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
19 views

Is the tangent bundle of an oriented surface plus a trivial bundle trivial?

Let $\Sigma$ be an oriented closed surface and $E$ be the direct sum of $T\Sigma$ with a trivial line bundle. Is $E$ a trivial rank $3$ vector bundle? For genus $0$ and $1$ the answer is yes since ...
4
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2answers
82 views

Fundamental group of the Poincaré Homology Sphere

I'm working on the Poincaré Homology Sphere $P_3$ and would like to compute it's Homology $H_1$ and fundamental group. I would like to identify it's fundamental group with the binary icosahedral group ...
0
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1answer
14 views

Determining images of points in a path homotopy.

Say the two paths $f_0$ and $f_1$ ae homotopic. Then $(1-t)f_0+tf_1$ is the homotopy between the two paths. Say $f_0,f_1\in\Bbb{R^2}$, and there is a point $(a,b)$ in $f_0$. How can we find which ...
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0answers
25 views

What is the complement of a loop?

My Algebraic Topology book says $A$ is a loop in the complement of another loop $B$ What does "in the complement of" mean here?
2
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0answers
29 views

Hopf Invariant Definitions

I have seen two definitions of the Hopf invariant given: (1) Cohomological Definition: Let $S^{n}$ denote the oriented $n$-sphere, where $n \geq 2$. Let there be given a map $f:S^{2n-1} \rightarrow ...
2
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2answers
56 views

Proving that the fundamental group of Klein bottle is generated by two elements without using covering spaces and van Kampen theorem, etc.

This is a problem in 'Topology and Geometry' by Bredon. I tried hard on this problem, but have no idea what to do. This question was onced asked by someone, but there was no satisfactory answer. Let ...
3
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1answer
72 views

Topological invariance of chern classes

Are Chern classes topological invariants? To be more precise: Given two complex manifolds $M$ and $N$. Does a homeomophism $f:M\to N$ map Chern classes to Chern classes?
1
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1answer
44 views

Fundamental Group of Klein Bottle?

Let $C^{*}=C \setminus \{ 0 \}$. What is the fundamental group of $C^{*}/H$, here $H=\{\psi^n;n \in \mathbb{Z}\}$ with $\psi(z) = 2 \bar{z}$?
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1answer
36 views

Question about degrees of maps from $S^1 \rightarrow S^1$

Note: Since the degree of a map is independent of the base-point I'll speak loosely and just say $\pi_1(S^1)$. One definition of the degree of a map $f$ from the circle to itself is the number $k$ ...
1
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1answer
29 views

Show that if $h,k: S^1\rightarrow S^1$ are homotopic, they have the same degree.

We define the degree of a continuous map $h: S^1 \rightarrow S^1$ as follows: Let $b_0$ be the point $(1,0)$ of $S^1$; choose a generator $\gamma$ for the infinite cyclic group $\pi_1(S^1,b_0)$. If ...
5
votes
4answers
526 views

Topological groups, why need them?

I'm reading through Munkres and Armstrong's books on topology. However, I find topological groups to be really complicated objects! I feel they are twice as hard to deal with then just groups and ...
4
votes
1answer
52 views

Show that if B is simply-connected, then p is a homeomorphism.

Let $p: E \rightarrow B$ be a covering map with $E$ path-connected. Show that if $B$ is simply-connected, then $p$ is a homeomorphism. I'm checking to see if my solution is flawed. Since $p$ is a ...
1
vote
2answers
104 views

Let $p: E\to B$ be a covering map. If $B$ is compact and $p^{-1}(b)$ is finite, then $E$ is compact. [duplicate]

So I start off and assume that some $\{U_\alpha\}$ is a cover of $E$. I want to reduce this cover to a finite subcover of $E$. Since $p$ is a covering map it is also an open map, therefore ...
1
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0answers
124 views

Help Understanding/Completing Proof of Prop 3.18/3.19 in Hatcher's Algebraic Topology

I apologize right away for the wall-o-text. I'm participating in a cohomology reading course, and I'll be leading the class through the following proposition later this week, but I'm having a hard ...
1
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1answer
26 views

A question about subexponential growth about group

Recall a group $\Gamma$ is said to have subexponential growth if lim sup$|E^{n}|^{1/n}=1$ for every finite subset $E\subset \Gamma. (E^{n}=\{g_{1}g_{2}...g_{n}: g_{i}\in E\}.)$ My question is: Can we ...
3
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0answers
34 views

Singular cohomology with compact support

If $X$ is a locally compact Hausdorff space, then for any $n \geq0$ is $H_c^n(X) \cong {\tilde H^n}({X^ + })$? ($H_c^n(X)$ is the Singular cohomology with compact support and $X^+$ is the one-point ...
1
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1answer
50 views

Mapping $\mathbb R^n - \{0\}$ to $S^{n-1}$

How might one map $\mathbb R^n - \{0\}$ to $S^{n-1}$ ? I found this in a primer on homology where it is proved that the to spaces are homotopy equivalent, as an example of removing a single point ...
0
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0answers
30 views

fundamental group of $\mathbb{C^*}/\{e,a\}$

I'm taking an intro to topology course, and am having trouble with this question. What is the fundamental group of $\mathbb{C^*}/\{e,a\}$, where $e$ is the identity homomorphism and $az=\overline{z}$. ...
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0answers
17 views

Presentations of fundamental groups regarding cones of simplicial subcomplexes

Let $L$ be a simplicial subcomplex of $K$. Let $CL$ be the cone on L. Let $X = CL \bigcup K$. Show that X is a simplicial complex and dscribe a presentation for $\pi_1(|X|,v)$ in terms of ...
4
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0answers
101 views
+100

Lefschetz duality for non-compact relative manifolds

I'd like to use the formulation of Lefschetz duality stated here, but I can't seem to find a reference for this particular version of it, and it doesn't seem quite right to me. The exact statement in ...
3
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1answer
24 views

The coeffcients of a generator of $H_0(X)$ sum to $\pm 1$?

I'm reading Theorem 4.14 (p. 70) of Rotman's Intro to Algebraic Topology. He proves that if $X$ is a nonempty path connected space, then $H_0(X)\simeq\mathbb{Z}$, and if $x_0,x_1\in X$, then ...
5
votes
1answer
29 views

Surprising constructions in algebraic topology that facilitate one's understanding of underlying theory

I am recently come into the world of algebraic topology and find it a fascinating place with lots of beautiful constructions that challenge one's intuition. Also, understanding these constructions are ...
2
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0answers
34 views

Surjectivity and exactness on higher homotopy groups

If $X$ is a normal algebraic variety over $\mathbb{C}$ and $U$ an open set of $X$ then we have surjectivity on the fundamental groups $ \pi_1(U)\rightarrow \pi_1(X)$. If $f\colon X \rightarrow Y$ is ...
1
vote
2answers
71 views

Fundamental Group of Punctured Plane

What is the fundamental group of $(\mathbb{C} \setminus {\{0\}})~/~\{e,a\}$, where $e$ is the identity homeomorphism and $az = -\bar{z}$? Clearly this is homeomorphic to the half cylinder , which is ...
0
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0answers
22 views

Edges and Vertices in relation to Free subgroups

Let $\Gamma$ be a finite connected graph (1-dimensional simplicial complex), with $V ( \Gamma)$ vertices and $E(\Gamma )$ edges (1-simplices). Show that $\pi_1(\Gamma, v)$ is a free group with ...
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0answers
33 views

Real Spectrum of a Commutative Ring

Could anyone explain briefly what is a real spectrum of a commutative ring? If it possible please give me some easy example. Many thanks in advance. $\text{Spec}_{r}(A)$
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0answers
24 views

John Hempel's proof of residual finiteness of surface groups

John Hempel proved (http://www.ams.org/journals/proc/1972-032-01/S0002-9939-1972-0295352-2/S0002-9939-1972-0295352-2.pdf ) that fundamental groups of 2-manifolds are residually finite. I want to ...
1
vote
1answer
29 views

A simple question about 1-norm

Let $\Gamma$ be a discrete group, if $\mu \in l^{1}(\Gamma)$, then what is the 1-norm of $\mu$, I mean $||\mu||_{1}=?$. As we know, $l^{1}(\Gamma)=\{(\alpha_{x})_{x\in\Gamma}: ...
1
vote
1answer
45 views

fundamental group of complex numbers?

Let $\mathbb{C}^*=\mathbb{C}-{0}$. What is the fundamental group $\mathbb{C}/G,$ where G is the group of homeomorphism $\{\phi^n ; n\in \mathbb{Z}\}$ with $\phi(z)=2z$? I think the fundamental group ...
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2answers
40 views

Simply Connected Points in Disk

Why is the set of all points $z \in D^2$ for which $D^2 \setminus \{z\}$ is simply connected just $S^2$?
0
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1answer
29 views

how to describe $f_*: \pi(S^1,1)\rightarrow \pi(S^1,1)$ in terms of the isomorphism of $\pi(S^1,1) \cong Z$

can someone help me with this question please? Let $f:S^1 \rightarrow S^1$ be the mapping defined by $f(z)=z^k$ for some integer $k$. Describe $f_*: \pi(S^1,1)\rightarrow \pi(S^1,1)$ in terms of the ...
3
votes
1answer
37 views

functoriality of $K(G,1)$ spaces in a particular situation involving complex elliptic curves

I apologize if the subject doesn't accurately describe my question. Let $F_2$ denote the free group on two generators. Suppose you have some group homomorphism $A : ...
-1
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0answers
38 views

Power set as a spectral space

Let $X$ be an infinite set. Is there way to show that $2^{X}$ is a spectral space ?. Thanks.
1
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1answer
62 views

A map $f: X\rightarrow Y$ is a homotopy equivalence if and only if $h\circ f,f\circ k$ are homotopy equivalences of $X,Y$ respectively.

Show that $f\colon X \rightarrow Y$ is a homotopy equivalence if and only if there exist maps $k,h\colon Y\rightarrow X$ such that $f\circ k$ is a homotopy equivalence of $Y$ to itself, and $h\circ f$ ...
11
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3answers
203 views

Homology Whitehead theorem for non simply connected spaces

(One version of) the Whitehead theorem states that a homology equivalence between simply connected CW complexes is a homotopy equivalence. Does the following generalisation hold true? Suppose ...
0
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1answer
38 views

A question about reduced C*-algebra of discrete group

There is a quotation below: Let $\Gamma$ be a discrete group and $\Lambda\subset \Gamma$ be a subgroup. The right cosets give a direct sum decomposition $$l^{2}(\Gamma)\cong\bigoplus l^{2}(\Lambda ...
0
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1answer
28 views

An easy (I guess) question about vector state in C*-algebra

I meet with some problems when I read a book about C*-algebra. Definition 2.5.10. Let $\phi:\Gamma \rightarrow \mathbb{C}$ be a function ($\Gamma$ is a discrete group here). We define a corresponding ...
3
votes
1answer
30 views

Appropriateness of using an interval in $\Bbb{R}$ as the parameter to a continuous path in space $X$.

The fundamental group $\pi_1(X,x)$ is usually defined using continuous paths $p : [0,1] \to X$. But... (1) Can you use other spaces besides a closed interval of $\Bbb{R}$ in the usual topo? (2) How ...
1
vote
1answer
37 views

Hatcher 1.1.12 and How to think about these problems

This problem was asked in a previous homework set: Show that every homomorphism $\pi_1(S^1)\rightarrow \pi_1(S^1)$ can be realized as the induced homomorphism $\varphi_*$ of a map $\varphi:S^1 ...
3
votes
1answer
74 views

Homotopy equivalence -> element in outer automorphism of fundamental group

Let $X$ be a path-connected topological space. for $x \in X, G = \pi_1(X,x)$ Show that a homotopy equivalence $f : X \to X$ gives a well-defi ned element $g \in Out(G)$. How might one begin on this ...
0
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1answer
63 views

History of five lemma

I am interested in the history of five lemma. Who was first to prove it and What was the purpose of proving it ? http://en.wikipedia.org/wiki/Five_lemma
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0answers
42 views

The homology group for $\mathbb{R}^n$ minus two points

This question is somehow related to the generalized Jordan curve theorem. I have already showed that if $h:S^k\to S^n$ is an embedding ($0\leq k<n$), then $\tilde{H}_i(S^n\setminus ...
1
vote
1answer
51 views

If $X \times \{0\} \cup A \times I$ is closed in $X \times I$. Then, is $A$ closed in $X$?

Let $X$ be a Hausdorff topological space, $A$ subset of $X$. Let $I$ be an interval $[0,1] \subset {\mathbb R}$. Suppose that $X \times \{0\} \cup A \times I$ is closed in $X \times I$. Then, is $A$ ...
0
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1answer
25 views

Mapping Class Group of $S^3$

I am wondering if we can compute $\pi_0(Homeo(S^3))$ (i.e. the group of hoemomorphisms of the three-sphere mod isotopy) or if anyone has a reference where I could find such information.
5
votes
1answer
113 views

What exactly is a dimension?

Maybe this is too broad a question, maybe I need to be more specific. I am just clearing my head here, feel free to ignore at your pleasure. In Linear Algebra, we learned that the dimension of a ...
1
vote
1answer
31 views

How can I compute $Tor\left(Z_{p},Z_{q}\right)$?

I am self-studying Vick's Homology theory, and now it is on the topic of free resolution. Since I am not familiar with it, I have little ideas about how to compute $$Tor\left(Z_{p},Z_{q}\right)$$ ...
0
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1answer
25 views

$[X,F] \to [X,E] \to [X,B] $ is exact sequence of pointed sets

how to show: if $F \to E \to B $ is a fibration then for any space $X$ the sequence $[X,F] \to [X,E] \to [X,B] $ is exact sequence of pointed sets. any hints, thanx.
0
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1answer
27 views

$\tilde{H}^i(\sum X) \cong \tilde{H}^{i-1}(X)$

I need help with this problem Let $\sum X =C^+X \cup_X C^-X$ be the union of two cones on $X$ with a common base. Show that $\tilde{H}^i(\sum X) \cong \tilde{H}^{i-1}(X)$. Dose this give an ...
4
votes
1answer
58 views

A characterization of wedges of 1-spheres and 2-spheres?

It is well known that if $X$ is a $1$-connected (i.e. path connected and simply connected) 2-dimensional finite simplicial complex, then $X$ is homotopy equivalent to a wedge of $2$-spheres. Consider ...
3
votes
1answer
174 views

Algebraic topology and homotopy in category theory

I've heard many times that for an algebraic topologist two spaces which are homotopy equivalent are essentially the same. But when the topological space is contractible then it is equivalent to a ...