Homotopical intersection theory, III: Multirelative intersection problems

Homotopical intersection theory I

We give a new approach to intersection theory. Our “cycles” are closed manifolds mapping into compact manifolds and our “intersections” are elements of a homotopy group of a certain Thom space. The results are then applied in various contexts, including fixed point, linking and disjunction problems. Our main theorems resemble those of Hatcher and Quinn [H-Q], but our proofs are fundamentally di...

Homotopical Intersection Theory, Ii: Equivariance

This paper is a sequel to [KW]. We develop here an intersection theory for manifolds equipped with an action of a finite group. As in [KW], our approach will be homotopy theoretic, enabling us to circumvent the specter of equivariant transversality. We give applications of our theory to embedding problems, equivariant fixed point problems and the study of periodic points of self maps.

Intersection Theory

I provide more details to the intersection theoretic results in [1]. CONTENTS 1. Transversality and tubular neighborhoods 1 2. The Poincaré dual of a submanifold 4 3. Smooth cycles and their intersections 8 4. Applications 14 5. The Euler class of an oriented rank two real vector bundle 18 References 20 1. TRANSVERSALITY AND TUBULAR NEIGHBORHOODS We need to introduce a bit of microlocal termino...

Intersection Theory Classes 20 and 21: Bivariant Intersection Theory

T Geometric Intersection Problems

We develop optimal algorithms for forming the intersection of geometric objects in the plane and apply them to such diverse problems as linear programming, hidden-line elimination, and wire layout. Given N line segments in the plane, finding all intersecting pairs requires O(N2) time. We give an O(N log N) algorithm to determine i to detect whether two simple plane polygons intersect. We employ...