Stable Pontryagin–Thom construction for proper maps

Abstract

AbstractWe will present proofs for two conjectures stated in Rot (Homotopy classes of proper maps out of vector bundles, 2020. arXiv:1808.08073). The first one is that for an arbitrary manifold W, the homotopy classes of proper maps $$W\times \mathbb {R}^n\rightarrow \mathbb {R}^{k+n}$$W×Rn→Rk+n stabilise as $$n\rightarrow \infty $$n→∞, and the second one is that in a stable range there is a Pontryagin–Thom type bijection for proper maps $$W\times \mathbb {R}^n\rightarrow \mathbb {R}^{k+n}$$W×Rn→Rk+n. The second one actually implies the first one and we shall prove the second one by giving an explicit construction.