  Quantum field theories with an energy gap can be approximated at long-range
by topological quantum field theories. The same should be true for suitable
condensed matter systems. For those with short range entanglement (SRE) the
effective topological theory is invertible, and so amenable to study via stable
homotopy theory. This leads to concrete topological invariants of gapped SRE
phases which are finer than existing invariants. Computations in examples
demonstrate their effectiveness.
