# 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.