Classifying spaces for projections of immersions with controlled singularities

Abstract

We give an explicit simple construction for classifying spaces of maps obtained as hyperplane projections of immersions. We prove structure theorems for these classifying spaces. IntroductionDefinition. Let M n and P n+k be smooth manifolds and f : M n → P n+k a smooth (C ∞ -) map. f is called a corank 1 map if rank df x ≥ n − 1 for all x ∈ M n . A stable corank 1 map is called a Morin map.Definition. Given a Morin map f we say that x ∈ M n is a Σ 1r,0 -point if there exists a regular curve γ : (R, 0) → (M, x) going through x that has∂t r+1 (0) = 0. Morin [4] showed that for a fix r all Σ 1r,0 germs are left-right equivalent (A-equivalent) and that for r = s the Σ 1r,0 germs are not equivalent to Σ 1s,0 germs.Definition. Given a Morin map f : M n → P n+k we denote by Σ(f ) the set of its singular points and we denote by Σ 1r,0 (f ) the set of its Σ 1r,0 -points. Definition.A Morin map is called a Σ 1r -map if it has no Σ 1s,0 -points with s > r. Definition.A corank 1 map f : M n → P n+k equipped with a trivialization of its kernel line bundle is called a prim map. by the National Research, Development and Innovation Office NKFIH (OTKA) Grant K 120697. A. Szűcs and T. TerpaiNote that a prim map is the composition of an immersion g : M n P n+k × R 1 with the standard projection pr : P n+k × R 1 → P n+k . 1We denote by CobPrimΣ 1r (P ) the cobordism group of prim Σ 1r -maps in a fixed target manifold P , and we denote by CobΣ 1r (P ) the cobordism group of all Σ 1r -maps in a fixed target manifold P . For the standard definitions of these groups see [13]. Analogous groups can be defined for the case of cooriented maps or maps with a quaternionic normal structure; we denote them by Cob SO PrimΣ 1r (P ), Cob SO Σ 1r (P ) and Cob Sp PrimΣ 1r (P ), Cob Sp Σ 1r (P ), respectively.