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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33 views

Structure space of a commutative ring with unity

I was reading the topic for the discussion of Stone-Cech compactification but stocked at some point: Suppose that $(R,+,.)$ is a commutative ring with unity. Let $\mathcal{M}(R)$ denotes the ...
3
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0answers
31 views

Proving that an $E$-oriented manifold has an $E$-oriented normal bundle

This is the setting we are working in: $M$ is a closed, smooth $n$-manifold embedded in $\mathbb{R}^{n+k}$ with a chosen embedding $e\colon M^n\to \mathbb{R}^{n+k}$. It is $E$-oriented, for $E$ a ...
3
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1answer
73 views

What is the topology of an infinite cylinder?

Consider an infinitely long straw. This is a genus 1, orientable manifold. It is not closed because it is infinitely long. Is there a way I can describe the property that it is "partially closed" or ...
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0answers
45 views

Fundamental group of cylinder

I calculated the fundamental group of the cylinder, $C$, using the following method: triangulate $C$ find max contractable subspace realise generators on remaining 1-simplices I found the ...
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0answers
29 views

Prove Thom isomorphism theorem using universal coefficient theorem

In Ralph Cohen's notes on the topology of fiber bundles pp.90, he claims that for the trivial bundle $p_{\xi}: X \times \mathbb{R}^n \to X$ Thom isomorphism follows from applying the universal ...
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2answers
123 views

Is $n$-dim manifold which is immersed in $\mathbb{R}^n$ diffeomorphic to a ball?

Let $M$ be an $n$-dimensional smooth compact oriented manifold with boundary which is immersed in $\mathbb{R}^n$ (codimension zero). Must $M$ be diffeomorphic to a ball with boundary (the closed unit ...
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1answer
79 views

Homotopy, Identity

Show that there is no homotopy between the identity and the function $f:S^1\to S^1$, $(x,y)\mapsto (x,-y)$ Hello, I have a problem with this task (the task got corrected), because I am not sure, ...
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0answers
39 views

A complement of Milnor's book on singularities with exercises/examples

I'm currently reading "Singular Point of Complex Hypersurfaces" by Milnor. This is really a great book, but I did realize I didn't saw any concrete examples, computations etc ... I was wondering if ...
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24 views

Classifying Unitaries on the Circle with K-Theory

Let $S^n$ be the $n$-sphere. I'm trying to understand the "meaning" of the $\mathbb{Z}$ factors in $$ K_0(C(S^{2n+1}))\cong\mathbb{Z}$$ and $$ K_0(C(S^{2n}))\cong\mathbb{Z}\oplus\mathbb{Z}$$ So $S^n$...
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2answers
72 views

Is there a non-trivial connex compact orientable topological manifold of Euler characteristic 1?

Is there a non-trivial connex compact orientable topological manifold of Euler characteristic $\chi = 1$? Remark: the point has $\chi = 1$, but it is trivial. The real projective plane has $\chi = ...
5
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1answer
80 views

Basic Algebraic Topology puzzler

I've been watching Norman Wildberger's lectures on Algebraic Topology and one of his problems really got me stuck. The question is to show how a double-holed torus with a line of infinite length ...
2
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1answer
44 views

Characteristic class invariant under bundle isomorphism

Let $c$ be a characteristic class for principal $G$-bundles and $p_1: E_1 \to X, p_2: E_2 \to X$ be isomorphic principal $G$-bundles, then $c(E_1) = c(E_2)$ Is this part of the defining naturality ...
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38 views

The complement $\mathbb{R^3}-A$ of a single circle $A$ deformation retracts onto the wedge of a Circle and a sphere. 2$

Here is my intuitive idea. We can pull all the points lying outside a sphere of some fixed radius containing the circle onto the sphere without disturbing the points inside the sphere. Now we have a ...
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2answers
326 views

Hatcher 2.1.10…

Hatcher asked a question in chapter $2$ (a) Show the quotient space of a finite collection of disjoint $2$-simplices obtained by identifying pairs of edges is always a surface,locally homeomorphic ...
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1answer
115 views

An application of Euler Characteristic to Tetrahedron Packing

The following is an application of Euler's equation to tetrahedron packing of any convex polyhedron. I related it to Euler formula; consequently, a third equation is obtained which is independent of ...
2
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1answer
51 views

Exact sequences of bundles and orientations

If we have an exact sequence of finite-dimensional vector spaces $$0\rightarrow E'\rightarrow E\rightarrow E''\rightarrow0$$ then an orientation of any two induces an orientation of the third. I have ...
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0answers
151 views

Why does excision imply this?

In exercise $4$, page 230 of Bredon, he asks for a proof of the Mayer-Vietoris sequence using a commutative braid diagram which substitutes some terms by others using excision. I've solved the ...
2
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1answer
367 views

Does there exist some relations between Cryptography and Algebraic Topology? [closed]

We know that there are many application of Cryptography in our real life. Are there any relation between Cryptography and Algebraic Topology? If yes, please suggest me some link or books. Thanks ...
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194 views

Homotopy of space of immersions, Smale-Hirsch theorem

If $M$ and $N$ are simply connected manifolds with $\dim M< \dim N$, we denote by $Imm\left(M,N\right)$ the space of immersions of $M$ in $N$. Let $M$ and $M'$ manifolds of dimensions $m>0$. ...
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1answer
56 views

Relative homotopy

Show, that the functions $g: S^1\to S^2$, $(x,y)\mapsto (x,y,0)$ and $h: S^1\to S^2$, $(x,y)\mapsto (x,-y, 0)$ are relative homotopies to $(1,0)\in S^1$ Hello, I have a question to this task. I ...
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50 views

Isomorphism in integral Cohomology gives isomorphism in rational cohomology

I was asking myself the question, if a map $f\colon X \to Y$ between CW complexes gives an isomorphism between $H^*(X,\mathbb{Z})$ and $H^*(Y,\mathbb{Z})$ does it already give an isomorphism between ...
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1answer
115 views

Poincare lemma for compact vertical supports in Bott & Tu

I'm trying to work out Proposition (6.16) from Bott and Tu which states that $H^*_{cv}(M\times\mathbb{R}^n)$ is isomorphic to $H^{*-n}(M)$. The first group being forms compactly supported along the ...
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2answers
51 views

Compact cohomology group of connected n-dimensional connected oriented manifold

I know how to show $H_c^n(M)\simeq\mathbb{R}$, where M is a oriented connected n-dimensional manifold, by showing the integration map is isomorphism. However, I found in the book that this is a ...
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2answers
942 views

Am I reading Bott - Tu right?

Summary: I'm finding Bott - Tu to be too brief and terse. I constantly have to look elsewhere to fill in details. This is not time-efficient. Am I missing something? If not - what other books do ...
0
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1answer
54 views

Coefficients of homology

I am wondering why people use different coefficients when defining homology of simplicial complex, like homology over $R$, $Z$, $Z/2$, etc? Is one better then the other and why? Moreover, which one(s)...
3
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1answer
38 views

Proof of :$H^0(E;\pi_0E)\cong \hom_{\pi_0}(H_0(E;E_0);\pi_0E)$ for $E$ a multiplicative spectrum.

Let $E$ be a multiplicative spectrum, connective, and assume $\pi_0E$ is cyclic. I want to prove that $$H^0(E;\pi_0E)\cong \hom_{\pi_0E}(H_0(E;\pi_0E);\pi_0E)$$ Recall that $H^0(E;\pi_0E):= [E; K(\...
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0answers
88 views

Deduce the Cup product structure $RP^n$ with $Z$ coefficient from $Z_2 $ coefficient

while I read Hatchers Algebraic topology book (pg:214) he says that " we can deduce the cup product structure of $RP^n$ with $Z$ coefficient from the cup product structure with $Z_2$ coefficient ? " ...
2
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1answer
52 views

Homeomorphism between N-disk and N-Projective Plane

I've just showed that: $D^n$, quotiented with this equivalence relation: $x\sim y \iff x=-y \text{ and } x,y\in\partial D^n$ (i.e. the antipodal points on the boundary of $D^n$ are identified) is ...
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0answers
16 views

Structure group reduction criterion in terms of classifying map lifting

I am looking for a proof of the following theorem which is given as an exercise in Ralph Cohen's notes on the topology of fiber bundles pp.74. But in view of its importance to the later chapters I ...
1
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1answer
50 views

$SO(3)$ homeomorphic to $\mathbb{R}P^3$

I'm doing some topological base-exercises, but I can't come up with this problem (That I suppose should be quite trivial): $SO(3)$ is homeomorphic to $\mathbb{R}P^3$. Any hints? thank you in Advance!...
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2answers
85 views

Reference for Algebraic Topology

I know undergraduate algebra (groups, rings, fields, Galois etc), undergraduate differential geometry, undergraduate real/complex analysis and now I feel as though to get to the next level, i should ...
2
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0answers
19 views

Pairing on the AHSS induced by cap product: why does it exists

This is my setting: Let $\xi \colon X \to B$ be a vector bundle. Let $E$ be a ring spectrum. Suppose given a natural cap product $$ \frown \colon E^s(X,X\setminus B)\otimes E_{s+t}(X,X\setminus B)...
3
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1answer
24 views

A spectrum $I$ is $E$-injective iff the map $i:I\rightarrow I\wedge E$ is an inclusion of a retract.

I was reading some notes on stable homotopy theory and I came across the statement in the title of this question. "Suppose $E$ is a ring spectrum, then $I$ is $E$-injective if and only if the ...
3
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0answers
22 views

Complexes $K$, $L$, imply $|K| \cap |L|$ is polyhedron.

I am using Armstrong's topology text, and have been really stumped on what this question is asking. It says If $K$ and $L$ are complexes in $\mathbb{E}^n$, show that $|K| \cap |L|$ is a ...
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0answers
9 views

Iff conditions for acyclic, free, positive chain complexes with augmentation

I have some doubts about the formulation of the following lemma (from Ferrario, Piccinini - Simplicial structures in topology) and its proof. (II.3.8, page 72) Lemma. Let $(C,\partial) $ be a ...
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1answer
26 views

Why is the quotient map $G\rightarrow G/T$ a fibration?

I have just learnt about fibrations and I saw somewhere the following. Given a compact lie group $G$, one can consider a maximal torus $T$ of $G$ then the quotient map $G\rightarrow G/T$ is a ...
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2answers
151 views

Intuitive or visual understanding of the real projective plane

If we take the definition of a real projective space $\mathbb{R}\mathrm{P}^n$ as the space $S^n$ modulo the antipodal map ($x\sim -x$), it is possible to see that $\mathbb{R}\mathrm{P}^1$ is ...
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0answers
42 views

Weibel 5.1.1 Exercise

I know this topic is already dealt on Total complex homology exact sequence, But have a question on the answer. The answer says that $$H_{p + q}(T) \cong \frac{\{(a,b) | d^v_{p-1,q+1}(a)+d^h_{p,q}...
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1answer
53 views

What is the classifying space G/Top?

I simply can't find the definition(except in one book on surgery where a definition was not actually given but instead they alluded to what the definition is) and I have spent an hour and half looking....
5
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1answer
674 views

Show that the $\Delta$-complex obtained from $\Delta^3$ by performing edge identifications deformation retracts onto a Klein bottle.

I am going through some exercises in Hatcher's Algebraic Topology. You have a $\Delta$-complex obtained from $\Delta^3$ (a tetrahedron) and perform edge identifications $[v_0,v_1]\sim[v_1,v_3]$ and $[...
2
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1answer
106 views

How often is a torus in a compact Lie group nullhomologous?

Minor nomenclature question: What is the standard name for the ring structure induced on the homology of an H-space by its multiplication? I've seen "homology ring" and "Pontrjagin ring." Hopefully ...
3
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2answers
59 views

Why $h$ has zero topological degree?

I am trying to prove that $f,g : M^n \to S^n$, both $C^1$ (indeed just $C^0$ is enough) with the same topological degree are homotopic. I saw on a book that the trick is as follows: Take $W = M\...
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1answer
164 views

Is every topological group the topological fundamental group of an space?

The fundamental group $\pi_{1}(X)$ of a path connected topological space $X$ is the image of $Hom(S^{1},X)$. So the fundamental group can be topologized with quotient topology where $Hom(S^{1},X)$,...
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1answer
42 views

What is the topological degree of the constant map?

What is the topological degree of the constant map? To me it does not make any sense, once $f$ being the constant map has no regular values. So, how to proceed?
5
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1answer
57 views

Explicit verification of signs in Morse complex

I'm trying to check by hand that the signs in the Morse complex, defined via choices of orientations on the unstable manifolds, lead to $\partial^2=0$. The books I've looked in seem to say either ...
3
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0answers
91 views

Fundamental Group, Piecewise Smooth Curves, Conservative Fields

Let M be compact Riemannian manifold. X be a vector field on M. I believe that work done by X any piecewise closed curve is zero, iff, the same is zero for a particular finite set of loops. I believe ...
10
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1answer
162 views

Explicit examples of (co)limit arguments in other fields

Over the past weeks, I have noticed that high level lecture notes in subjects like algebraic geometry, algebra, and algebraic topology often sketch proofs in the following form: Proof sketch ...
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1answer
29 views

Covering map associated with open cover

Let $ \left\{U_i \right\}$ be an open cover of $X$. On some online sources and some MSE questions, the map $\coprod _iU_i\rightarrow X$ is given as an example for a local homeomorphism which is not a ...
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1answer
50 views

Subgroup of Finite Index Containing a Given Finitely Generated Subgroup of a Free Group: Problem 12 in 1A in Hatcher.

Problem 1A.12 (Hatcher) Let $F$ be a finitely generated free group and $H$ be a finitely generated subgroup of $F$. Let $x\in F-H$. Show that there is a finite index subgroup $K$ of $F$ such that $H\...
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0answers
39 views

Homotopy, topology

Let $X$ be a topological space. Let $w$ be path in $X$. $\overline{w}(t)=w(1-t)$ and $\iota_{x}(t)=x$ for every $t\in[0,1]$ and $x\in X$. Give homotopies $H$ and $K$, with $w\ast\iota_{...