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Questions tagged [poincare-duality]

For questions involving or related to Poincaré duality.

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For a compact 4-manifold, no 2-torsion in $H_1(M;\Bbb Z)$ implies no 2-torsion in $H_n(M;\Bbb Z)$ for all $n$

Let $M$ be a topological compact connected oriented 4-manifold with nonempty boundary, and suppose that each boundary component of $M$ is a rational homology 3-sphere. Is it true that if $H_1(M;\Bbb Z)...
user302934's user avatar
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Twisted Poincaré duality reduces to usual one when manifold is oriented

Theorem 12.15 of Differential Forms in Algebraic Topology by Bott and Tu states that when $M$ is a smooth of dimension $n$ and $\mathfrak U$ is a a good cover of $M$ satisfying the condition $(*)$, ...
JerryCastilla's user avatar
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A proof of a form of the Poincare Lemma:

If $\omega$ is a closed $k-$current on the interior of $S_n:=\{x \in \Bbb{R}^n: 0 \leq x_i \leq 1, 0 \leq x_1+...+x_n \leq 1\}.$ Then $\omega = d \eta$ where $\eta$ is an extendible $(k-1)-$current or ...
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Class in $H^1(T^2)$ that is not dual to closed 1-dim submanifold of $T^2$

The definition for Poincaré dual $\omega$ of $N\subset M$ a $k$-dimensional submanifold is that given $i:N\to M$ we have $\int_N i^*\theta=\int_M \theta\wedge\omega$ for every $k$-form $\theta$. I ...
Michael Wang-Wakamatsu's user avatar
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1 answer
54 views

Why is the bilinear form on $H^d(X,\mathbb Q_{\ell})$ afforded by Poincaré duality alternating when $d = \dim(X)$ is odd?

Let $X$ be a smooth projective irreducible variety of pure dimension $d$ over an algebraically closed field of positive characteristic $p$. Let $\ell \not = p$ be a prime number. There is an ...
Suzet's user avatar
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Fix points of Galois action on etale homology

Let $k = \mathbb{F}_q$ be a finite field of characteristic $p$ and $C$ a projective, smooth curve over $k$. Denote by $\bar C$ the base change of $C$ to a separable closure $\bar k$ of $k$. Let $\ell$ ...
Erich's user avatar
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Definition of Lefschetz Number in Bott&Tu and in Gullemin&Pollack Differs by a sign?

In Bott and Tu's Differential Forms in Algebraic Topology, the Lefschetz number of a map $f:M\to M$ between an oriented compact manifold $M^m$ is defined, as in any algebraic topology text, to be $L(f)...
Tianyi Wang's user avatar
0 votes
1 answer
79 views

Why is the signature of a manifold homotopy invariant?

I'm going through some applications of Poincaré duality for de Rham cohomology in Greub's Connections, Curvature and Cohomology. I was wondering about the homotopy invariance of the signature of a ...
dwhydtea's user avatar
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Intersection of submanifolds, cup products, and Poincaré duality

Recently I have been thinking and inquiring about how "cup products are dual to intersection of submanifolds", and wanted to verify whether the following is accurate (and to find a source ...
user0134's user avatar
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An example of a non hausdorff space

Here is the question I am trying to understand its solution: Show that there exist nonorientable $1$-dimensional manifold if the Hausdorff condition is dropped from the definition of a manifold. I ...
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Let $A$ be a 2-torsion finite abelian group . Let $f: A\to (\Bbb{Q}/\Bbb{Z})^n$ be an arbitrary group homomorphism. $image(f)\le 2^n$?

Fix a natural number $n$. Let $A$ be a 2-torsion finite abelian group . Let $f: A\to (\Bbb{Q}/\Bbb{Z})^n$ be an arbitrary group homomorphism. How can I prove $image(f)\le 2^n$? My try: From structure ...
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The inverse map of the Poincare duality map

Let $M$ be an oriented compact smooth manifold of dimension $n$. Let $[M]$ be the fundamental class of $M$, that is, $[M]\in H_n(M, \mathbb Z)$. Then, the Poincare duality map is the isomorphism given ...
Flavius Aetius's user avatar
5 votes
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118 views

Proof of Poincaré duality

I am working through a proof of Poincaré duality. I don't understand the one step marked in bold. Let $R$ be a ring. Pick an $R$-orientation $(o_x; x\in\mathbb{R}^m)$ of $\mathbb{R}^m$. Pick $r\in \...
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Faces of the cap product

Let $X$ be a topological space and $A\subseteq X$ an open subspace. Let $R$ be an associative unital ring. Define the cap product $$\cap\colon S^q(X,A;R)\otimes S_{p+q}(X,A;R)\rightarrow S_p(X;R)$$ on ...
Margaret's user avatar
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If $M$ is compact connected non-orientable 3-manifold, then $H_1(M)$ is infinite.

Let $M$ be a compact connected non-orientable 3-manifold. The goal is to show that $H_1(M)$, the first integral homology group, is infinite. There is a proof of this assuming $\partial M = \emptyset$. ...
Luke's user avatar
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3 votes
2 answers
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Prove/disprove that $X$ is a manifold

Let $X$ be the 2-dimensional CW complex where 2-cells are glued as follows: The goal is to prove/disprove that $X$ is a manifold. Here's my attempt to disprove this. Suppose $X$ is a manifold. In ...
Luke's user avatar
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Correspondence between Euclidean and non-Euclidean geometries

I would like to ask correspondence between Euclidean and non-Euclidean geometries. In the science and hypothesis, Poincare says that non-Euclidean geometry can be translated into Euclidean geometry ...
Atsu's user avatar
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Composition of Gysin and restriction maps on $\ell$-adic cohomology

I follow the notations of Milne's lectures notes on etale cohomology, most specifically the section titled "The Gysin map" in chapter 24, p. 145. Let $k$ be an algebraically closed field, ...
Suzet's user avatar
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2 votes
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66 views

Intersection number of submanifolds with boundary

Let $N$ be an $n$-dimensional compact oriented manifold without boundary, and let $X$, $Y$ be compact oriented submanifolds of complementary dimensions $k$ and $n - k$, with boundary but such that $X \...
NPG's user avatar
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How Bockstein homomorphism maps under Poincarè duality?

Given a manifold $X$ and short exact sequence of abelian groups $$ 1\rightarrow A_1\overset{\iota}{\rightarrow} A_2\overset{\pi}{\rightarrow} A_3\rightarrow 1 $$ we get the Bockstein map in cohomology ...
Andrea Antinucci's user avatar
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How to construct the action on chains of the Bockstein homomorphism for homology

Given a manifold $X$ and short exact sequence of abelian groups $$ 1\rightarrow A_1\overset{\iota}{\rightarrow} A_2\overset{\pi}{\rightarrow} A_3\rightarrow 1 $$ we get the Bockstein map in cohomology ...
Andrea Antinucci's user avatar
1 vote
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122 views

Relative Intersection Pairing (Poincaré-Lefschetz duality)

Let $M$ be a $2$-dimensional orientable compact manifold. (For example, a Riemann surface.) We have an intersection pairing $$H_1(X,\def\Z{\mathbb{Z}}\Z) \times H_1(X,\Z)\to \Z$$ which is unimodular, ...
Johann Birnick's user avatar
3 votes
1 answer
179 views

On Poincaré Inequalities

I am trying to prove the following: Lemma. If a metric measure space $(X,d,\mu)$ supports a $p$-Poincaré inequality for some $p \in [1;\infty)$, then it supports a $q$-Poincaré inequality for all $q \...
Philippe Knecht's user avatar
3 votes
1 answer
189 views

Poincaré duality and coefficients in the circle group

In Hatcher, Poincaré duality is stated for coefficients in a ring rather than a general abelian group. I am wondering whether it also holds when taking coefficients in the circle group, which is not a ...
mathieu's user avatar
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Poincare Inequality for Mean-Zero Functions

I'm studying Sobolev spaces these days and came across a property of Poincare Inequality for mean-zero functions that I don't understand; $u(x,y)=-u(x,-y) \implies \int_\Omega u=0$ and that Poincare ...
mathlearner's user avatar
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Trouble with a classical fact about homology of manifold minus disk.

Let $M$ be a compact connected (smooth) $n$ dimensional manifold, and let $D$ be an $n$-dimensional disk embedded in $M$. Let $W:=\overline{M\setminus D}$, then $H_{n}(W;Z/2)=0$. I see the following ...
Espace' etale's user avatar
2 votes
1 answer
136 views

Geometric interpretation of intersection product

Let $E,E^*$ be a pair of dual $n$-dimensional real vector spaces and let $e,e^*$ be a pair of dual basis vectors of $\bigwedge^n E$ and $\bigwedge^n E^*$, respectively. For $u,v\in\bigwedge E$, we can ...
blargoner's user avatar
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2 votes
1 answer
221 views

Poincare Duality: $H^k(M) = H_{n-k}(M)^*$ or $H^k(M) = H^{n-k}(M)^*$?

There are $2$ versions of Poincare duality, in topology or in differential geometry. Theorem(Topological Version) Let $M$ be a closed oriented $n$-manifold with fundamental class $[M]\in H_1(X)$. ...
Hydrogen's user avatar
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How to prove two partial different equations are dual

I have $m(t, x)$ and $n(t, x)$, where $t \geq 0$ is the time variable and $x \in \mathbb{R}$ is the space variable. I also have a speed $x \mapsto v(x)$. We consider the following models: $$ \frac{\...
Sisislas's user avatar
1 vote
1 answer
187 views

Poincare duality for algebraic de Rham cohomology with integrable connection coefficients

I am reading "The Gauss-Manin Connection and Tannaka Duality" (here is the link to the paper). I am specifically interested in the proof of Proposition 2.2. In this proof, the authors use ...
Khainq's user avatar
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2 votes
0 answers
137 views

Bockstein sends fundamental class to Poincare dual of $w_1$

Suppose $X$ is an $n$-dimensional manifold, and consider the Bockstein homomorphism $\beta \colon H_k(X; \mathbb{Z}_2) \to H_{k-1}(X; \mathbb{Z})$ induced by the sequence $$0 \to \mathbb{Z} \...
YoungMath's user avatar
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2 votes
0 answers
97 views

Version of Thom isomorphism theorem for fibers with Poincaré duality

Let $F$ be a path-connected Poincaré duality space (using $\mathbb{Z}$ coefficients for singular co/homology) of dimension $n$. In particular, we have $H^n(F;\mathbb{Z}) \approx \mathbb{Z} \approx H_n(...
Indraneel Tambe 2's user avatar
4 votes
0 answers
70 views

Mayer-Vietoris Sequence and Poincare dual

Given a $4$-dimensional simply connected manifold $M$ and open sets $U,V\subseteq M$ such that $U\cup V=M$ we can compute the deRham cohomology in terms of the Mayer-Vietoris sequence: \begin{align*} ...
Bigolini's user avatar
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1 answer
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Euler characteristic of odd dimensional manifold - Hatcher

I ran into some trouble while reading through Hatcher's proof of the following: Corollary 3.37. A closed manifold of odd dimension has Euler characteristic zero. There is only one part of the proof ...
jasnee's user avatar
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2 votes
0 answers
119 views

algebraic intersection number and Poincare duality

I am trying to understand the algebraic intersection number in terms of Poincare dual and the cup product. This is: Let $M$ be a compact oriented $m$-dimensional smooth manifold together with a ...
WhenYouHaveNoClue's user avatar
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0 answers
57 views

Poincaré-Bendixson theorem in $\mathbb{R}^{1}$

The theorem of Poincaré-Bendixson states that: For an open set $U \subset \mathbb{R}^{2}$ and a continuously differentiable vector field $F:U \rightarrow \mathbb{R}^2$ and a compact set $K \subset U$, ...
illuminatitruthseeker's user avatar
2 votes
0 answers
162 views

Sources on twisted/local (co)homology

Any suggestions for sources to learn about (co)homology with twisted/local coefficients? I've read Hatcher's section 3.H and have previously been directed to Sakasai's paper A Survey of Magnus ...
Ethan Dlugie's user avatar
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7 votes
2 answers
1k views

Analogous of Poincaré Duality for relative homology and relative cohomology

I am studying Morse Theory on finite dimensional and compact manifolds using homology groups and relative homology groups on $\mathbb{Z}$. I want to show that this theory could be developed using De ...
Dylan 's user avatar
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4 votes
1 answer
111 views

Definition of Poincaré homomorphism

In the following question $H^i_c(M;G)$ denotes the compact support cohomology, $M$ an $n-$oriented manifold and $G$ an abelian group. The definition I have of Poincarè homomorphism $P_M : H^i_c(M;G) \...
jacopoburelli's user avatar
3 votes
0 answers
259 views

Poincaré Duality Theorem (Massey proof)

I'm trying to understand the Case $1$ of Poincaré Duality Theorem (Massey p.$361$). Theorem: Let $M$ be an $n-$oriented manifold and $G$ an arbitrary abelian group. Then the homomorphism $$P : H_c^q(...
jacopoburelli's user avatar
2 votes
0 answers
148 views

A duality generalizing Lefschetz duality of a non-compact manifold

I am trying to solve out Exercise 3.3.35 of Hatcher's Book on Algebraic Topology: I have two basic questions: How to define the long exact sequence in the first row? For example, can we first find a ...
Smart Yao's user avatar
  • 574
3 votes
1 answer
298 views

Is compactly supported de Rham cohomology always finite dimensional?

I know that for a compact smooth manifold $M$, $\dim H^p_{dR}(M) < \infty$. I am trying to prove Poincaré duality, and in one of my steps so far, the compactly supported cohomology groups being ...
Daniel Waters's user avatar
1 vote
0 answers
123 views

Induced map on cohomology is preimage in geometry

My question is about p. 69 of Bott Tu. I don't understand how the commutative diagram implies that if $\omega$ is the cohomology class on $M$ representing $S$, then $f^*\omega$ represents $f^{-1}(S)$. ...
boink's user avatar
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8 votes
0 answers
347 views

Help with proof of noncompact Poincaré Duality by Hatcher.

I am trying to follow Hatcher's proof of Poincaré Duality on p. 248. Suppose $M$ is an $R$-oriented manifold. We have first defined $H^k_c(M;R)$ to be the direct limit of groups $H^k(M,M\setminus K;R)$...
Jarne Renders's user avatar
2 votes
1 answer
232 views

If $X$ is an orientable connected non compact $n$- manifold then $H_n(X) = 0$

To prove the assert I led back to prove that $H_c^0(X) = 0$ thanks to Poincarè-duality, using the fact that $H_c^0(X) = \text{lim}H^0(X,X\setminus K,\mathbb{Z}) = \text{lim} 0 = 0$, since here we ...
jacopoburelli's user avatar
7 votes
1 answer
336 views

Poincaré's take on Poincaré duality before the advent of cohomology?

Poincaré duality was first stated, without proof, by Henri Poincaré in 1893. It was stated in terms of Betti numbers: The $k$th and $n-k$th Betti numbers, $b_k$ and $b_{n-k}$ of a closed orientable n-...
wonderich's user avatar
  • 5,969
0 votes
1 answer
253 views

Different approach of Poincaré Duality

I am studying Algebraic Topology on my own. First I would like to tell something about me. Mainly I am focused about Hyperbolic Geometry, Geometry and Topology of 3-manifolds, knot theory, etc. I am ...
user avatar
1 vote
0 answers
123 views

Lefschetz duality pairing on images of projections from long exact sequence

I am redoing this question as the first rendition has been a bit of a mess. Throughout this question, we want to only consider homology with coefficients in $\mathbb{Q}$, so we can ignore issues that ...
Finn Klein's user avatar
5 votes
1 answer
1k views

What is the essence of ``the naturality of the cap product''?

Associated to a continuous map $f : X → Y$, there are natural pushforward and pullback maps on homology and cohomology, respectively, denoted $f_∗ : H_∗(X) → H_∗(Y)$ and $f^*: H^* (Y) → H^* (X)$. ...
wonderich's user avatar
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3 votes
1 answer
743 views

the difference between chains and cochains

I have heard the following definitions, for concreteness I will refer a simplicial complex $K$ and a ring $R$, though the definitions can be extended: A chain is a formal linear combination of ...
Kai's user avatar
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