  Higher Dimensional Automata (HDA) are higher dimensional relatives to
transition systems in concurrency theory taking into account to which degree
various actions commute. Mathematically, they take the form of labelled cubical
complexes. It is important to know, and challenging from a
geometric/topological perspective, whether the space of directed paths
(executions in the model) between two vertices (states) is connected; more
generally, to estimate higher connectedness of these path spaces.
  This paper presents an approach for such an estimation for particularly
simple HDA modelling the access of a number of processors to a number of
resources with given limited capacity each. It defines a spare capacity for a
concurrent program with prescribed periods of access of the processors to the
resources. It shows that the connectedness of spaces of directed paths can be
estimated (from above) by spare capacities. Moreover, spare capacities can also
be used to detect deadlocks and critical states in such a HDA.
  The key theoretical ingredient is a transition from the calculation of local
connectedness bounds (of the upper links of vertices of an HDA) to global ones
by applying a version of the nerve lemma due to Anders Bj\"orner.
