Extensions of codimension one immersions

Codimension one immersions and the Kervaire invariant one problem

Codimension, Multiplicity and Integral Extensions

Let A ⊂ B be a homogeneous inclusion of standard graded algebras with A0 = B0. To relate properties of A and B we intermediate with another algebra, the associated graded ring G = grA1B(B). We give criteria as to when the extension A ⊂ B is integral or birational in terms of the codimension of certain modules associated to G. We also introduce a series of multiplicities associated to the extens...

Geometric Formulas for Smale Invariants of Codimension Two Immersions

We give three formulas expressing the Smale invariant of an immersion f of a (4k− 1)-sphere into (4k + 1)-space. The terms of the formulas are geometric characteristics of any generic smooth map g of any oriented 4k-dimensional manifold, where g restricted to the boundary is an immersion regularly homotopic to f in (6k − 1)-space. The formulas imply that if f and g are two non-regularly homotop...

Order One Invariants of Immersions

We classify all order one invariants of immersions of a closed orientable surface F into R, with values in an arbitrary Abelian group G. We show that for any F and G and any regular homotopy class A of immersions of F into R, the group of all order one invariants on A is isomorphic to G0 ⊕ B ⊕ B where G0 is the group of all functions from a set of cardinality א0 into G and B = {x ∈ G : 2x = 0}....

Codimension One Symplectic Foliations

We define the concept of symplectic foliation on a symplectic manifold and provide a method of constructing many examples, by using asymptotically holomorphic techniques.