from the localization (w.r.t. $\pi_0$)

Let $R$ be a ring and $S$ a given multiplicative subset of $R$. Suppose we know the multiplication structure of $S$. If we know the ring structure of $R[S^{-1}]$, the localization of $R$ with respect ...
1
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0answers
40 views

Existence of Shafarevich maps(theorem 3.6) on Kollar 's book

I have some problem when reading Theorem 3.6 of Kollar's book Shafarevich Maps and Automorphic Forms, page 41 (Corollary 3.5 of this article ), which states that Let $X$ be a normal variety, ...
1
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0answers
31 views

a question about Fenchel's theorem(differential geometry)

I am an undergraduate student studying differential geometry right now. I am just finishing reading how to prove Fenchel's theorem:The total curvature of a smooth closed curve in 3-dimensional space ...
1
vote
0answers
31 views

natural map to the homotopy fibre

In the paper Homology fibrations and group completion theorem, McDuff-Segal, page 280, paragraph 4, line 2-line 3 and Configuration spaces of positive and negative particles, McDuff, page 105, line ...
5
votes
1answer
56 views

Is simply connectedness preserved after deleting a high codimension set

Suppose $X$ is a complex manifold of complex dimension $n$, $Z$ is a subvariety of complex codimension at least $2$. Suppose $\pi_1(X)=0$, do we have $\pi_1(X-Z)=0$? Do we have $\pi_1(X-Z)=\pi_1(X)$ ...
2
votes
0answers
48 views

Fundamental group two torus minus a single point?

So, if I take one torus and take of one single point, what will be its fundamental group? I think that one single point will not change the topology in this sense. Or will? If yes, how?
0
votes
0answers
25 views

Homotopy Equivalence and Local Coefficient Systems

Suppose I am computing the (co)homology of a nice space M using a local coefficient system G, i.e. $H_{*}(M, G)$. If M is homotopy equivalent to N, then M and N should have isomorphic (co)homology. ...
1
vote
1answer
57 views

Question about simply connected spaces.

I am reading Hatcher's Topology and in it, it is noted that a space is simply connected, by definition, if and only if it is path connected and has trivial fundamental group. Can someone provide some ...
2
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0answers
51 views

when will homology and direct limit commute?

Question: Let a sequence of maps between topological spaces $$ X_1\to^{f_1}X_2\to^{f_2}X_3\to^{f_3}\cdots $$ The mapping telescope is denoted by $T$. Under what conditions will $H_*(T)$, the ...
0
votes
1answer
22 views

action of a monoid on a mapping telescope

In the paper Homology fibrations and group completion theorem, McDuff-Segal, page 281, line 14-line 15: For a topological monoid $M$, if $\pi_0(M)=\{0,1,2,3,......\}$, then the action of $M$ on ...
0
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0answers
54 views

Prerequisites “Homology Theory of Algebraic Varieties” by Wallace

I bought this book because the title was very interesting, the description as well and the price very cheap. You can read it here in PDF. Unfortunately, I realized after reading the first lines I was ...
2
votes
0answers
28 views

Action of $H^1$ on spin structures

Since the set of spin structures on a principal $SO(n)$-bundle on a manifold $X$ is in one to one correspondence with $H^1(X,Z_2)$, this group admits an action on spin structures. I wanted to know if ...
1
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0answers
50 views

Long exact homology sequence in singular homology

I am trying to understand/develop the proof of the following theorem: Let $R$ be a commutative ring with 1. Suppose $(C_*, c_*), (D_*, d_*), (E_*, e_*)$ are $R$-chain complexes and $i_*: C* ...
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0answers
19 views

Definition of structure group associated with fiber bundle.

I was studying fiber bundle from Spanier book.In that book there is a definition of structure group associated with a fiber bundle.But I am not able to understand the definition properly.Could ...
1
vote
1answer
47 views

Homology of $S^2/x\sim -x$ for $x$ on the equator

Let $X$ be the quotient space of $S^2$ under the identifications $x\sim -x$ for $x$ in the equator $S^1$. Compute the homology groups $H_i(X)$. I wrote my solution/attempt below and I would like ...
7
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0answers
77 views

How to prove this is a fibre bundle?

Let $M^{2n}$ be $2n$ dimensional toric manifold over a simple polytope $P^n$. Let $\pi : M^{2n} \longrightarrow P^n$ be the orbit map of the torus action. Let $F^k$ be a $k$ dimensional face of $P^n$. ...
2
votes
1answer
25 views

Why is the induced homomorphism an injection?

I am reading Hatcher's Algebraic Topology. One of the propositions says that if a space X retracts to a subspace A, the the homomorphism i# induced by the inclusion i: A --> X is injective. It is ...
2
votes
0answers
61 views

Singular homology: Change of coefficients

Let $f: X \to Y$ be a map of topological spaces which induces isomorphisms $H_*(f;\mathbb{Z})$ on singular homology with $\mathbb{Z}$-coefficients. Show that $f$ induces isomorphisms ...
1
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0answers
28 views

A function between covering spaces.

Given $p_1:\bar X_1\rightarrow X$ and $p_2:\bar X_2\rightarrow X$ covering maps. Proof that if exist $f:\bar X_1\rightarrow \bar X_2$ continuos and surjective then $f$ is a covering map. I don't know ...
2
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1answer
54 views

How can I get a cohomology of hypersurfaces by using their equation?

While studying about complex projective hypersurfaces, I attempts to find a cohomology of this hypersurface : $$X_n=\{(x_0:x_1:x_2:x_3) \in \mathbb{C}\mathbb{P}^3~|~x_0^n+x_1^n+x_2^n+x_3^n=0\}$$ I ...
1
vote
1answer
29 views

The preimage of a curve in the projective plane by the quotient map.

Let $q:S^n \rightarrow \mathbb{R}P^n$ the quotient map between the $n$-sphere and the $n$-dimensional projective plane. Prove that if $\alpha$ is a curve in the projective plane then $p^{-1}\alpha$ ...