Since d F p is a nonsingular linear map, this last degree is easily seen to be 0, or ±1.
Let ϕ : A → A be a smooth map with smooth extension f : ( X, A ) → ( X, A ). Suppose that p ∈ A is an isolated fixed point of f and that d ϕ p − I A : T p A → T p A is a nonsingular linear transformation.
Given a smooth map ϕ : ∂ X → ∂ X and a smooth map f : ( X, ∂ X ) → ( X, ∂ X ) extending ϕ, suppose that p ∈ ∂ X is an isolated fixed point of f and that d ϕ p − I ∂ X : T p ( ∂ X ) → T p ( ∂ X ) is a nonsingular linear transformation.
Theorem 6 Given a smooth map ϕ : S n − 1 → S n − 1 and a smooth map f : ( B n, S n − 1 ) → ( R n, S n − 1 ) extending ϕ, suppose that p ∈ S n − 1 is an isolated fixed point of f and that d ϕ p − I : T p ( S n − 1 ) → T p ( S n − 1 ) is a nonsingular linear transformation.
A smooth map ϕ : S n − 1 → S n − 1 with finitely many fixed points is transversely fixed if d ϕ p − I : T p ( S n − 1 ) → T p ( S n − 1 ) is a nonsingular linear map for each fixed point p. For F = { p 1, …, p r } a fixed point class of ϕ, let i ( F ) = ∑ j = 1 r i ( S n − 1, ϕ, p j ).
Let A be a nonsingular matrix.
Let A be a nonsingular nonrandom matrix.
Lemma 2.3 Let A = (a ij ) be a nonsingular d.d.
Let A be a nonsingular matrix with nonzero diagonal entries.
Proof Let S = (X, Y ) be a nonsingular matrix with.
