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The de Rham isomorphism and the $L_p$-cohomology of non-compact Riemannian manifolds

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Speaker: 
Nikolai Nowaczyk
Affiliation: 
U Bonn
Date: 
Mon, 2011-03-28 15:00 - 16:00
Location: 
MPIM Lecture Hall
Parent event: 
Topics in Topology

The classical de Rham isomorphism between the singular cohomology
of a smooth manifold $M$ and its de Rham cohomology is a well-known theorem.
In this talk we will consider Riemannian manifolds that admit a certain
triangulation $h: K \to M$, where $K$ is a simplicial complex in $\mathbb{R}^n$.
We will establish the notion of $L_p$-norms for differential forms on $M$
as well as for simplicial cochains on $K$. These define two cochain complexes
$L_p(K)$ and $L_p(M)$. Its cohomologies $H_p(K)$ and $H_p(M)$ are
the $L_p$-cohomology of $M$. We will discuss a variant $H_p(K) \to H_p(M)$
of the de Rham isomorphism that holds in this particular setting. We will
also sketch some other aspects of $L_p$-spaces. A full script is available
at http://math.nikno.de/

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