We study the question of existence of a Riemannian metric of positive scalar curvature
metric on manifolds with the Sullivan–Baas singularities. The manifolds we consider
are
and simply connected. We prove an analogue of the Gromov–Lawson Conjecture for
such manifolds in the case of particular type of singularities. We give an affirmative
answer when such manifolds with singularities accept a metric of positive scalar
curvature in terms of the index of the Dirac operator valued in the corresponding
“–theories
with singularities”. The key ideas are based on the construction due to Stolz, some
stable homotopy theory, and the index theory for the Dirac operator applied to the
manifolds with singularities. As a side-product we compute homotopy types of the
corresponding classifying spectra.
Keywords
Positive scalar curvature, Spin manifolds, manifolds with
singularities, Spin cobordism, characteristic classes in
$K$–theory, cobordism with singularities, Dirac operator,
$K$–theory with singularities, Adams spectral sequence,
$\mathcal{A}(1)$–modules
