 Morse cell-attachment theorem for geodesic energy

On a complete compact Riemannian manifold, nondegenerate critical points of fixed-endpoint geodesic energy yield a CW complex model of the path space, with one cell per geodesic and dimension equal to its Morse index. Finite-dimensional broken-geodesic models prove the assertion at bounded energy, and exhaustion gives the full path space. On $S^n$ with distinct nonantipodal endpoints the cell dimensions are $j(n-1)$, one in each such dimension.