Bordism groups of immersions and classes represented by self-intersections

Abstract

A well-known formula of R J Herbert's relates the various homology classes represented by the self-intersection immersions of a self-transverse immersion. We prove a geometrical version of Herbert's formula by considering the self-intersection immersions of a self-transverse immersion up to bordism. This clarifies the geometry lying behind Herbert's formula and leads to a homotopy commutative diagram of Thom complexes. It enables us to generalise the formula to other homology theories. The proof is based on Herbert's but uses the relationship between self-intersections and stable Hopf invariants and the fact that bordism of immersions gives a functor on the category of smooth manifolds and proper immersions.

Multiplicative properties of Morin maps

“…(3) It seems that the same results could be obtained using techniques of Eccles and Grant from [4]. (4) We would like to point out that Theorem 2.3 is a nontrivial generalization of the oriented case of Theorem A in Byun and Yi [3], which considers the case of n D k .…”

Singularities and stable homotopy groups of spheres. II

“…Recall also that the space ΓTξ i is the classifying space of immersions equipped with a pullback of their normal bundle fromξ i (such immersions will be called ξ i -immersions). That is, if we denote by Immξ i (m) the cobordism group of immersions of m-manifolds into R m+ci+k with the normal bundle induced fromξ i , the following well-known proposition holds: [9]) Immξ i (m) ∼ = π m+ci+k (ΓTξ i ) ∼ = π S m+ci+k (Tξ i ). Hence there is a spectral sequence in which the starting page is given by the cobordism groups ofξ i -immersions:…”

On realizing homology classes by maps of restricted complexity

“…The elements of this group are equivalence classes of triples (M n−k , f, v) under a suitable bordism relation, where M is a closed manifold, f : M n−k N n is an immersion of codimension k, and v : ν f → ζ is a bundle map isomorphic on fibres. We refer the reader to [5] for more details.…”

“…There is a natural transformation Θ : I(N ; γ k ) → H k (N ) that sends the bordism class of an immersion f : M n−k N n to the cohomology class it realizes (see [5,Section 3.2]). Representing these functors homotopically, we have the following diagram:…”