Classifying Closed 2-Orbifolds With Euler Characteristics

Abstract

Abstract. We determine the extent to which the collection of -Euler-Satake characteristics classify closed 2-orbifolds. In particular, we show that the closed, connected, effective, orientable 2-orbifolds are classified by the -Euler-Satake characteristics corresponding to free or free abelian and are not classified by those corresponding to any finite set of finitely generated discrete groups. These results demonstrate that the -Euler-Satake characteristics corresponding to free abelian constitute new invariants of orbifolds. Similarly, we show that such a classification is neither possible for non-orientable 2-orbifolds nor for non-effective 2-orbifolds using any collection of groups .2010 Mathematics Subject Classification. Primary 57R20, 57S17; Secondary 22A22, 57P99.

$\Gamma $-extensions of the spectrum of an orbifold

“…It is also of interest to consider the Γ-sectors of the orbifolds O 1 and O 2 for other free groups Γ. Since K 1 and K 2 are abelian, the Z 2 -sectors and the F 2sectors coincide; see [11]. The fact that the Z 2 -sectors coincide with the Z-sectors of the Z-sectors computed above (see [16, As the isotropy group of each point in O 1 and O 2 is abelian and can be generated by two elements, the Z ℓ -sectors of each O i for ℓ > 2 will simply yield multiple copies of the Z 2 -sectors of O i .…”

“…It is also of interest to consider the Γ-sectors of the orbifolds O 1 and O 2 for other free groups Γ. Since K 1 and K 2 are abelian, the Z 2 -sectors and the F 2sectors coincide; see [11]. The fact that the Z 2 -sectors coincide with the Z-sectors of the Z-sectors computed above (see [16,Theorem 3.1]) makes it straightforward to compute the Z 2 -sectors of O 1 and O 2 .…”

Untitled

“…In [Duval et al 2010], it was demonstrated that the collection of ‫ޚ‬ l -extensions of the Euler-Satake characteristic determine the diffeomorphism type of a closed, effective, orientable 2-dimensional orbifold and that infinitely many were required to do so. In addition, it was demonstrated that the χ ES corresponding to any collection of finitely generated discrete groups do not distinguish between certain effective, nonorientable 2-orbifolds.…”