  We study sparsity constrained nonlinear optimization (SCNO) from a
topological point of view. Special focus will be on M-stationary points from
Burdakov et al. (2016). We introduce nondegenerate M-stationary points and
define their M-index. We show that all M-stationary points are generically
nondegenerate. In particular, the sparsity constraint is active at all local
minimizers of a generic SCNO. Some relations to other stationarity concepts,
such as S-stationarity, basic feasibility, and CW-minimality, are discussed in
detail. By doing so, the issues of instability and degeneracy of points due to
different stationarity concepts are highlighted. The concept of M-stationarity
allows to adequately describe the global structure of SCNO along the lines of
Morse theory. For that, we study topological changes of lower level sets while
passing an M-stationary point. As novelty for SCNO, multiple cells of dimension
equal to the M-index are needed to be attached. This intriguing fact is in
strong contrast with other optimization problems considered before, where just
one cell suffices. As a consequence, we derive a Morse relation for SCNO, which
relates the numbers of local minimizers and M-stationary points of M-index
equal to one. The appearance of such saddle points cannot be thus neglected
from the perspective of global optimization. Due to the multiplicity phenomenon
in cell-attachment, a saddle point may lead to more than two different local
minimizers. We conclude that the relatively involved structure of saddle points
is the source of well-known difficulty if solving SCNO to global optimality.
