Re: [Q] Generalization of Poincare-Hopf Theorem

From: Daniel Asimov (asimov.nospam_at_msri.org)
Date: 07/29/04 

Date: 29 Jul 04 08:43:53 -0400 (EDT)

On 16 Jul 2004 08:20:58 -0400, theos ek mechanes wrote:

<<
...
The Poincare-Hopf theorem counts the zeros, with
the proper multiplicities, of a vector field on a given
manifold and equates this number with the Euler
charcteristic of this manifold.

However, see for example [1], if the Euler characteristic
of a manifold is zero, then most vector fields have no
zeroes but may have periodic orbits.
>>

I would like to know how you arrived at this last statement. In what
sense do you mean "most" ? On any compact manifold, there are
certainly open sets U of vector fields (in the C^1 topology) where
every member of U has a zero.

<<
The Reidemeister torsion counts these closed orbits, at least for some
families of vector fields.
...
Is there a generalization of the Poincare-Hopf theorem
which calculates, for a manifold of vanishing Euler
characteristic, the Reidemeister torsion of a manifold
in terms of the number of closed orbits of a "generic"
vector field on this manifold?
...
Are there generalizations of the Poincare-Hopf theorem
which calculate generalizations of the Euler characteristic,
Reidemeister torsion... of a manifold in terms of the
number of closed $S^0$'s, closed $S^1$'s, closed $S^2$'s,
closed $S^3$'s...of a generic vector field on this manifold?

[1] L. Nicolaescu, "Notes on the Reidemeister Torsion,"
     http://www.nd.edu/~lnicolae/Torsion.pdf
>>

For a smooth non-singular vector field V with n (finite) closed orbits
{C_k, k = 1,...,n} on a compact manifold M, define the "homology index
of V" h(V) as

             h(V) := sum from 1 to n of i(C_k)*[C_k]

where i(C_k) is the Poincaré index of the Poincaré map for C_k, and
[C_k} is the homology class in H_1(M: Z) of C_k.

See my paper "Flaccidity for geometric index for vector fields"
(Comm. Math. Helv., v.52, no. 2, 1977), on the web at
<http://snipurl.com/83ls>, for examples showing that the index
h(V) among *structurally stable* nonsingular vector fields on a
manifold of Euler characteristic = 0 can take any value in H_1(V).

Yea, this is true even if all these vector fields are restricted to
the same homotopy class among non-singular vector fields on M.

This shows that in the C^1 topology on vector fields, there are open
sets of vector fields on each of which the index h is constant, but
among these open sets it varies -- implying that this index is not
well-defined on "generic" vector fields in the C^1 topology.

For a more sophisticated closed orbit index, see the paper of David
Fried, "Lefschetz formulas for flows", Contemporary Math. 58 (1987),
19-69.

Dan Asimov

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