L ^p cohomology

In mathematics , cohomology is a cohomology theory for simplicial complexes or smooth manifolds . It is mainly used to study "geometry at infinity". ${\displaystyle L^{p}}$

Simplicial L ^p cohomology

Let be a finite-dimensional simplicial complex of bounded geometry (i.e. there is one such that every simplex has at most neighbors). We provide the length metric in which each simplex is isometric to the standard simplex . For let be the set of -Simplices of . Define the chains of by ${\displaystyle X}$ ${\displaystyle K>0}$ ${\displaystyle K}$ ${\displaystyle X}$ ${\displaystyle k\in \mathbb {N} }$ ${\displaystyle X_{k}}$ ${\displaystyle k}$ ${\displaystyle X}$ ${\displaystyle l_{p}}$ ${\displaystyle X}$

${\displaystyle C_{p}^{k}(X):={\Bigl \{}f\colon X_{k}\to \mathbb {R} \;{\Big |}\;\sum _{\sigma \in X_{k}}|f(\sigma )|^{p}<\infty {\Bigr \}}}$.

Together with the norm, they form a topological vector space . ${\displaystyle L^{p}}$

The Korand operator is defined by for all . Then one defines the cohomology of by ${\displaystyle \delta _{k}\colon C_{p}^{k}(X)\to C_{p}^{k+1}(X)}$ ${\displaystyle \delta f(\sigma ):=f(\partial \sigma )}$ ${\displaystyle \sigma \in X_{k+1}}$ ${\displaystyle L^{p}}$ ${\displaystyle X}$

${\displaystyle l_{p}H^{k}(X):=\ker(\delta _{k})/\operatorname {im} (\delta _{k-1})}$

and the reduced cohomology ${\displaystyle L^{p}}$

${\displaystyle l_{p}{\overline {H}}^{k}(X):=\ker(\delta _{k})/{\overline {\operatorname {im} (\delta _{k-1})}}}$.

Both are topological vector spaces with the topology induced by the norm. ${\displaystyle L^{p}}$

properties

Invariance among quasi-isometries

Let be a quasi-isometry between uniformly contractible simplicial complexes, then and are isomorphisms of topological vector spaces. (A metric space is uniformly contractible, if for every a are such that each -Ball in a -Ball contractible is.) ${\displaystyle F\colon X\to Y}$ ${\displaystyle F^{*}\colon l_{p}H^{k}(Y)\to l_{p}H^{k}(X)}$ ${\displaystyle {\overline {F}}^{*}\colon l_{p}{\overline {H}}^{k}(Y)\to l_{p}{\overline {H}}^{k}(X)}$ ${\displaystyle r>0}$ ${\displaystyle R>r}$ ${\displaystyle r}$ ${\displaystyle R}$

Geometric group effects

If a group acts geometrically on an equally contractible simplicial complex , then is ${\displaystyle \Gamma }$ ${\displaystyle X}$

${\displaystyle l_{p}H^{*}(X)=H^{*}(\Gamma ,l^{p}\Gamma )}$.

Additionally, if the center of is infinite, applies for all and . This is especially the case for infinite nilpotent groups . ${\displaystyle \Gamma }$ ${\displaystyle l_{p}H^{k}(X)=0}$ ${\displaystyle p}$ ${\displaystyle k}$

Dualities

For that is -Kohomologie dual to homology . ${\displaystyle {\tfrac {1}{p}}+{\tfrac {1}{q}}=1}$ ${\displaystyle l_{p}}$ ${\displaystyle l_{p}H^{*}(X)}$ ${\displaystyle l_{q}}$ ${\displaystyle l_{q}H_{*}(X)}$

For Riemannian manifolds of the quasi-isometric dimension to a simplicial complex of bounded geometry, one also has the Poincaré duality . ${\displaystyle n}$ ${\displaystyle l_{p}H^{k}(X)=l_{p}H_{n-k}(X)}$

Definition using differential forms

For differentiable manifolds it can be defined equivalently as the quotient space of the closed forms modulo the differentials of forms with . ${\displaystyle M}$ ${\displaystyle l_{p}H^{k}(M)}$ ${\displaystyle k}$ ${\displaystyle \alpha \in L^{p}}$ ${\displaystyle (k-1)}$ ${\displaystyle \beta \in L^{p}}$ ${\displaystyle d\beta \in L^{p}}$

Examples

Hyperbolic space

Let the -dimensional hyperbolic space . Then applies for or each and for each . ${\displaystyle X}$ ${\displaystyle n}$ ${\displaystyle p<{\tfrac {n-1}{k}}}$ ${\displaystyle p>{\tfrac {n-1}{k-1}}}$ ${\displaystyle l_{p}H^{k}(X)=l_{p}{\overline {H}}^{k}(X)=0}$ ${\displaystyle {\tfrac {n-1}{k}}<p<{\tfrac {n-1}{k-1}}}$ ${\displaystyle l_{p}H^{k}(X)=l_{p}{\overline {H}}^{k}(X)\not =0}$

Heintze groups

For Heintze groups with and applies if and only if . ${\displaystyle X=\mathbb {R} ^{n}\rtimes _{\alpha }\mathbb {R} }$ ${\displaystyle \alpha (t)=diag(e^{\lambda _{1}t},\ldots ,e^{\lambda _{n}t})}$ ${\displaystyle 0<\lambda _{1}\leq \ldots \leq \lambda _{n}}$ ${\displaystyle l_{p}H^{k}(X)=0}$ ${\displaystyle p>{\tfrac {\lambda _{1}+\ldots +\lambda _{n}}{\lambda _{n-k}+\ldots +\lambda _{n}}}}$

Manifolds of negative curvature

For a simply connected, complete Riemannian manifold the section curvature is for all . ${\displaystyle -1\leq K\leq -\delta <0}$ ${\displaystyle l_{p}H^{k}(X)=0}$ ${\displaystyle 1<p\leq 1+{\tfrac {n-k-1}{k}}{\sqrt {\delta }}}$

literature

• Michail Leonidowitsch Gromow : Asymptotic invariants of infinite groups in "Geometric Group Theory", Cambridge University Press, 1993.

Web links

• Pierre Pansu : L ^p cohomology