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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1answer
26 views

The line between the centers of two k-simplices crosses through the centers of lower dimensional simplices

"The line between the centers of two k-simplices crosses through the centers of lower dimensional simplices." Is this always true? The barycentric centers are equidistant from the vertices. k=2 So ...
1
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1answer
29 views

Identifying letters up to homotopy

I already identified the letters of the alphabet up to homeomorphism and the useful characteristic was cut-points and their preservation under homeomorphism. As a visual representation you can imagine ...
0
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1answer
39 views

Is a Covering Space of a Topological Space always Hausdorff?

Is a Covering Space of a Topological Space always Hausdorff? I can separate two different points from the same fiber, but what about two arbitrary points?
1
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2answers
47 views

A more general definition of branched covering.

If $f:X\longrightarrow Y$ is a holomorphic map between two compact Riemann surfaces, then $f$ is called also a branched covering map. This because the branched points of $f$ form a finite set ...
2
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1answer
83 views

Solving an exercise in Milnor-stasheff's “characteristic classes”

I am trying to solve the following exercise (which is an exercise in Milnor-Stasheff's book). It basically says the following: Let $ M =S^n $ be the $n$-sphere and let $TM$ be its tangent ...
10
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1answer
103 views

How can I understand the three-dimensional space forms?

Here is what I know: A space form is defined as a manifold admitting a Riemannian manifold of constant sectional curvature A classical result of Cartan states that a manifold is a space form if and ...
1
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0answers
100 views

Refining homotopy commutative maps of spectra to maps of E_{\infty}-ring spectra

In Adams' blue book (page 54) we have a map in the homotopy category of ring spectra $f: MU \rightarrow K$ where $K$ is complex $K$-theory such that $g_*x^{MU} = (u^K)^{-1}x^K$ where $x^E$ denote ...
6
votes
1answer
69 views

De Rham cohomology of $T^*\mathbb{CP}^n$

I am a bit rusty on my de Rham cohomology, and I'm hoping that someone here could help me. I want to find the cohomology of $T^*\mathbb{CP}^n$ (seen as a real manifold). Now, this should be equal to ...
1
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1answer
43 views

Can we compare cohomology rings with different coefficients?

I have an example sheet that asks me to compute the cohomology rings for two spaces, say X and Y, with coefficients in $\mathbb{Z}$ and $\mathbb{Z}_d$ respectively. It then asks whether X and Y are ...
1
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1answer
33 views

Poincare lemma on current

The Poincare lemma on current states that: If $U$ is a star-shaped open set in $\mathbb R^n$ and $T$ is a $k$-current on $U$ such that $dT=0$, then there is a $k-1$-current $S$ on $U$ such that $dS = ...
2
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1answer
47 views

$S^{1}$-bundles over $\mathbb{RP}^2$

How many $S^1$-bundles over $\mathbb{RP}^2$ do exist? Is it true that there exist only two bundles - trivial and not?
2
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0answers
35 views

Hatcher Theorem 3.26 - orientability

I am reading Hatcher, the beginning of the chapter on Poincare duality. I am trying to understand how theorem 3.26 is deduced from lemma 3.27 and I must admit I find Hatcher's proof very esoteric. ...
0
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0answers
38 views

Euler characteristics and 2-manifold mesh

I would like to ask you for a simple definition of two-manifold surfaces and correlation with Euler characteristics $\chi=2-2g$. Is this characteristics always valid for 2-manifolds? More particularly ...
0
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1answer
24 views

Elementry collapses implies same homotopy type.

Let $\Delta$ be a simplicial complex, and suppose that $\sigma \in \Delta$ is a proper face of exactly one maximal simplex $\tau \in \Delta$. A simplicial collapse of $\Delta$ is the removel of all ...
1
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0answers
34 views

how does dunce cap has simplicial structure

Dunce cap is a example of space which is contractible but not collapsible, but collapsibility is only defined for simplicial complexes. Can anyone explain how does dunce hat has simplicial structure?
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2answers
54 views

When do homology groups have torsion?

Let $C_q(K)$ be a group of q-chains on given simplicial complex K. Since this is a free abelian group, its subgroup must be a free abelian group especially $Z_q(K),B_q(K)$. Then we define the homology ...
7
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2answers
118 views

Is there any point-set definition of simple connectedness?

The definition of path-connectedness refers to the set of real numbers, $\mathbb{R}$. (More precisely, the interval $[0,1]$) On the other hand, connectedness is defined "purely" in terms of points and ...
0
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1answer
50 views

Manifolds Resulting from Gluing Tori

I'm trying to show that if solid tori $T_1, T_2; T_i=S^1 \times D^2$ ,are glued by homeomorphisms between their respective boundaries, then the homeomorphism type of the identification space depends ...
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0answers
31 views

Fundamental groups of configuration spaces

In a previous answer see here by Samuel Reid, I read the following: "The configuration space of $n$ points in a topological space $X$ is usually defined to be, $$C_{\hat{n}}(X) = \{(z_{1},...,z_{n}) ...
0
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1answer
26 views

Is it possible to compute homology groups of a space given the Pontryagin ring?

Or similarly, given the cohomology ring of a space, is it possible to compute its cohomology groups? I'm mainly interested in integer and mod 2 homology and cohomology.
1
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0answers
21 views

homotopy - maintaining curvature signs

I have the following dilemma! Say, $f_1=\sqrt{1-x^2}$, and $f_2=-\sqrt{1-x^2}$ are two continuous functions on $[-1,1]$ Lets define another function by $F = tf_1 + (1-t)f_2$ where $t=[0,1]$ ...
0
votes
1answer
55 views

Why is $*$ defined only for homotopy classes, and not individual paths between points?

Why is the operation $*$ well-defined on homotopy classes, and not all continuous paths from $[0,1]$ to $X$ in general? I suppose "well-defined" means that if $a=b$ and $c=d$, then $a*c=b*d$. I feel ...
4
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0answers
46 views

Fiber Bundle of Manifolds

I want to show the following: Let $E \rightarrow B$ be a fiber bundle with fiber $F$. Show that if $B$ and $F$ are manifolds, then so is $E$. Solving this problem seems easy enough simply by ...
2
votes
1answer
34 views

Mapping Class Group of Simply Connected Spaces

I was wondering the following: If we take $M$ to be some orientable, simply-connected $n$-manifold. What can be said about $\pi_0(Homeo(M))$? We know that $\pi_1(M)=0$ and I know that the group is ...
0
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1answer
36 views

Correspondence of Grassmannian cells

I have shown that the correspondence $f: X \rightarrow \mathbb{R} \oplus X$ defines an embedding of the Grassmannian $\mathrm{Gr}_{k}(\mathbb{R}^{n+k})$ into $\mathrm{Gr}_{k+1}(\mathbb{R} \oplus ...
1
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0answers
18 views

What is the topological relation between $ C^\infty (E, S^1) $ and $ E $

What is the topological relation between $ C^\infty (E, S^1) $ and $ E $? Here, $ C^\infty (E, S^1) $ is the algebra of all continuous functions from $ S^1 $ to $ E $. $ E $ is a four dimensional ...
2
votes
1answer
36 views

Product of simplicial complexes?

Given two abstract simplicial complexes $\mathcal{K}$ and $\mathcal{L}$, what is the definition of their product $\mathcal{K} \times \mathcal{L}$, as another abstract simplicial complex? Basically I'm ...
3
votes
1answer
42 views

Isomorphism of Grassmannians

I want to prove that two CW complexes $\mathrm{Gr}_{n}(\mathbb{R}^{n+k})$ and $\mathrm{Gr}_{k}(\mathbb{R}^{n+k})$ are isomorphic to one another. I'm pretty sure I can just show that the number of ...
6
votes
1answer
62 views

Cohomological Whitehead theorem

Let $X$ and $Y$ be CW complexes (resp. Kan complexes) and let $f : X \to Y$ be a continuous map (resp. morphism of simplicial sets). The following seems to be a folklore result: Theorem. The ...
0
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1answer
50 views

A doubt regarding the need for lemma 52.3 in Munkres' “Topology”.

Munkres defines a simply connected space $X$ as: A path connected space in which $\pi_1(X,x_0)$ is the trivial one-element group for some $x_0\in X$, and hence for every $x_0\in X$. He then goes ...
0
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0answers
40 views

English edition of Vol 9 of Dieudonné's Foundations of Modern Analysis?

I have found the first 8 volumes of Dieudonné's Foundations of Modern Analysis in English translation, but I'm having difficulty locating volume 9. I have searched the catalogues of numerous libraries ...
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3answers
46 views

Why is $\pi_1(\Bbb{R}^n,x_0)$ the trivial group in $\Bbb{R}^n$?

My Algebraic Topology book says Let $\Bbb{R}^n$ denote Euclidean n-space. Then $\pi_1(\Bbb{R}^n,x_0)$ is the trivial subgroup (the group consisting of the identity alone). I wonder why that is. ...
2
votes
1answer
50 views

If M is a non-orientable closed connected 3 manifold prove H1(M) is an infinite group.

This is an example from a question sheet (non-assessed) of a university class. If M is a non-orientable, closed, connected 3 manifold, prove $H_1(M;\mathbb{Z})$ is an infinite group. I know that since ...
6
votes
1answer
56 views

Relation between different definitions of degree in complex geometry

Consider a holomoprhic map from a Riemann surface $$ f: \Sigma_g \to \mathbb{CP}^n. $$ This is given by some homogeneous polynomials in some variables. How can we show that the homogeneous degree $d$ ...
2
votes
1answer
41 views

$\mathrm{Homeo}(S^1)$ and the Mapping Class Group

Is there a full description of $\mathrm{Homeo}(S^1)$ (i.e. the group of self-homeomorphisms of the circle)? By full description I mean a presentation/list of subgroups ect. Basically anything ...
1
vote
1answer
51 views

Computing the Todd class of projective space.

As an exercise I'm trying to verify that for $X=\Bbb{P}_k^n$, where $k$ is an algebraically closed field, we have $$\operatorname{td}(X)=\left(\frac{\epsilon}{1-e^{-\epsilon}}\right)^{n+1},$$ where ...
0
votes
1answer
49 views

Isomorphic homology and cohomology groups

Let $X$ be a CW-complex of finite dimension and $F$ be a field. Do we have that $H^q(X;F)=H_q(X;F)$ for each $q\leq n$? I know that with filed coefficients the universal coefficient theorem simplifies ...
3
votes
1answer
47 views

Does $\pi_0$ preserve infinite products?

The functor $\pi_0\colon \operatorname{sSets}\to \operatorname{Sets}$ has value on a simplicial set $X$ the coequalizer of the two boundary morphisms $d_0,d_1\colon X_1\to X_0$. It is easy to see ...
0
votes
1answer
39 views

using covering space technique,prove that $[G:H \cap K] \leq [G:H][G:K]$.

using covering space technique,prove that if $G$ is a group with subgroups $H$ and $K$ then $$[G:H \cap K] \leq [G:H][G:K]$$ I couldn't understand the relation between them and the covering space,so ...
3
votes
0answers
79 views

Application of Lefschetz duality to prove Lefschetz hyperplane theorem

I'm trying to understand the proof of the Lefschetz hyperplane theorem in Milnor's book "Morse Theory", page 41 but I can't understand his use of Lefschetz duality. At this point it has been proven ...
2
votes
1answer
51 views

vector bundles and their cross-sections

Let $B$ be a compact Hausdorff space and $C^0(B)$ be the ring of continuous real valued functions on $B$. For any vector bundle $\xi$ over $B$, let $\Gamma(\xi)$ denote the $C^0(B)$-module consisting ...
1
vote
1answer
38 views

Covering of a Topological Group(Use of fundamental theorem of covering spaces)

Suppose we have two path-connected spaces $G$ and $H$. Suppose also that $G$ is a topological group with an identity element $e$ and there is a covering $$ p: H \rightarrow G $$ The problem asks that ...
1
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0answers
59 views

Showing a subcategory of $\mathbf{Top}$ is Cartesian-Closed

We start with some preliminary definitions (necessary because there is not much literature on this): a test map is a continuous function $\varphi:V\rightarrow X$ where $V$ is an open subspace of ...
1
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1answer
43 views

what is the Cayley complex of dihedral group $D_{4}$?

what is the Cayley complex of dihedral group $D_{4}$? I am aware of Cayley graph of $D_{4}$,can you explain to me how I should I attach 2-cell complexes to the loops to make it covering space? I ...
0
votes
0answers
40 views

How to deform a curve in specific manner

I am wondering whether we can deform a path in specific ways continuously i mean if there is a closed piece wise $C^1$ smooth path which has to be deformed to another piece wise $C^1$ smooth path. Let ...
1
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0answers
36 views

existence of a cofiber sequence

can anyone help me with this problem. thanx. Show that there are cofiber sequence $S^{n+3} \to S^{n+2} \to \sum^{n}\mathbb{C}P^2$ for each $n \in \mathbb{Z}^+$. Conclude that a space of the form ...
2
votes
1answer
45 views

What are the formulas for topological transformations? How to obtain them?

I'm reading Flegg's From Geometry to Topology, the author says that in Euclidean geometry, translation and rotation are: $$T:(x,y)\to(x+a,y+b)$$ $$R:(x,y)\to(x \cos \phi - y \sin \phi, x \sin \phi +y ...
7
votes
1answer
123 views

Explanation for a line from a MathOverflow answer

Sometimes I see questions answered on MathOverflow in such a way that I don't really understand the answers. Sometimes I work out what they mean, and other times I can't. I'd like to ask for more ...
0
votes
3answers
46 views

if $p:\widetilde{X}\rightarrow X$ is a covering space and $\widetilde{X}$ is path connected ,show that $p^{-1}(A)$ is path connected.

if $p:\widetilde{X}\rightarrow X$ is a covering space and $\widetilde{X}$ is path connected ,also $A\subset X$ is a path connected subset,show that $p^{-1}(A)$ is path connected. I suppose that ...
0
votes
1answer
35 views

Hyperbolic surface

Let $X=S_1\times S_1−Δ$ , where $Δ=\{(x,y)∈S_1\times S_1|x=y\}$. I know that this is (one of the) usual model for the cylinder , but how to proof that it is a hyperbolic surface? Any help is ...