which we verify two topological conjectures using Ricci flow. First,

we classify the homotopy type of every 3-dimensional spherical space

form. This proves the Generalized Smale Conjecture and gives an

alternative proof of the Smale Conjecture, which was originally due to

Hatcher. Second, we show that the space of metrics with positive

scalar curvature on every 3-manifold is either contractible or

empty. This completes work initiated by Marques.

Our proof is based on a new uniqueness theorem for singular Ricci

flows, which I have previously obtained with Kleiner. Singular
Ricci

flows were inspired by Perelman's proof of the
Poincaré and

Geometrization Conjectures, which relied on a
flow in which

singularities were removed by a certain surgery
construction. Since

this surgery construction depended on various
auxiliary parameters,

the resulting flow was not uniquely
determined by its initial

data. Perelman therefore conjectured
that there must be a canonical,

weak Ricci flow that
automatically "flows through its singularities"

at an
infinitesimal scale. Our work on the uniqueness of singular

Ricci
flows gives an affirmative answer to Perelman's conjecture and

allows the study of continuous families of singular Ricci flows

leading to the topological applications mentioned above.