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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9 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
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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 ...
3
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1answer
183 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
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
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
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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 ...
2
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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
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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
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1answer
492 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
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0answers
601 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 ...
6
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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 ...
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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
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 ...
1
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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
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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 ...
3
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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 ...
4
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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 ...
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
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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
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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
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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 ...
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
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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?
1
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1answer
66 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
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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 ...
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 ...
1
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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
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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 ...
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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) := ...
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 ...
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 ...
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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
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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 ...
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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 ...
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0answers
33 views
+50

Convention of a continued fraction presentation of a lens space

I want to clarify the following two conventions on a surgery description of a lens space. Let $p$ and $q$ are relatively prime integers. Express $$ ...
1
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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\{ ...
9
votes
6answers
2k views

What is the difference between homotopy and homeomorphism?

What is the difference between homotopy and homeomorphism? Let X and Y be two spaces, Supposed X and Y are homotopy equivalent and have the same dimension, can it be proved that they are homeomorphic? ...
11
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5answers
1k views

Wedge sum of circles and Hawaiian earring

The (countably infinite) wedge sum of circles is quotient of disjoint countable union of circles $\amalg S_i$, with points $x_i\in S_i$ identified to a single point, while the Hawaiian earring H is ...
0
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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) ...
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.
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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 ...
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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}} ...
1
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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 ...
3
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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 ...
3
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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 ...
0
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1answer
28 views

Homotopy Invariance: Cone Construction and Prisms Operators

I'm looking at different approaches to proving the homotopy invariance of homology. Rotman and Dieck both mention "the cone construction", but hatcher only introduces the prism operators and does not ...
2
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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
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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 ...
1
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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
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 ...