2020/04/12 by Santiago R. Simanca, Simanca, Santiago R
Computer Science · Mathematics · #53C20 #57R20 #Differential Geometry (math.DG) #FOS: Mathematics #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2004.05719
openalex publication_date 2020/04/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
If M is a closed manifold, and K is a smooth triangulation of M, Whitney proved that all of the Stiefel-Whitney classes are specified as cochains on the dual cell complex (K')^* assigning the value 1 mod 2 to each dual cell. We provide the pair (M,K) with an arbitrary Riemannian metric g, and use Whitney's criteria to show that there are associated representatives of all the Stiefel-Whitney classes w1(M), … , wn(M). The representative of w1(M) is determined by detgij, the gijs computed in a frame that is locally defined at each dual 1-cell; the representatives of the even classes w2k(M) are determined by the Chern-Gauss-Bonnet density 2k-form of locally defined totally geodesic oriented 2k manifolds with boundary associated to each dual 2k-cell; and the representatives of the odd classes w2k+1(M) are determined by the hypersurface area form of the boundary sphere of a locally defined totally geodesic oriented (2k+1) manifold with boundary associated to each dual (2k+1)-cell. If (M,J,g) is Hermitian, we prove that the metric representative of w2k(M) so obtained is the ℤ/2 reduction of the k-th Chern class ck(M,J) induced by the coefficient homomorphism, and that the metric representative of any odd degree class w2k+1(M) so obtained is trivial in cohomology.