From: foltinek@math.utexas.edu (Kevin Foltinek)
Subject: Re: Poincare duality
Date: 15 Oct 1999 14:58:14 -0500
Newsgroups: sci.math

In article <38077BAF.1763E032@physik.stud.uni-erlangen.de> Markus
Krapf <markus.krapf@physik.stud.uni-erlangen.de> writes:

> On a compact m-dimensional manifold M $H^r(M)$ denotes the r-th
> cohomology-module.  I can not see that the inner product $<.,.> :
> H^r(M)\times H^{m-r}(M)\rightarrwow \mathbb {R}$ is well defined by
> $<a,b>=\int_M a\wedge\b $

Write this a little bit more precisely: let [a] be in H^r(M), [b] be
in H^{m-r}(M); then [a] has a representative r-form a and b has a
representative (m-r)-form b, both of which are closed.

Then, \int_M (a+d\delta) \wedge b =
  \int_M a\wedge b + \int_M d\delta \wedge b
so, as you say, you need to show that \int_M d\delta \wedge b = 0.

From  here it is a familiar calculus technique: integration by parts.

Compute d(\delta \wedge b).  (Remember that b is closed.)

What famous theorem do you know that says something about
  \int_M d\phi?

Kevin.
==============================================================================

From: Boudewijn Moonen <Boudewijn.Moonen@ipb.uni-bonn.de>
Subject: Re: Poincare duality
Date: Mon, 18 Oct 1999 11:57:58 +0200
Newsgroups: sci.math
To: Markus Krapf <markus.krapf@physik.stud.uni-erlangen.de>

Markus Krapf wrote:
> 
> Kevin Foltinek wrote:
> 
> >
> > From here it is a familiar calculus technique: integration by parts.
> >
> > Compute d(\delta \wedge b).  (Remember that b is closed.)
> >
> > What famous theorem do you know that says something about
> >   \int_M d\phi?
> >
> 
> Okay, let me see:
> \int _M d\delta \wedge \beta= \int_M d(\delta \wedge \beta) =\int
> _{\partial M} \delta \wedge \beta
> Using d is a derivation , d\beta=0 and Stokes' Theorem. But what about
> the Boundary-Integral, why does this vanish only assuming \beta is
> closed?
> 
> Markus


You will have to assume that M is not only compact, but closed,
i.e. has no boundary. Then for all beta

        int_M d beta = int_{boundary M} beta = 0

since boundary M = emptyset.


Regards,


-- 
    Boudewijn Moonen
    Institut fuer Photogrammetrie der Universitaet Bonn
    Nussallee 15
    
    D-53115 Bonn
    
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    e-mail: Boudewijn.Moonen@ipb.uni-bonn.de
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