For every connected manifold with corners there is a homology theory called conormal
homology, defined in terms of faces and orientation of their conormal bundle and
whose cycles correspond geometrically to corner cycles. Its Euler characteristic (over
the rationals, dimension of the total even space minus the dimension of the total odd
space),
${\chi}_{cn}:={\chi}_{0}{\chi}_{1}$,
is given by the alternating sum of the number of (open) faces of a given
codimension.
The main result of the present paper is that for a compact connected manifold with
corners
$X$,
given as a finite product of manifolds with corners of codimension less or equal to
three, we have that:
1) If
$X$
satisfies the Fredholm perturbation property (every elliptic pseudodifferential boperator
on
$X$ can be
perturbed by a bregularizing operator so it becomes Fredholm) then the even Euler corner
character of
$X$
vanishes, i.e.,
${\chi}_{0}\left(X\right)=0$.
2) If the even periodic conormal homology group vanishes, i.e.,
${H}_{0}^{pcn}\left(X\right)=0$, then
$X$ satisfies the
stably homotopic Fredholm perturbation property (i.e., every elliptic pseudodifferential
boperator on
$X$
satisfies the same named property up to stable homotopy among elliptic
operators).
3) If
${H}_{0}^{pcn}\left(X\right)$
is torsion free and if the even Euler corner character of
$X$ vanishes,
i.e.,
${\chi}_{0}\left(X\right)=0$,
then
$X$
satisfies the stably homotopic Fredholm perturbation property. For example, for
every finite product of manifolds with corners of codimension at most two the
conormal homology groups are torsion free.
The main theorem behind the above result is the explicit computation
in terms of conormal homology of the Ktheory groups of the algebra
${\mathcal{K}}_{b}\left(X\right)$ of bcompact
operators for
$X$
as above. Our computation unifies the known cases of codimension zero (smooth
manifolds) and of codimension one (smooth manifolds with boundary).
