degreesI

DegreesI is a family of integer-valued topological invariants defined for continuous maps between oriented manifolds, indexed by a distinguished submanifold I of the domain. It generalizes the usual notion of degree by measuring how a map f: M -> N covers a chosen target value when restricted to points lying on I. The invariant is defined for appropriate transversality and orientation data and is stable under homotopies that preserve the pair (M, I) in a suitable sense.

Definition and construction: Given a smooth map f: M -> N between oriented manifolds of the same dimension, choose a regular value y in N. If the intersection f^{-1}(y) ∩ I is transverse, deg_I(f) is defined as the algebraic count of these intersection points, with signs determined by the ambient orientations. This count is invariant under smooth deformations of f that keep I fixed, making deg_I a homotopy invariant relative to I.

Relations and special cases: If I equals M and M is connected, deg_I(f) reduces to the standard Brouwer degree deg(f). If I is a boundary component suitable for a relative degree, deg_I agrees with a relative degree notion in relative homology. There is a functorial property: for a map g: N -> P with nonzero degree and a map f: M -> N, under compatibility conditions, deg_I(g ∘ f) = deg(g) · deg_I(f).

Examples and applications: For the identity map id_M with I = M, deg_I(id_M) = 1. If a map factors through a contractible piece away from I, deg_I can vanish. In fixed-point theory and intersection theory, deg_I informs existence and counting of intersection points constrained to I, and in dynamical systems it can capture behavior on invariant submanifolds.