On spherical Milnor Classifying Spaces I: differential geometry

Abstract

We compare the original product-subset diffeology on the Milnor join of Magnot--Watts with the locally finite final diffeology on a signed spherical model. The canonical squaring map from the spherical space to the Magnot--Watts join is smooth, whereas the positive square-root section is generally not; we also construct a different smooth injection from the locally finite simplicial model whose inverse is not smooth. For Hausdorff Fréchet--Lie groups whose smooth real-valued functions separate points, we compute the full algebraic internal tangent space, exhibiting separate first-order branches labelled by elements of the group at every inactive coordinate. The spherical weighted energy descends plotwise and restricts to a Riemannian metric on each support stratum. Smooth normalized strictly positive-definite kernels yield non-canonical ambient tangent metrics; compact Lie groups admit an explicit realization through their heat kernels. We characterize global diffeological forms by descent, construct plotwise Hausdorff Jacobians and varying-degree complementary forms, and show why these data do not define a canonical Hodge star across support changes. Stratumwise Hodge--Dirac operators remain available, but an explicit example obstructs their automatic gluing. We prove smooth contractibility of the total spherical space for any diffeological group, as well as equivariant uniqueness of maps to it. For any diffeological group we establish a classifying theorem for $D$-numerable principal bundles, without regularity hypotheses. For Lie groups we construct barycentric connections and their Chern--Weil forms, and distinguish sign local systems from gerbes. The hypotheses and proofs specify which conclusions are global, stratumwise, or conditional.

arXiv commentThis new version contains new results and corrected proofs