In this paper, we study the relationship of the Brouwer degree of a vector field with the dynamics of the induced flow. Analogous relations are studied for the index of a vector field. We obtain new forms of the Poincar é–Hopf theorem and of the Borsuk and Hirsch antipodal theorems. As an application, we calculate the Brouwer degree of the vector field of the Lorenz equations in isolating blocks of the Lorenz strange set.

In this paper, we construct a generalized degree theory of Browder-Petryshyn or Petryshyn type for a class of semilinear operator equations involving a Fredholm type mapping with infinite dimensional kernel.

A correspondence between the equivariant degree introduced by Ize, Massabó, and Vignoli and an unstable version of the equivariant fixed point index defined by Prieto and Ulrich is shown. With the help of conormal maps and properties of the unstable index, a sum decomposition formula is proved for the index and consequently also for the degree. As an application, equivariant homotopy groups are decomposed as direct sums of smaller groups of fixed orbit types, and a geometric interpretation of each summand is given in terms of conormal maps.

Let N be a closed s-Hopfian n-manifold with residually finite, torsion free π1 (N) and finite H1(N). Suppose that either πk(N) is finitely generated for all k ≥ 2, or πk(N) ≅ 0 for 1 < k < n – 1, or n ≤ 4. We show that if N fails to be a co-dimension 2 fibrator, then N cyclically covers itself, up to homotopy type.

Given a continuous map δ from the circle S to itself we want to find all self-maps σ: S → S for which δ o σ. If the degree r of δ is not zero, the transformations σ form a subgroup of the cyclic group Cr. If r = 0, all such invertible transformations form a group isomorphic either to a cyclic group Cn or to a dihedral group Dn depending on whether all such transformations are orientation preserving or not. Applied to the tangent image of planar closed curves, this generalizes a result of Bisztriczky and Rival [1]. The proof rests on the theorem: Let Δ: ℝ → ℝ be continuous, nowhere constant, and limx→−∞ Δ(x) = −∞, limx→−∞ Δ(xx) = +∞; then the only continuous map Σ: R → R such that Δ o Σ = Δ is the identity Σ = idℝ.

Properties of fixed points of equivariant maps have been studied by several authors including A. Dold (cf. [2], 1982), H. Ulrich (cf. [9], 1988), A. Marzantowicz (cf. [7], 1975) and others. Closely related is the work of R. Rubinsztein (cf. [8], 1976) in which he investigated homotopy classes of equivariant maps between spheres. There have been many attempts to introduce and effectively apply these concepts to nonlinear problems. In particular we mention the work of E. Dancer (cf. [1], 1982) in which some applications to nonlinear problems are given.