The
–invariant is the
simplest
–manifold
invariant defined by configuration space integrals. It is actually an invariant of rational
homology spheres equipped with a combing over the complement of a point. It can be
computed as the algebraic intersection of three propagators associated to a given combing
in the
–point configuration
space of a
–sphere
. These propagators
represent the linking form of
so that
can be thought of as the cube of the linking form of
with respect to
the combing
.
The invariant
is the sum of
and
,
where denotes the
Casson–Walker invariant, and
is an invariant of combings, which is an extension of a first relative Pontrjagin
class. In this article, we present explicit propagators associated with
Heegaard diagrams of a manifold, and we use these “Morse propagators”,
constructed with Greg Kuperberg, to prove a combinatorial formula for the
–invariant
in terms of Heegaard diagrams.
Keywords
configuration space integrals, finite type invariants of
$3$–manifolds, homology spheres, Heegaard splittings,
Heegaard diagrams, combings, Casson–Walker invariant,
perturbative expansion of Chern-Simons theory,
$\Theta$–invariant
