  The Conley index for flows is a topological invariant describing the behavior
around an isolated invariant set $S$. It is defined as the homotopy type of a
quotient space $N/L$, where $(N,L)$ is an index pair for $S$. In the case of a
discrete dynamical system, i.e., a continuous self-map $f\colon X\to X$, the
definition is similar. But one needs to consider the index map $f_{(N,L)}\colon
N/L\to N/L$ induced by $f$. The Conley index in this situation is defined as
the homotopy class $[f_{(N,L)}]$ modulo shift equivalence. The shift
equivalence relation is rarely used outside this context and not well
understood in general. For practical purposes like numerical computations, one
needs to use weaker algebraic invariants for distinguishing Conley indices,
usually homology. Here we consider a topological invariant: the homotopy type
of the mapping torus of the index map $f_{(N,L)}$. Using a homotopy type offers
new ways for comparing Conley indices -- theoretically and numerically. We
present some basic properties and examples, compare it to the definition via
shift equivalence and sketch an idea for its construction using rigorous
numerics.
