William S. Massey

Homotopy classification of 3-component links of codimension greater than 2

Abstract Let L be a link consisting of spheres of dimensions p 1 , p 2 , and p 3 respectively imbedded in R m such that m − p i > 2 for i = 1, 2, and 3. For such links we define an invariant β ( L ) ϵ π p 1+ p 2+ p 3 ( S 2 m −3 ) which is invariant under link homotopy. This invariant is shown to be the stable suspension of a certain link concordance invariant introduced by Haefliger and Steer in 1964. For a certain range of dimensions such links are classified up to link homotopy by the new invariant β( L ) and the link homotopy classes of their 2-component sublinks.

Sufficient conditions for a local homeomorphism to be injective

Abstract Let U be an open connected subset of Rn whose closure, Ū, is compact and whose boundary, ∂U, has only finitely many components. Assume f:Ū→Rn is a continuous map such that f|U is a local homeomorphism (e.g. f|U has continuous partial derivatives and nonvanishing Jacobian). Under certain hypotheses on ∂U and f|∂U, it is proved that f must map U homeomorphically onto its image.

Products in Homology and Cohomology

The most important product is undoubtedly the so-calledcup product: It assigns to any elements u ∈H p (X; G1) and v ∈ Hq(X ; G2) an elementu ∪ v ∈ H p + q (X ;G1⊗ G2). This product is bilinear (or distributive), and is natural with respect to homomorphisms induced by continuous maps. It is an additional element of structure on the cohomology groups that often allows one to distinguish between spaces of different homotopy types, even though they have isomorphic homology and cohomology groups. This additional structure also imposes restrictions on the possible homomorphisms which can be induced by continuous maps.

Homology with Arbitrary Coefficient Groups

This chapter is more algebraic in nature than the preceding chapters. In §2 we discuss chain complexes. This discussion mainly puts on a formal basis many facts that the reader must know by now. Nevertheless, there is some point to a systematic organization of the ideas involved, and certain new ideas and techniques are introduced. The remainder of the chapter is concerned with homology groups with arbitrary coefficients. These new homology groups are a generalization of those we have considered up to now. In the application of homology theory to certain problems they are often convenient and sometimes necessary.

Definitions and Basic Properties of Homology Theory

This chapter gives formal definitions of the basic concepts of homology theory, and rigorous proofs of their basic properties. For the most part, examples and applications are postponed to Chapter III and subsequent chapters.

Homology of CW-complexes

The purpose of this chapter is to develop a systematic procedure for determining the homology groups of a certain class of topological spaces. The class of topological spaces chosen consists of the CW-complexes of J. H. C. Whitehead. The procedure developed is a natural generalization and extension of the method used in the preceding chapter to determine the homology groups of graphs and compact 2-manifolds.

Determination of the Homology Groups of Certain Spaces : Applications and Further Properties of Homology Theory

In this chapter, we will actually determine the homology groups of various spaces : the n-dimensional sphere, finite graphs, and compact 2-dimensional manifolds. We also use homology theory to prove some classical theorems of topology, most of which are due to L. E. J. Brouwer. In addition, we prove some more basic properties of homology groups.

Cup Products in Projective Spaces and Applications of Cup Products

In this chapter we will determine cup products in the cohomology of the real, complex, and quaternionic projective spaces. The cup products (mod 2) in real projective spaces will be used to prove the famous Borsuk—Ulam theorem. Then we will introduce the mapping cone of a continuous map, and use it to define the Hopf invariant of a map f : S 2n-1 → S n . The proof of existence of maps of Hopf invariant 1 will depend on our determination of cup products in the complex and quaternionic projective plane.

The Homology of Product Spaces

If two or more spaces are related to each other in some way, we would naturally expect that their homology groups should also be related in some way. Some of the most important theorems in the preceding chapters bear out this expectation: If A is a subspace of X, the exact homology sequence of the pair (X, A) describes the relations between the homology groups of A and the homology groups of X. If the space X is the union of twp subspaces U and V, then the Mayer-Vietoris sequences gives relations between the homology groups of U, V, U ⋂ V, and X.

Background and Motivation for Homology Theory

Homology theory is a subject whose development requires a long chain of definitions, lemmas, and theorems before it arrives at any interesting results or applications. A newcomer to the subject who plunges into a formal, logical presentation of its ideas is likely to be somewhat puzzled because he will probably have difficulty seeing any motivation for the various definitions and theorems. It is the purpose of this chapter to present some explanation, which will help the reader to overcome this difficulty. We offer two different kinds of material for background and motivation. First, there is a summary of some of the most easily understood properties of homology theory, and a hint at how it can be applied to specific problems. Secondly, there is a brief outline of some of the problems and ideas which lead certain mathematicians of the nineteenth century to develop homology theory.