Let $M$ be a compact, smooth, $2$-connected,
$2m$-dimensional manifold with $\partial M$ simply connected. If $M$ has the homotopytype of an $m$-dimensional
$CW$-complex, then it supports a smooth, self-indexed function, maximal, constant and regular on $\partial M$
with at most $\mbox{cat}(M)+2$ critical points, all of which are of a certain ‘reasonable’ type. To such a
critical point there corresponds, homotopically, the attachment of a cone. Conversely, to a cone attachment we
may associate, under certain dimensionality and connectivity conditions, a ‘reasonable’ critical point.
1991 Mathematics Subject Classification: primary 55M30, 57R70; secondary 58E05, 55P50, 55P62.