Two Gauss–Bonnet and Poincaré–Hopf theorems for orbifolds with boundary

Abstract

DateThe final copy of this thesis has been examined by the signatories, and we find that both the content and the form meet acceptable presentation standards of scholarly work in the above mentioned discipline.iii Seaton, Christopher W. (Ph.D., Mathematics)Two Gauss-Bonnet and Poincaré-Hopf Theorems for Orbifolds with Boundary Thesis directed by Assoc. Prof. Carla FarsiThe goal of this work is to generalize the Gauss-Bonnet and Poincaré-Hopf Theorems to the case of orbifolds with boundary. We present two such generalizations, the first in the spirit of [20], which uses an argument parallel to that contained in [22]. In this case, the local data (i.e. integral of the curvature in the case of the Gauss-BonnetTheorem and the index of the vector field in the case of the Poincaré-Hopf Theorem)is related to Satake's orbifold Euler characteristic, a rational number which depends on the orbifold structure.For the second pair of generalizations, we use a more recent orbifold cohomology[3] to express the local data in a way which can be related to the Euler characteristic of the underlying space of the orbifold. This case applies only to orbifolds which admit almost-complex structures.

Characteristic classes of bad orbifold vector bundles

“…As H * orb (Q) and H * orb (E) are defined to be sums of the de Rham cohomology of the sectors of the respective orbifolds, the sum, ι * , is then clearly an isomorphism of additive groups. ✷ Finally, we note that any characteristic class c (see [2] Proposition 4.3.4) or orbifold characteristic class c orb (see [5] Appendix B) defined by the ChernWeil construction for vector bundles can be extended to the case of bad orbifold vector bundles. Note that the definition of orbifold characteristic classes c orb (E) applies the Chern-Weil construction to connections on the bundleẼ overQ.…”

“…In previous work (see [5] and [6]), we have defined an orbifold Euler class in Chen-Ruan orbifold cohomology (note that in [5], the definition of an orbifold vector bundle was taken to be that of a good orbifold vector bundle). In [6], it was demonstrated that when all of the local groups are cyclic, this class acts as a complete obstruction to the existence of nonvanishing tangent vector fields.…”

“…In Section 3, we apply these results to the Euler class and extend the obstruction property of the Euler class to the case of bad orbifold vector bundles. The reader is referred to [3], [4], [2], [1], and [5] for relevant background material. In particular, [2] contains as an appendix a modern introduction to orbifolds; for the most part, we follow its notation.…”

Nonvanishing vector fields on orbifolds

“…If the orbifold Q is closed, we let χ ES (Q) denote the Euler-Satake characteristic of Q (see [15] where this number is called the Euler characteristic of Q as a V -manifold or [16] where this quantity is denoted 1 by χ orb (Q)). Recall that the Euler-Satake characteristic is most easily defined in terms of a simplicial decomposition T of Q such that the isomorphism class of the isotropy group is constant on the interior of each simplex (it is shown in [10] that such a simplicial decomposition always exists).…”

“…Satake's Euler characteristic as a V -manifold) is a trivial consequence of Satake's Poincaré-Hopf Theorem in [15]. In [16,Corollary 3.4], the second author offered a different Poincaré-Hopf theorem, demonstrating that a nonvanishing vector field also implies that the Euler characteristic of the underlying topological space of Q vanishes. However, the converse of both of these statements is false; it is easy to construct examples of orbifolds such that both characteristics vanish, yet whose singular strata force any vector field to vanish (see [17]).…”