Journal of the Ramanujan Mathematical Society
Volume 34, Issue 1, March 2019 pp. 109–132.
The Homotopy obstructions in complete intersections
Authors:
Satya Mandal and Bibekananda Mishra
Author institution:University of Kansas, Lawrence, Kansas 66045, USA
Summary:
Let A be a regular ring over a field k, with 1/2 ∈ k, and
dim A = d. We discuss the Homotopy Obstruction Program, in the
complete intersection case. Fix an integer n ≥ 2. A local
orientation is a pair (I, ω), where I is an ideal and
ω: A n → I/I2 is a surjective map. The goal is to define
and detect homotopy obstructions, for ω to lift to a
surjective map A n → I. Denote the set of all local
orientations by LO(A, n). A homotopy relations on
LO (A, n) is induced by the maps
LO (A, n) ← T = 0 LO (A[T], n) T = 1 →
LO (A, n)}. The homotopy obstruction set
π 0(LO(A, n)) is defined to be the set of all
equivalence classes. Assume 2n ≥ d+2. We prove that
π 0 (LO(A, n)) is an abelian group. We also
establish a surjective map ρ : En(A) → π 0(LO (A, n)),
where En(A) denotes the Euler class group. When
2n ≥ d + 3, and A is essentially smooth, we prove ρ is an
isomorphism. This settles a conjecture of Morel.
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