= Morse cell-attachment theorem for geodesic energy
{c}
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.
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