Basic cohomology

In the mathematics which is base-like cohomology or basic cohomology (engl .: basic cohomology ) a cohomology for describing Foliations , in particular for the investigation of Riemann Foliations is used. The name refers to the fact that the base-like cohomology describes the topology of the space of the leaves and, in particular in the case of a fiber bundle, agrees with the De Rham cohomology of the base space of the bundle. The definition goes back to Reinhart.

definition

Let be a -dimensional foliation of a -dimensional manifold . The complex of basic forms is defined as a sub-complex of the complex of differential forms by ${\displaystyle {\mathcal {F}}}$ ${\displaystyle q}$ ${\displaystyle n}$ ${\displaystyle \Omega ^{*}(M)}$

${\displaystyle \Omega ^{*}(M,{\mathcal {F}}):=\left\{\alpha \in \Omega ^{*}(M):i_{X}\alpha =0,i_{X}d\alpha =0\ \forall \ X\in T_{x}{\mathcal {F}},x\in M\right\}}$.

Here denotes the inner product and the outer derivative . ${\displaystyle i_{X}}$ ${\displaystyle d}$

The outer derivative maps to . The base cohomology is defined as the cohomology of the complex ${\displaystyle d}$ ${\displaystyle \Omega ^{q}(M,{\mathcal {F}})}$ ${\displaystyle \Omega ^{q+1}(M,{\mathcal {F}})}$ ${\displaystyle H^{*}(M,{\mathcal {F}})}$

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

Properties and examples

• If the foliage is through the fibers of a fiber bundle , then applies . ${\displaystyle {\mathcal {F}}}$ ${\displaystyle E\rightarrow B}$ ${\displaystyle H^{*}(M,{\mathcal {F}})=H^{*}(B;\mathbb {R} )}$
• It applies if and only if there is a Riemannian metric for which all leaves are minimal areas. ${\displaystyle H^{n-q}(M,{\mathcal {F}})\simeq \mathbb {R} }$ ${\displaystyle M}$
• Poincaré duality only holds for the basic cohomology of a foliation if it holds, which in general is not necessarily the case for Riemann folios either. ${\displaystyle H^{n-q}(M,{\mathcal {F}})\simeq \mathbb {R} }$
• For the weak stable (or the weak unstable) (m + 1) -dimensional foliation of an Anosov river on a (2m + 1) -dimensional manifold holds . ${\displaystyle H^{m+1}(M,{\mathcal {F}})=0}$
• The base-like cohomology is invariant under - diffeomorphisms , but generally not under homeomorphisms . ${\displaystyle C^{1}}$

See also

• Scrolling cohomology

literature

• Tondeur, Philippe: Foliations on Riemannian manifolds. University text. Springer-Verlag, New York, 1988. ISBN 0-387-96707-9
• Molino, Pierre: Riemannian foliations. Translated from the French by Grant Cairns. With appendices by Cairns, Y. Carrière, É. Ghys, E. Salem, and V. Sergiescu. Progress in Mathematics, 73. Birkhauser Boston, Inc., Boston, MA, 1988. ISBN 0-8176-3370-7

Individual evidence

1. ↑ Reinhart, Bruce L .: Harmonic integrals on foliated manifolds. Amer. J. Math. 81: 529-536 (1959).
2. ^ Masa, Xosé: Duality and minimality in Riemannian foliations. Comment. Math. Helv. 67 (1992) no. 1, 17-27.