1.Elementary Morse theory associates to a Riemannian manifold and a Morse
function a cochain complex of finite dimensional vector spaces, known as
the geometric complex, which is fundamental both in topology and geometric
analysis.
2.If $f$ is a Morse Bott function, the geometric complex can still be
defined but is not finite dimensional; its underlying vector space is the
same as the space of differential forms of the critical manifolds. The
boundary operators however are not even pseudo-differential but very
natural and easy to describe. This complex is particularly important since
in many situations there are no Morse functions around but only Morse Bott
functions (for example in the case of $G-$ manifolds, $G$ a compact Lie
group).
3. From analysis perspective the complex has all the features of an
elliptic complex unfortunately none of the basic technics work. The
spectral geometry of this complex is particularly relevant geometrically
and requires an "elliptic theory " but the operators involved are not even
pseudo-differential and the Hilbert space theory (based on Sobolev norms)
is likely of little use.
This is a report on my work (with S. Haller) on this complex. |
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