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

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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 ...
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 $X$?
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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?
2
votes
1answer
26 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
24 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
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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 ...
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0answers
27 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) := ...
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0answers
42 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
55 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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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
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1answer
41 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
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2answers
45 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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1answer
23 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
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0answers
18 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
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0answers
27 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
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0answers
13 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
49 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
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1answer
74 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
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1answer
30 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
116 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
64 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
45 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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0answers
18 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
23 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 ...
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votes
1answer
54 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
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0answers
17 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
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0answers
15 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
59 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 ...
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2answers
44 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 ...
1
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1answer
63 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
68 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 ...
8
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2answers
203 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 ...
1
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1answer
43 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
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1answer
29 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 ...
1
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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 ...
1
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1answer
53 views

Mapping torus with homotopic homeomorphisms

Suppose I define the mapping torus $M_f$ in the usual way by identifying $(x, 0)$ and $(f(x), 1)$. If I have a homeomorphism $f: X \rightarrow X$ and another homeomorphism $f': X \rightarrow X$ that ...
5
votes
2answers
71 views

Why are contact structures studied from a cohomological, rather than homological, perspective?

As far as I know, contact structures are studied from a "cohomology/dual" perspective, meaning mostly from the perspective of contact forms and their respective kernels, instead of from a ...
0
votes
1answer
14 views

If the reduced homology of K is nonzero, k is evasive.

Where can I find a proof for this theorem of Kahn, Saks and Sturtevant? K is a simplicial complex. Theorem $\textbf{10.1}.$ If $\tilde H_*(K)\neq 0$, where $\tilde H_*(K)$ denotes the reduced ...
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0answers
19 views

CW approximation

I was reading several proofs of the CW approximation theorem. If $X$ is a space then the idea is to make $n$-equivalences $f_n:K_n \to X$ where $K_n$ is a $n$-dimensional CW-complex. This goes by ...
2
votes
1answer
39 views

Direct Sum of Homology Groups and Connected Sums - Something's gone wrong.

I know that for any surfaces, $K, L$ forming $K$#$L$ (where # denotes the connected sum) is done by deleting a disk from each and gluing together the boundary. I also know a few facts about the ...
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0answers
45 views

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 $$ ...
2
votes
0answers
33 views

Definition of CW complex

In Hatcher, a CW complex is defined by inductively attaching cells, where we begin with $X^0$, a discrete space and then attach $1$-cells etc. We then get spaces $X^0,X^1,\cdots$ where ...
1
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1answer
22 views

Lifting properties of Serre fibrations

Suppose that $p:X\rightarrow B$ is a Serre fibration. I want to prove that $p$ has the right lifting property with respect to all maps of the form: $$S^{n-1}\times ...
1
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0answers
42 views

De Rham Cohomology of the complement of an ellipse

I'll call $A$ the complement of an ellipse in $\mathbb{R}^2$. I know that $H^0(A)=\mathbb{R}^2$ and $H^k(A)=0$ if $k \ge 3$. What about $k=1,2$? I know that $A$ has two connected components, so ...
1
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1answer
37 views

Degree of an antipodal map

Let $f:S^n\to S^n$ a continuous map, $n>0$; we consider the induced homomorphism $f_* : H_n(S^n)\to H_n(S^n)$, and, recalling $H_n(S^n)\simeq\mathbb Z$, define $deg(f)\doteq f_*(1)$. I'm asked to ...
1
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0answers
28 views

cohomology of labelled configuration space & relation with braid space [closed]

Let: $M$ be a manifold (if we want, we can let $M=S^2$ , $S^1\times \mathbb{R}$, etc.); $(X,*)$ be a pointed topological space. $F(M,k)$ be the ordered configuration space of $k$-tuples on $M$; ...
3
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1answer
41 views

Prove that there is no entire function such that for every $w$ in the complex plane $f^{-1}(w)$ consists of exactly 2 points.

So I came across the following problem. Prove that there is no entire function such that for every $w$ in the complex plane $f^{-1}(w)$ consists of exactly 2 points. So here is what I was thinking. ...
1
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1answer
49 views

Adjoining an $(n+1)$-cell is an $n$-equivalence

Suppose $X$ is a topological space and $x_0 \in X$. Let $$ X' = X \cup e^{n+1} $$ be obtained from adding a $(n+1)$-cell (so $X'$ is the pushout of the map $\partial e^{n+1} \to e^{n+1}$ and the ...
0
votes
1answer
106 views

Fundamental group of complement of circle with finite number of lines through origin

In $\mathbb{R}^3$, consider the space missing a circle and a finite number of distinct lines passing through the center of the circle. In the case of one line, one can show that it deform retracts to ...
3
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0answers
58 views

Calculating the first homology group

Suppose all vertices on a polygon are identified and the polygon is $abcb^{-1}a^{-1}c$. Is it enough to simply switch to additive notation, get $2c$ and realize that $H_1(X) = \mathbb{Z}_2 * ...