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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0
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1answer
240 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 ...
1
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0answers
76 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
53 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
29 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
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1answer
73 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
86 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
51 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
55 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 ...
3
votes
1answer
768 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
57 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
170 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
63 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 ...
5
votes
1answer
140 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 ...
1
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3answers
59 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. ...
1
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2answers
910 views

A contractible space is path connected.

Let the space be $X$ and $\rm{id} \simeq x_0$ where $\rm{id}$ is the identity map on $X$ and $\rm x_0$ is some fixed point in $X$. How do I show that for any two points $a,b \in X$ there is a ...
8
votes
1answer
270 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
69 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
105 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 ...
2
votes
1answer
209 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
72 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
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1answer
96 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
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1answer
48 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
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1answer
265 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
90 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
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1answer
76 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 ...
2
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0answers
115 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
88 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
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0answers
46 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
47 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
109 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 ...
9
votes
1answer
259 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
76 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
89 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 ...
1
vote
1answer
48 views

Connectivity and Euler characteristic for surfaces

I learn the concept of connectivity from Hilbert's Geometry and the Imagination as follows: A polyhedron is said to have connectivity $h$ (or to be $h$-tuply connected) if $h-1$, but not $h$, ...
5
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0answers
69 views

Reference about history of characteristic classes

I'm looking for a good reference about history of characteristic classes. For example, I would like to know how Chern define the Chern class at first in history, or who rewrote the characteristic ...
4
votes
1answer
149 views

Do the composition of two Knots always yield a distinct knot (ignoring orientation)?

I would greatly appreciate if I could get some help in clarifying my understanding. (This is a special topic I am studying as a 2nd year University student - I haven't taken topology yet - so please ...
3
votes
1answer
84 views

What are other examples of characteristic numbers?

Be warned, this may be a ridiculous question. I understand characteristic classes of principal $G$-bundles (and associated vector bundles) over a space $X$ arise from the classifying maps $f\colon X ...
1
vote
2answers
125 views

Cell Complex: Proposition 5.5 in John Lee's book “Introduction to Topological Manifolds”

The proposition reads: "Suppose $X$ is an $n$-dimensional CW complex. Then every $n$-cell of $X$ is an open subset of $X$." The proof first shows that the intersection of any $n$-cell of $X$ with the ...
2
votes
1answer
293 views

How to prove that two curves are not path homotopic

I have a unit circle around origin.And another unit circle around $(2,0)$. Consider the domain $R^2 / \{(0,0)\}$. I am able to clearly see that both are not homotopic but i am unable to prove it ...
2
votes
2answers
117 views

homotopy between two functions

Let us discuss this problem: Let $A=\{a_{1},a_{2},\ldots,a_{n}\}$, $B=\{b_{1},b_{2},\ldots,b_{n}\}$ and $C=\{c_{1},c_{2},\ldots,c_{n-1}\}$ be discrete finite sets embedded in a unit sphere ...
1
vote
1answer
122 views

Cohomology to compute number of holes?

Can one use cohomology to compute the number of holes in a space $E$, where $E=R\times R$, $R$ is a Riemann surface of genus $g$, - i.e., is $\dim(H^n(E))$, and by Künneth's formula, $H^{n}(E) \cong ...
0
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0answers
115 views

References Request (Poincare Duality via Unimodular pairing, de Rham isomorphism)

Chapter 0 of Griffith's Harris contains a substantial amount of topological results that I have not seen elsewhere, namely: Poincare duality as a unimodular intersection pairing on homology. Also ...
3
votes
0answers
290 views

3-fold connected covers of punctured torus

I am interested in finding all the connected surfaces (up to homeomorphism) that can be described as $3$-fold covering spaces of the torus with a disc deleted (in both cases that the disc is closed or ...
9
votes
6answers
1k views

Isomorphic fundamental groups result in homeomorphism?

I know that the fundamental group of homeomorphic spaces are isomorphic. Is the converse true? I mean, can we say the two spaces with isomorphic fundamental groups are homeomorphic?
3
votes
1answer
102 views

Why are there cubics in a Calabi-Yau manifold?

I heard from a recent talk that the "number of $n$-degree curves" in a Calabi-Yau manifold is an invariant of the space. But what does that mean? (Specifically I would like to ask the following.) ...
4
votes
2answers
196 views

Why is $\pi_1(X,x_0)$ a group?

I want to show that $\pi_1(X,x_0)$ is a group. I am told that $e(t) := x_0$ is the identity element. Now, I am struggling to show that it is an identity element, and also that the inverse of an ...
1
vote
1answer
39 views

$H_0(X) \cong \tilde{H}_0(X) \oplus \mathbb{Z}$

I can't seem to figure out why $H_0(X) \cong \tilde{H}_0(X) \oplus \mathbb{Z}$. In my notes it says: ''...Since $\varepsilon \partial_1 = 0$, $\varepsilon$ vanishes on $im \partial_1$ and hence ...
3
votes
0answers
67 views

Path-homotopic definition.

Given two paths $f,g: [0,1] \mapsto X $ Then their product is defined by $f\cdot g := \begin{cases} f(2t) , \space 0\leq t \leq \frac{1}{2}\\ g(2t-1) ,\space \frac{1}{2} \leq t \leq 1\\ \end{cases} ...
1
vote
1answer
105 views

Problems about exact sequence in Vick's homology theory.

I am self studying algebraic topology through vick's homology theorey, but I don't know how to prove 2.3 proposition on page 38~39. (which the author stated only and said it is easy) Here's the ...
4
votes
1answer
196 views

A group-like topological monoid is a loop space

I am looking for an elementary reference for the following fact. Let $M$ be a topological monoid and suppose moreover that it is group-like, ie. $\pi_{0}(M)$ is a group. Then the canonical map $M ...