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Riemannian metric representatives of the Stiefel-Whitney classes

Santiago R Simanca

TL;DR

The paper links Whitney's dual-cell cochain description of the Stiefel-Whitney classes to explicit metric representatives on smooth triangulated closed manifolds. By constructing locally defined totally geodesic blocks attached to each dual cell and using curvature data (including the Chern-Gauss-Bonnet density in higher degrees), it provides concrete $\mathbb{Z}/2$-valued cochains $\mathbf w^g_i$ representing all $w_i(M)$. In the Hermitian case, the even-degree representatives reduce to mod $2$ reductions of the corresponding Chern classes $c_k(M,J)$ and all odd-degree classes become trivial in cohomology. This work thus yields a geometric readout of the Stiefel-Whitney classes from a chosen Riemannian metric and clarifies their relation to complex geometry via Chern classes.

Abstract

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 $w_1(M), \ldots , w_n(M)$. The representative of $w_1(M)$ is determined by $\det{g_{ij}}$, the $g_{ij}$s computed in a frame that is locally defined at each dual $1$-cell; the representatives of the even classes $w_{2k}(M)$ 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 $w_{2k+1}(M)$ 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 $w_{2k}(M)$ so obtained is the $\mathbb{Z}/2$ reduction of the $k$-th Chern class $c_k(M,J)$ induced by the coefficient homomorphism, and that the metric representative of any odd degree class $w_{2k+1}(M)$ so obtained is trivial in cohomology.

Riemannian metric representatives of the Stiefel-Whitney classes

TL;DR

The paper links Whitney's dual-cell cochain description of the Stiefel-Whitney classes to explicit metric representatives on smooth triangulated closed manifolds. By constructing locally defined totally geodesic blocks attached to each dual cell and using curvature data (including the Chern-Gauss-Bonnet density in higher degrees), it provides concrete -valued cochains representing all . In the Hermitian case, the even-degree representatives reduce to mod reductions of the corresponding Chern classes and all odd-degree classes become trivial in cohomology. This work thus yields a geometric readout of the Stiefel-Whitney classes from a chosen Riemannian metric and clarifies their relation to complex geometry via Chern classes.

Abstract

If is a closed manifold, and is a smooth triangulation of , Whitney proved that all of the Stiefel-Whitney classes are specified as cochains on the dual cell complex assigning the value mod to each dual cell. We provide the pair with an arbitrary Riemannian metric , and use Whitney's criteria to show that there are associated representatives of all the Stiefel-Whitney classes . The representative of is determined by , the s computed in a frame that is locally defined at each dual -cell; the representatives of the even classes are determined by the Chern-Gauss-Bonnet density -form of locally defined totally geodesic oriented manifolds with boundary associated to each dual -cell; and the representatives of the odd classes are determined by the hypersurface area form of the boundary sphere of a locally defined totally geodesic oriented manifold with boundary associated to each dual -cell. If is Hermitian, we prove that the metric representative of so obtained is the reduction of the -th Chern class induced by the coefficient homomorphism, and that the metric representative of any odd degree class so obtained is trivial in cohomology.

Paper Structure

This paper contains 10 sections, 8 theorems, 27 equations.

Table of Contents

  1. The Stiefel-Whitney classes
  2. Smooth triangulations, barycentric subdivisions, and dual cell complexes: The class $w_0(M)$
  3. The Stiefel-Whitney class $w_0(M)$
  4. Smoothly triangulated Riemannian manifolds: The classes $w_1(M)$, $w_2(M)$, and $w_3(M)$
  5. The Stiefel-Whitney class $w_1(M)$
  6. The Stiefel-Whitney class $w_2(M)$
  7. The Stiefel-Whitney class $w_3(M)$
  8. Metric representatives of higher degree Stiefel-Whitney classes
  9. The Stiefel-Whitney class $w_{2k}(M)$
  10. The Stiefel-Whitney class $w_{2k+1}(M)$

Key Result

Theorem 1

The cochain ${\bf w}_1^g$ in (sw1g) represents the first Stiefel-Whitney class $w_1(M)$.

Theorems & Definitions (8)

  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Theorem 4
  • Theorem 5
  • Corollary 6
  • Theorem 7
  • Theorem 8