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

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17 views

Homology groups of $S^1\times (S^1 \vee S^1)$ [duplicate]

I'm trying to calculate the homology groups of $S^1\times (S^1 \vee S^1)$. This complex has two 2-cells, three 1-cells and one 0-cell, so using cellular homology, I have deduced that $H_2 = ...
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1answer
39 views

Notation: determinant of Jacobian matrix

Given a function $f:\Delta^n \to Y$ from a simplex into a riemmannian manifold. Furthermore given a point $x \in \Delta^n$ we can send an orthonormal basis at $x$ using $D_x f$ to a set of vectors ...
5
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0answers
62 views

“Toys” spaces in algebraic topology

I did follow a course of algebraic topology last semester and I still want to continue to do some computations. But in many books it's all the time the same examples which comes back for computing ...
5
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1answer
116 views

Non-orientable one dimensional manifold.

I was trying to solve a question from Hatcher's book in section 3.3. Question is: Show that there exist a non-orientable 1-dimensional manifold if Hausdroff condition is droped from the definition ...
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0answers
24 views

quotient map (induced from a special group action) is a covering map

I have a question related to Proposition 1.40 a) (page 72, chapter 1) in Hatcher's book http://www.math.cornell.edu/~hatcher/AT/AT.pdf, that the Quotient map is a covering map, if a group acts freely ...
3
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1answer
66 views

Are all maps into path-connected spaces homotopic?

Context: A problem I'm solving asks "Prove that any two maps $S^m \rightarrow S^n$, where $n>m$ is homotopic. [Hint: use the Simplicial Approximation theorem] My first thoughts were that if we ...
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1answer
65 views

General Linear Group over the quaternions is a a topological group

How to show that General Linear Group over the quaternions is a a topological group?
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0answers
36 views

an orientable surface with infinite cyclic fundamental group must be annulus?

It is claimed in this accepted answer http://mathoverflow.net/questions/79929/how-to-rigorously-prove-that-simple-closed-curves-on-a-surface-are-primitive-clo that an orientable surface with infinite ...
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1answer
91 views

Calculate the homology group of $S^3/G$, an Harvard qualifying exam problem with “unclear” solution

Problem Suppose that $G$ is a finite group whose abelianization is trivial. Suppose also that $G$ acts freely on $S^3$. Compute the homology groups (with integer coeffcients) of the orbit space ...
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0answers
22 views

How do the direct and inverse image sheaf functors interact with homotopy?

The direct image sheaf functor $f_\ast$ and inverse image sheaf functor $f^\ast$ (here I mean the usual inverse image sheaf functor often denoted by $f^{-1}$) form a well-known adjunction for ...
2
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1answer
53 views

Show that $f:X\to \mathbb{RP}^n$ factors through $S^n\to\mathbb{RP}^n$ iff $f^*:H^1(\mathbb{RP}^n,Z_2)\to H^1(X,Z_2)$ is zero

Problem Show that (1) $f:X\to \mathbb{RP}^n$ factors through $S^n\to\mathbb{RP}^n$ iff $f^*:H^1(\mathbb{RP}^n,Z_2)\to H^1(X,Z_2)$ is zero (2) $f:X\to \mathbb{CP}^n$ factors through ...
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1answer
38 views

Homotopy type of intersection of complement of hyperplanes in projective space.

Let $U_i = \{x=(x_0 :… :x_n) \in \mathbb{P}^n(\mathbb{C}); x_i \neq 0 \}$ be the usual trivialization of the complex projective space. I have been trying to compute the homotopy type of all the ...
6
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2answers
106 views

Constructing a compact manifold with chosen homology groups

Let $G_1,G_2,\ldots,G_k$ be $k$ finitely presented abelian groups. It's possible to construct a $(2k+3)$-dimensional manifold $X$ s.t $H_i(X) = G_i$ in following way: consider $k$ copies of the Moore ...
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0answers
41 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 ...
3
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1answer
31 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 ...
6
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2answers
98 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 ...
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0answers
27 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
27 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 ...
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1answer
43 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
52 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 ...
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0answers
16 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: ...
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1answer
22 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
90 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 ...
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0answers
47 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
37 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
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0answers
59 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
33 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 ...
4
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1answer
116 views

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 ...
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0answers
58 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 ...
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2answers
61 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, ...
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0answers
38 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
49 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 ...
3
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1answer
35 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
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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 ...
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5answers
66 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
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1answer
62 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
29 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
30 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 ...
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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
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1answer
69 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
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1answer
37 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.
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0answers
23 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 ...
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0answers
41 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, ...
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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
57 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
51 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?
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0answers
29 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. ...
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1answer
58 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
52 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
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1answer
24 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 ...