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
1answer
14 views

If the fibers of a quotient map are all discrete, is this map a covering map?

If $p:\tilde{X}\rightarrow X$ is a covering projection then I know that for every point $x \in X$ the fibre above $x$, i.e $p^{-1}(x)$, has the discrete topology. Here $p$ being a covering map means ...
1
vote
0answers
6 views

co H-space on CW-complex

Does someone have any idea how to prove these? If X is (n-1)-connected CW-complex of dimension <2n then there exists homotopy co-multiplication on X. If X is (n-1)-connected CW-complex and ...
0
votes
1answer
31 views

Is the stable homotopy group of sphere a commutative ring? If not, are there easy examples?

Is the stable homotopy group of spheres a commutative ring? If not, are there easy examples? In the Adams spectral sequence converging to the stable homotopy group of spheres, it seems that any page ...
2
votes
2answers
16 views

Homology groups of orientable surfaces.

I am trying to show that the second (simplicial) homology group or an orientable surface is ismormophic to $\mathbb Z$. I can show that this group is non-trivial by triangulating the surface, and ...
2
votes
1answer
15 views

Homotopically equivalent to Čech nerve?

I see a theorem without proof on Gelfand & Manin: Suppose $\mathfrak U=\{U_\alpha\}_\alpha$ is a locally finite open covering of the topological space $X$ such that each finite intersection ...
6
votes
2answers
45 views

Counter example to Mostow's rigidity theorem for 2-manifolds.

I am trying to understand a counter-example to Mostow's rigidity theorem. Here is the counter example I want to understand. Take two non-isometric octagons with the sum of interior angles equal to ...
8
votes
2answers
197 views

What's the difference between cohomology theories of varieties and topological spaces

There is defined several cohomology theories for algebraic varieties, but in the situation is very different for topological spaces (up to homotopy) for which there is only one cohomology theory for ...
0
votes
1answer
26 views

about a sequence of isometries' convergency.

Let $M$ be a compact metric space, let $(i_n)$ be a sequence of isometries: $M \rightarrow M$. I've already showed that there exists a subsequence $(i_{n_k})$ that converges to $i$ which is also a ...
6
votes
1answer
31 views

Books or texts on singularity theory

So a friend is doing his PhD in maths (algebraic topology) and his advisor wants him to publish something on singularities (of which, as fas as I understand, he knows next to nothing). I want to give ...
2
votes
2answers
89 views

Computing $\pi_4(S^3)$ using Serre spectral sequence

I'm following Davis & Kirk's computation that $\pi_4(S^3)=\mathbb Z/2$ using the Serre spectral sequence but I'm having problems at the very end. We consider a homotopy fibration $X\to S^3 \to ...
1
vote
1answer
35 views

Fundamental group of two circles joined by an arc

What is the fundamental group of two circles joined by an arc? In other words, let $S_1$ and $S_2$ be two standard circles. Let $p_1$ and $p_2$ be two points in $S_1$ and $S_2$ respectively. Join ...
2
votes
0answers
44 views
+50

Quotient Groups and Covering Spaces in Painting Hanging

Consider the $1$-out-of-$n$ painting hanging problem: Given $n$ nails in a wall, how can we hang a painting such that upon removal of any nail, it falls. This has a nice interpretation as a problem in ...
20
votes
10answers
881 views

Surprising applications of topology [on hold]

Today in class we got to see how to use the Brouwer Fixed Point theorem for $D^2$ to prove that a $3 \times 3$ matrix $M$ with positive real entries has an eigenvector with a positive eigenvalue. The ...
4
votes
1answer
59 views

A seemingly wrong definition of convergence of spectral sequences in Bott & Tu?

After introducing exact couples, Bott & Tu defines spectral sequences as follows: A sequence of differential groups $\{E_r,d_r\}$ in which each $E_r$ is the homology of its predecessor ...
2
votes
1answer
17 views

Boundary maps of the projective plane as a $\Delta$-complex (homology)

Hi, very simple question here. In Hatcher's 'Algebraic Topology' the diagram above is used to describe the projective plane as a $\Delta$-complex(see p.102). Later the 2-boundary maps are given by ...
4
votes
1answer
39 views

Poincare duality isomorphism problem in the book “characteristic classes”

This is the problem from the book, "characteristic classes" written by J.W. Milnor. [Problem 11-C] Let $M = M^n$ and $A = A^p$ be compact oriented manifolds with smooth embedding $i : M \rightarrow ...
5
votes
1answer
132 views

Why is SO(3) not $S^1 \times S^2$? (Where is the mistake?)

I was trying to calculate the fundamental group of SO(3). In order to represent the group I reasoned the following way: In order to build the 3X3 orthogonal matrix I need an orthonormal positive ...
-2
votes
0answers
40 views

Simplicial homology [on hold]

Let $\Delta$ be the simplicial complex on vertex set [5] whose Stanley-Reisner ideal is $I_{\Delta}=(x_{1}x_{4},x_{1}x_{5},x_{2}x_{5},x_{1}x_{2}x_{3},x_{3}x_{4}x_{5})$. Write the augmented oriented ...
3
votes
1answer
47 views

the top chern class of the holomorphic tangent bundle is the euler class

Is the following true? Let X be a complex manifold of complex dimension d and let V denote its holomorphic tangent bundle (ie it's $T^{1,0} \subset T \otimes_R C$, where T is the tangent bundle of ...
18
votes
4answers
1k views

Motivating Cohomology

Question: Are there intuitive ways to introduce cohomology? Pretend you're talking to a high school student; how could we use pictures and easy (even trivial!) examples to illustrate cohomology? Why ...
2
votes
1answer
25 views

The Oriented Universal Bundle in Characteristic classes by J.W. Milnor.

I have a problem to understand the section "The Oriented Universal Bundle" in the page 145 of "Characteristic classes" written by J.W. Milnor. The content in that page is like below, ...
2
votes
3answers
47 views

Universal Cover of $\mathbb{R}P^{2}$ minus a point

I've already calculated that the fundamental group of $\mathbb{R}P^{2}$ minus a point as $\mathbb{Z}$ since we can think of real projected space as an oriented unit square, and puncturing it we can ...
2
votes
1answer
581 views

Group action and covering spaces

Let $X$ be a path-connected and locally path-connected topological space. The action of a topolgical group $G$ on $X$ is a covering space action. For any subgroup $H < G$, we have a composition ...
-2
votes
1answer
42 views

what does Homotopy Tell?

What is the Homotopy geometrically?And what is path-homotopy? If two real valued functions are homotopic to each other, then what can we say about the geometry behind this? It need not be equal ...
0
votes
0answers
23 views

Proof of the Borsuk antipodal theorem

In the proof of the Borsuk antipodal theorem about antipodal functions on the $n$-sphere one usually changes from the $\ell_2^n$-sphere to the $\ell_1^n$-sphere. Fair enough, there are homeomorphic by ...
0
votes
1answer
27 views

Universal covers of lattice complements.

Background: I would like to construct a continuous map (in particular, a covering map) $$ f ~\colon \mathbb{D} \longrightarrow \mathbb{C} \setminus \left( \mathbb{Z} \oplus \mathbb{Z}[i] \right) $$ ...
3
votes
1answer
187 views

What is $\mathbb{P}^{\infty}$?

Can we look at a complex projective space $\mathbb{P}^{\infty}$? I am curious to know what would it be. What is the right intuition to think about it? I know $\mathbb{P}^{n}$ is a space of ...
1
vote
1answer
65 views

fundamental group of the complement of a circle

This should be a quick question. I am reading Hatcher's Algebraic Topology book. At page 46, in the example of computing the fundamental group of the complement in $\mathbb{R^3}$ of a single circle, ...
2
votes
1answer
21 views

Seperating points in the complex plane

Given a finite set of points say $p_1,p_2, \ldots, p_n$ in the complex plane, how do I find another point $q$ such that ray $R_i$ joining $q$ to $p_i$ are all distinct. I would be happy with any kind ...
2
votes
0answers
17 views

Who first computed the Euler characteristic of a generalized flag manifold?

Let $G$ be a compact Lie group and $H$ a closed subgroup containing a maximal torus $T$ of $G$. Then the Euler characteristic of $G/H$ satisfies $$\chi(G/H) = \frac{|W_G|}{|W_H|},$$ where $W_G = ...
0
votes
1answer
494 views

Cellular homology of projective space $\mathbb{R}P^n$

For the projective space the cell decomposition is $e_0 \cup \dots \cup e_n$ and the attaching map is $a_1 a_1 \dots a_k a_k$ for the $k$-th cell. So for $k \leq n$ I thought that this means that the ...
8
votes
0answers
603 views

Cohomology ring of Grassmannians

I'm reading a paper called An Additive Basis for the Cohomology of Real Grassmannians, which begins by making the following claim (paraphrasing): Let $w=1+w_1+ \ldots + w_m$ be the total ...
1
vote
2answers
32 views

Induced subgroup of $\pi_1(S^1)$ by $p_n$

Consider the following covering map $p_n: S^1 \to S^1, z \mapsto z^n$. Why is the subgroup of $\pi_1(S^1)$ induced by $p_n$ isomorphic to $n\mathbb{Z}$? I know that $\pi_1(S^1) \cong \mathbb{Z}$ but ...
0
votes
1answer
52 views

Homotopy equivalence?

Can someone explaine what this means mathematicaly : "Let us denote by $h: X\rightarrow Y$ a homotopic equivalence map for which $h|_{Y}$ is the identity " Remark: $Y$ is include in $X$ Please ...
1
vote
1answer
33 views

cohomology of permutation group with mod 2 coefficient

Let $S_n$ be the permutation group of order $n$. Let $\mathbb{Z}_2=\mathbb{Z}/2\mathbb{Z}$. What is the cohomology algebra $$H^*(S_n;\mathbb{Z}_2)?$$ For $n=2$, $BS_2=\mathbb{R}P^\infty$ hence I ...
3
votes
2answers
41 views

Fundamental group of $\mathbb{R}^n\backslash \{0\}$

I am wondering about what the fundamental group of $\mathbb{R}^n \backslash \{0\}$ or more generally $\mathbb{R}^n \backslash U$ where $U$ is a subset of $\mathbb{R}^n$ for $n>1$. For $n=1$ I ...
4
votes
0answers
36 views

How to show that a leaf is topologically a cone.

I am trying to understand the topological behaviour of foliations around irreducible singularities, specially in the case of singularities in the Poincaré domain. I am using the third chapter of this ...
2
votes
2answers
38 views

Simplicial homology of a wedge product

If $X$ and $Y$ are triangulated topological spaces, how do I prove that $$H_n(X \vee Y) \cong H_n(X) \oplus H_n(Y),$$ where $H_n(X)$ is the $n$th reduced simplicial homology group of $X$?
1
vote
1answer
32 views

cohomology of semi-direct product of groups

Let $G, H$ be groups. Let $G\rtimes _\phi H$ be a semidirect product. The product is twisted. Let $BG$, $BH$, and $B(G\rtimes_\phi H)$ be the classifying spaces of $G$, $H$, and $G\rtimes _\phi H$. ...
0
votes
0answers
19 views

Ambient isotopy of based surface knots

Let $S$ be a smooth closed surface of genus $\ell$. Let $p$ be a point of $S$ and $a_i$, $b_i$ with $i=1,\ldots,\ell$ be $2\ell$ curves embedded in $S$ based at $p$ smooth everywhere except perhaps ...
1
vote
1answer
21 views

augmented chain complex

From Hatcher's Algebraic Topology, I know that a continuous map induces a morphism of chain complexes $f :C(X) → C(Y)$ by invariance of homotopy, but how would I show that $f$ also induces a ...
2
votes
1answer
25 views

Number of connected components of a real variety

Let $f_1,\ldots,f_k\in\mathbb{R}[X_1,\ldots,X_n]$ with $d_i:=\deg f_i$ and suppose that $V:=\{x\in\mathbb{R}^n\,:\, f_1(x)=f_2(x)=\ldots=f_k(x)=0\}$ is of dimension $n-k$. I would like to bound the ...
1
vote
1answer
36 views

How does one see connectedness of a covering space?

Something can be proven about the loops (or their possible lifts?) in the base space which will ensure connectivity of the cover?
1
vote
1answer
69 views

Reference request: Zero set of global section

Things are in the complex algebraic setting. Assume that a vector bundle $V$ of rank $n$ over a $\mathbb{P}^n$ has a global section $\sigma$. Is it true that the zero set of $\sigma$ is a ...
0
votes
0answers
14 views

Does the singular cohomology theory agree with Alexander-Spanier's for compact metric spaces?

From http://en.wikipedia.org/wiki/Alexander%E2%80%93Spanier_cohomology, we know that the Alexander–Spanier cohomology groups coincide with Cech's for compact metric spaces, and coincide with singular ...
2
votes
2answers
67 views

The singular homology and cohomology of manifolds vanishes in high dimensions

Let $M$ be an $n$-manifold. It seems that there are two results that the $p$-th singular homology and cohomology of $M$ are zero if $p>n$. But I can not find them in my books of algebraic ...
1
vote
2answers
43 views

The top singular cohomology group with coefficient $\mathbb{Z}_2$ for non-compact, non-orientable manifolds without boundary

I want to know the top singular cohomology group with coefficient $\mathbb{Z}_2$ for non-compact, non-orientable manifolds without boundary. I think Poincare and Lefschetz duality may help. However, ...
1
vote
0answers
15 views

use homology groups to obtain minimal cell structure

On Hatcher's book Algebraic Topology, Section 4.C Prop. 4C.1, for a simply-connected CW-complex $X$, if $H_*(X;\mathbb{Z})$ is known as a graded module over $\mathbb{Z}$, then the minimal cell ...
0
votes
0answers
26 views

Acyclic model type result [closed]

If $\sigma: \Delta_n \to X$, define $\overline{\sigma}: \Delta_n \to X$ by$$\overline{\sigma}(t_0, \dots, t_n) := \sigma(t_n, \dots, t_0).$$Define a map $T: C_n(X) \to C_n(X)$ by $T(\sigma) := ...
5
votes
1answer
110 views

Why does Seifert-Van Kampen not hold with $n$-th homotopy groups?

My question concerns the Seifert-Van Kampen theorem, in the following form. Let $X$ be an arch-wise connected topological space, consider a poin $x_{0}\in X$, and let $\{U_{i}\}_{i\in I}$ be an open ...