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)

0
votes
0answers
10 views

Universal covers of lattice complements.

Background: I would like to construct a continuous map (in particular, a covering map) $$ f ~\colon \mathbb{D} \longrightarrow \mathbb{C} \setminus \left( \mathbb{Z} \oplus \mathbb{Z}[i] \right) $$ ...
1
vote
0answers
13 views

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 ...
2
votes
1answer
16 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 ...
1
vote
0answers
21 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 ...
1
vote
1answer
51 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
0answers
14 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 = ...
-1
votes
0answers
20 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 ...
0
votes
1answer
44 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 ...
2
votes
2answers
33 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
34 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 ...
3
votes
1answer
35 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 ...
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
2answers
25 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
0answers
13 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
30 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 ...
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
35 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?
2
votes
1answer
23 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
19 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 ...
1
vote
0answers
13 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
24 views

Acyclic model type result [on hold]

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) := ...
0
votes
0answers
41 views

How to convert an object into a sphere?

I'm not sure if I understand it enough, but doesn't the Poincare conjecture prove any shape can be turned into a sphere? How would I go about transforming such an object? Like let's say I have a ...
5
votes
0answers
53 views

Is local homology group on a manifold a sheaf?

Let $X$ be a manifold of dimension $n$, and define $\mathcal{F}(U) = H_n(X,X-U)$. Then clearly $\mathcal{F}$ is a presheaf. I am thinking whether $\mathcal{F}$ is a sheaf. According to Lemma 3.27 in ...
-1
votes
0answers
30 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
1answer
38 views

Requirements for Mayer-Vietoris

This question might be a duplicate -- but as I don't find an entry (maybe because of the lack of a good keyword) I open this question. Besides, this questions arises when trying to prove Proposition ...
1
vote
2answers
36 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
1answer
21 views

Why is this proof that the group operation on $\pi_1(X,x_0)$ is well defined?

Let $g_1,g'_1, g_2, g'_2$ be loops on a topological space $X$ at $x_0 \in X$. Suppose that $[g_1]=[g'_1]$ and $[g_2]=[g'_2]$. Then let a map $F: [0,1]\times [0,1] \to X$ be defined as $$ F = \Big\{ ...
0
votes
0answers
12 views

An inverse of a ring automorphism $Sq$ which is a total Steenrod squaring operation and Wu's formula.

The question comes from the Problem 11-E in the book "Characteristic classes" written by Milnor. Problem 11-E) Prove the following version of Wu's formula. Let $\overline{Sq} : H^{\prod}(M) ...
1
vote
0answers
25 views

Homology CW complex

I have strong intuition that the following fact is true: If $X = \bigcup_{n\in \mathbb N} X_n$ is a CW-complex (and $X_n$ its $n$-skeleton) then $$ \tilde H_n (X) = \tilde H_n(X_{n+1}). $$ ($\tilde ...
1
vote
0answers
12 views

$S^1$ a p-local complex?

Let $p$ be a prime. Is $S^1$ a p-local CW-complex? Meaning, for any reduced homology theory $\overline{E}_*$, do we have $\overline{E}_*(S^1)=\overline{E}_*(S^1) \otimes_{\mathbb{Z}} ...
5
votes
0answers
41 views

Intuitive Aproach of Dolbeault Cohomology

I would like to understand an intuitive approach to the definitions of Dolbeault Cohomology (using $\partial$ and $\bar{\partial}$) similar to the one given here. All suggestions are welcome.
3
votes
1answer
63 views

Intuitive Approach to de Rham Cohomology

The intuition behind homology may be summarized in a sentence: to find objects without boundary which are not the boundary of an object. This has geometric meaning and explains the algebraic boundary ...
1
vote
1answer
27 views

How do I give a homeomorphism between $\mathbb R P^n$ and the space obtained by identifying antipodal points of $S^{n+1}$?

Suppose that $Y$ is the quotient space obtained by identifying the antipodal points of $S^ {n+1}$. I'm trying to give a homeomorphism between $\mathbb R P^n$ and $Y$. I think that the map ...
5
votes
1answer
79 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 ...
3
votes
1answer
52 views

Riemann-Hurwitz formula generalization in higher dimension

In "Basic algebraic geometry 2", Shafarevich finds a relation between the Euler characteristic and the genus of the curve. At page 139 he says that there's no analogue for varieties of dimension ...
2
votes
0answers
41 views

Reduced homology and colimits

I would like to prove that colimits commute with reduced homology, i.e that if $ X = \operatorname{colim}\limits_{n \in \Bbb{N}} X_n$, then $$ \displaystyle\tilde H_k(X) = ...
0
votes
0answers
13 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 ...
0
votes
1answer
21 views

a generalization of punctured cylinder

Let $S^1\times \mathbb{R}$ be the infinite cylinder. Pucture it, we have $S^1\times \mathbb{R}-*$. Then $(S^1\times \mathbb{R}-*)\simeq Skeleton^1(S^1\times \mathbb{R})\simeq S^1\vee S^1$. How ...
0
votes
1answer
43 views

Punctured $3$-Space Fundamental Group Calculation [closed]

How do I calculate the fundamental group of $\mathbb R^3\setminus \{(0,0,0)\}$, at base point $(1,0,0)$?
-1
votes
1answer
48 views

Mayer-Vietoris sequence [closed]

How do I compute the homology of the space obtained by taking three copies of $D^n$ and identifying their boundaries with each other?
0
votes
0answers
15 views

Mapping cylinder of punctured plane reflection

Suppose I define the mapping cylinder of a reflection about the $x$-axis for a punctured plane missing $(x, y)$ and $(x, -y)$. Obviously this quotient map would be homeomorphic to a one hole ...
0
votes
0answers
14 views

Explanation of CW Complexes

We recently studied about CW complexes in algebraic topology class, and I find it hard to understand how can I think of one. For example, can you please tell me how to find the CW complex of a torus? ...
0
votes
1answer
56 views

Which of these are homotopy equivalent? $S^1, \mathbb{R}, \{*\}$

Which of these spaces are homotopy equivalent: $S^1, \mathbb{R}, \{*\}$? I found a homotopy equivalence between $\mathbb{R}$ and the one point space $\{*\}$, so they are homotopy equivalent. The ...
1
vote
2answers
42 views

Proving that a continuous map is homotopic to the constant map

How can I prove that a continuous map $f : \mathbb{R}P^2 \to S^1$ is homotopic to the constant map? I know that in the projective space every point is a line but I do not get why the above has to be ...
0
votes
0answers
17 views

cohomology ring of punctured spaces [closed]

What is the cohomology ring (algebra) of $$ H^*(S^{1}\vee S^1;\mathbb{Z})? $$ $$ H^*(\vee_k S^n;\mathbb{Z})? $$ The punctured torus $$ H^*(\prod_n S^1-*;\mathbb{Z})? $$ and $$ H^*(\prod_n ...
1
vote
1answer
57 views

Quotient space of $S^n$ and the projective plane

The quotient space on $S^n \times I$ obtained from equating $(x, 0) \sim (-x, 1)$ seems like it might have the same fundamental group as the projective plane, but I'm not entirely sure how to prove ...
2
votes
2answers
62 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 ...
6
votes
1answer
124 views
+200

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 ...
1
vote
1answer
41 views

cohomology ring of a quotient space

what is the cohomology ring $$ H^*((S^3\times S^3\setminus \{e\})/(a,b)\sim (ab,b^{-1});\mathbb{Z}_2)? $$ Here the unit $e$ and the product $ab$ is of the Lie group $S^3=Sp(1)$.
1
vote
1answer
28 views

cohomology ring of some spaces [closed]

What is the cohomology ring $$ H^*(\mathbb{R}^2\times S^1\setminus (0,1);\mathbb{Z})? $$ $$ H^*(\mathbb{R}^2\times S^3\setminus (0,e);\mathbb{Z})? $$ Here $0=(0,0)\in \mathbb{R}^2$, $1=\exp(i0)\in ...