Scrolling cohomology

In mathematics , scrolling cohomology is a cohomology theory for describing scrolling .

It is a modification of the De Rham cohomology , in which the differential forms and differentials are only viewed along leaves. Compared to the De Rham cohomology, it has a much more complex structure, for example the cohomology groups are often infinitely dimensional even with compact manifolds.

definition

Be a smooth -manifold and a -Blätterung of codimension with and . The sub- bundle of the tangential bundle that is tangential to the leaves of is denoted by and the dual bundle is denoted by . ${\displaystyle M}$ ${\displaystyle n}$ ${\displaystyle {\mathcal {F}}}$ ${\displaystyle C^{r}}$ ${\displaystyle q=n-p}$ ${\displaystyle 1\leq r\leq \infty }$ ${\displaystyle 0\leq p\leq n}$ ${\displaystyle T{\mathcal {F}}}$ ${\displaystyle {\mathcal {F}}}$ ${\displaystyle TM}$ ${\displaystyle T^{*}{\mathcal {F}}}$

The space of foliation differential shapes (i.e. the differential shapes defined along leaves) is

${\displaystyle \Omega ^{*}(M,{\mathcal {F}}):=\Gamma ^{r}(M,\Lambda ^{*}(T^{*}{\mathcal {F}}))}$,

thus the space of - cuts in the outer algebra of . Is equivalent ${\displaystyle C^{r}}$ ${\displaystyle T^{*}{\mathcal {F}}}$

${\displaystyle \Omega ^{*}(M,{\mathcal {F}})=\Omega ^{*}(M)/I^{*}({\mathcal {F}})}$

With

${\displaystyle I^{*}({\mathcal {F}}):=\left\{\omega \in \Omega ^{k}(M)\vert \omega (v)=0\ \forall \ v\in \Gamma ^{r}(M,\Lambda ^{k}(T{\mathcal {F}}))\right\}}$.

According to Frobenius' theorem , the outer derivative maps onto itself and thus defines a well-defined differential ${\displaystyle I^{*}({\mathcal {F}})}$

${\displaystyle d^{*}\colon \Omega ^{*}(M,{\mathcal {F}})\to \Omega ^{*+1}(M,{\mathcal {F}})}$.

A foliation differential form can be used locally in a foliation map as

${\displaystyle \omega =\sum _{i_{1},\ldots ,i_{k}}f(x_{1},\ldots ,x_{p},y_{1},\ldots ,y_{q})dx_{i_{1}}\vee \ldots \vee dx_{i_{k}}}$

describe, where the local coordinates are in the direction of the leaves and the coordinates are in the transverse direction. In such coordinates the differential is described by ${\displaystyle x_{1},\ldots ,x_{p}}$ ${\displaystyle y_{1},\ldots ,y_{q}}$

${\displaystyle d\omega =\sum _{i_{1},\ldots ,i_{k}}\sum _{j}{\frac {\partial f}{\partial x_{j}}}(x_{1},\ldots ,x_{p},y_{1},\ldots ,y_{q})dx_{i_{1}}\vee \ldots \vee dx_{i_{k}}}$.

The scrolling cohomology is then defined as

${\displaystyle H^{*}(M,{\mathcal {F}}):=\operatorname {Kern} (d^{*})/\operatorname {Bild} (d^{*+1})}$.

The cohomology groups are Frechet spaces , which in general do not have to be Hausdorff-like . One therefore also considers the reduced scrolling cohomology

${\displaystyle {\overline {H}}^{*}(M,{\mathcal {F}}):=\operatorname {Kern} (d^{*})/{\overline {\operatorname {Bild} (d^{*+1})}}}$.

Examples

• For the scrolling of the by points is and for . ${\displaystyle \mathbb {R} ^{1}}$ ${\displaystyle H^{0}(\mathbb {R} ,{\mathcal {F}})=C^{\infty }(\mathbb {R} )}$ ${\displaystyle H^{k}(\mathbb {R} ,{\mathcal {F}})=0}$ ${\displaystyle k>0}$
• For from a subset induced foliation of a locally homogeneous space is where the Lie algebra of and iher Lie algebra cohomology with coefficients in is. ${\displaystyle H\subset G}$ ${\displaystyle M=\Gamma \backslash G}$ ${\displaystyle H^{*}(M,{\mathcal {F}})=H^{*}({\mathfrak {h}},C^{\infty }(M))}$ ${\displaystyle {\mathfrak {h}}}$ ${\displaystyle H}$ ${\displaystyle H^{*}({\mathfrak {h}},C^{\infty }(M))}$ ${\displaystyle C^{\infty }(M)}$

properties

• The scrolling cohomology is invariant under tangential homotopias.
• There is a natural Mayer-Vietoris sequence for foliage cohomology.
• For Riemann foliations , the foliation cohomology can be calculated using the transversal Hodge theory of a bundle-like metric.

See also

• Basic cohomology

literature

• B. Mümken: A coincidence formula for foliated manifolds , Dissertation University of Münster, 2002.
• C. Peters: Foliage of Nile Manifolds , Dissertation University of Düsseldorf, 2003.
• S. Maßberg: The scrolling cohomology of node scrolling of the spheres , dissertation University of Düsseldorf, 2008.