The normal cycle of a compact definable set

Abstract

An elementary construction of the normal cycle of a compact definable set in Euclidean space (and more generally of a compactly supported constructible function) is given. Here "definable" means definable in some o-minimal structure. The construction is based on the notion of support function and uses only basic o-minimal geometry.

On the normal cycles of subanalytic sets

“…The points x for which i(x, ξ) = 0 should be viewed as critical points of the function −ξ : X → R; see [15, §5.4] or Appendix C. Thus, the slice N ξ records both the collection of critical points of −ξ| X and their Morse indices. (b) Our sign conventions are different from the ones used in [1,9], but they coincide with the conventions in [3].…”

“…Very recently, A. Berning [1] has proposed a very ingenious and elegant elementary construction of the normal cycle of a subanalytic set using the recent advances in o-minimal topology and basic facts about currents. Unfortunately there is a flaw in a key existence result, [1, Lemma 6.4]; see Remark 4.2(a) for more detail;s. The present paper grew out of our attempts to fix that flaw.…”

“…We construct the normal cycle in Section 3 via an approximation method. Section 4 describes how a combination of facts proved in this paper and in [1] leads to an alternate proof of the existence part of [1, Thm. 6.2].…”

“…1 In particular, for any ζ ∈ Z the fiber Y ζ is contained in the finite union of planes Ξ ζ × V . (c6) For any ζ ∈ Z the slice S, Ψ, ζ is well defined.…”

“…1 The definability of the Euler characteristic [25, §4.2] implies that the cardinality of Ξ ζ is bounded from above by a constant independent of ζ.…”