### Abstract:

Let M be a closed n-dimensional manifold with a
flow () that has a global cross section Sigma ~= D^(n-1), and let h be
the (piecewise continuous) first return map for Sigma. Our primary
examples of such flows are minimal ones. We study how the return
map captures topological properties of the flow and of the manifold.
For a given map h if there exists an M, () such that h is a first return
map over some cross section then we call M, () the suspension of
h.
As an application, we give several (piecewise continuous) maps
of D^2 and a (piecewise continuous) map on D^3 which have suspensions.
The suspension manifold of the map h3 from Figure 6
is homotopic to S^3. Hence, if there exists a suspendable minimal
map of D^2 which is cell conjugate to h3 then it induces a minimal
flow on this homotopy--S^3. We also discuss ways to test if the suspension
manifold is the suspension of a map on a closed manifold,
as in the case of an irrational flow on T^2, and when it is not, as in
the case of any flow on S^3 (Refer to PDF file for exact formulas).