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

learn more… | top users | synonyms (1)

2
votes
0answers
11 views

Cross product of cohomology classes: intuition

Let $X$ and $Y$ be topological spaces and consider cohomology over a ring $R.$ Hatcher (in his standard Algebraic Topology text) defines the cross product of cohomology classes $$H^k(X) \times ...
4
votes
4answers
507 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 ...
0
votes
0answers
10 views

is simplicial approximation of $gof$ equal to the combination of simplicial approximation of $g$ with simplicial approximation of $f$?

suppose $M,L,K$ are complex and $f:|K| \rightarrow |L|$ and $g:|L| \rightarrow |M|$ are continues maps,can we consider combination of simplicial approximation of $f$ and $g$ as a simplicial ...
3
votes
1answer
61 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
vote
0answers
12 views

Explain the terms k-simplex and simplical complex geometrically?

I m new to algebraic topology .so confused with these terms pls suggest simple books
10
votes
2answers
153 views
+50

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 ...
1
vote
2answers
24 views

Group acts freely on a closed surface

My question is as follows: Let G be a finite group which acts freely as a group of homeomorphisms of a closed surface S (so the only element with fixed points is the identity) Then: Show the orbit ...
4
votes
2answers
227 views

About covering maps and sections!

If $q: E\rightarrow X$ is a covering map that has a section $(i.e. f: X\rightarrow E, q\circ f=Id_X)$ does that imply that $E$ is a 1-fold cover?
4
votes
2answers
73 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
votes
0answers
19 views

Let $p$ be a covering space and $X, Y$ be path connected. Show there exists a map $q$ such that $q\circ p=1_{X}$ iff $p$ is a homeomorphism.

Let $p\colon X\rightarrow Y$ be a covering map where $X$ and $Y$ are path connected. Show that there exists a map $q\colon Y\rightarrow X$, such that $q\circ p=1_{X}$ if and only if $p$ is a ...
0
votes
0answers
25 views

Mayer-Vietoris sequence confusion

In the Mayer-Vietoris exact sequence $$... \rightarrow H_n(A) \oplus H_n(B) \rightarrow H_n(X) \rightarrow H_{n-1}(A \cap B) \rightarrow H_{n-1}(A) \oplus H_{n-1}(B) \rightarrow...$$ I am confused ...
3
votes
0answers
43 views

does the pullback of a covering space correspond to the pullback of the corresponding representations of $\pi_1$?

Say you have a covering space $C \rightarrow X$ corresponding to some homomorphism $\pi_1(X)\rightarrow S_n$. Suppose you have an arbitrary (continuous) map $f : Y\rightarrow X$. Then we may pull back ...
0
votes
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 ...
0
votes
1answer
10 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 ...
3
votes
1answer
167 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 ...
0
votes
0answers
24 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
votes
0answers
25 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
votes
2answers
43 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 ...
1
vote
1answer
41 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 $\psi(z) = 2 \bar{z}$?
1
vote
2answers
91 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 ...
3
votes
1answer
334 views

covering map with finite fibres and preimage of a compact set

Let $f:X\to Y$ be a covering map (covering maps are surjective) , Y be compact set. And suppose that $f^{-1}(y) $ is finite for each $y\in Y$. Prove that $X$ is also compact. I think that this ...
4
votes
1answer
47 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 ...
0
votes
1answer
35 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
vote
1answer
28 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 ...
1
vote
2answers
79 views

Homology of orientable surface of genus $g$

I came across the problem of computing the homology groups of the closed orientable surface of genus $g$. Here Homology of surface of genus $g$ I found a solution via cellular homology. This seems ...
1
vote
1answer
50 views

Inequivalent Model Categories

A Model Category is, informally, a category where a "reasonable" notion of homotopy can be developed. I'm curious to know when two model categories are considered equivalent to each other. Thanks for ...
0
votes
0answers
119 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
vote
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
votes
0answers
32 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
vote
1answer
61 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 $h\circ f$ ...
3
votes
1answer
36 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
vote
2answers
66 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 ...
5
votes
2answers
78 views

Is there an analogue of the universal cover for higher homotopy groups?

The universal cover $U$ of a topological space $X$ is a simply-connected covering space of $X$. As the 'universal' moniker implies, this space is universal in the category of covering spaces of $X$ ...
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 ...
0
votes
1answer
43 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 ...
6
votes
1answer
227 views

Exercise 4-A “Characteristic Classes” by Milnor and Stasheff

Exercise 4-A of Milnor and Stasheff's book "Characteristic Classes" reads: Show that the Stiefel-Whitney classes of a Cartesian product are given by $w_k(\xi\times\eta) = \sum^k_{i=0} ...
0
votes
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 $az=\overline{z}$. ...
2
votes
0answers
58 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 ...
0
votes
0answers
15 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 ...
3
votes
0answers
45 views

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
votes
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 ...
28
votes
10answers
4k views

A really complicated calculus book

I've been studying math as a hobby, just for fun for years, and I had my goal to understand nearly every good undergraduate textbook and I think, I finally reached it. So now I need an another goal. ...
0
votes
1answer
36 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 ...
2
votes
0answers
32 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 ...
0
votes
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 ...
8
votes
1answer
118 views

Let $A$, $B$ be subsets of $S^n, n≥2$. Show that if $A$ and $B$ are closed, disjoint, and neither separates $S^n$, then…

Let $A,B$ be subset of $S^n$, $n\geq 2$. Show that if $A$ and $B$ are closed, disjoint, and neither separates $S^n$, then $A\cup B$ does not separate $S^n$. I've thought to do it by contradiction and ...
4
votes
1answer
88 views

Working out an example in Hatcher vol. $2$: Pullbacks of the Möbius Bundle

I'm working out the examples made by Hatcher to shows some pullbacks (definition here for clarity) and this ("simplified" version with $n=2$ or $n=3$) gave me an hard time: $$\times \times \times ...
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
2answers
45 views

Reference request for bounded cohomology

I want to read Gromov's IHES paper Volume and bounded cohomolgy. I have a decent background in algebraic topology at the level of Hatcher. What other background is required to understand the landmark ...
1
vote
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. $\text{Spec}_{r}(A)$