Bordism of constrained Morse functions

We call a Morse function $f$ on a closed manifold $k$-constrained if neither $f$ nor $-f$ has critical points of indefinite Morse index $< k$. In this paper we study bordism groups of $k$-constrained Morse functions, and thus interpolate between the case $k = 1$ of bordism groups of Morse functions (computed by Ikegami) and the case $k \gg 1$ of bordism groups of special generic functions (computed by Saeki). We employ Levine's elimination of cusps, Stein factorization, the two-index theorem of Hatcher-Wagoner, and a handle extension theorem for fold maps due to Gay-Kirby to show that the notion constrained bordism is strongly related to so-called connective bordism. As an application of our results we show that the oriented bordism group of constrained Morse functions detects exotic Kervaire spheres in certain dimensions.