J. H. C. Whitehead

A proof and extension of Dehn's lemma

Publisher Summary C. D. Papakyriakopoulos has recently proved Dehns lemma. His proof has the merit that the basic construction, the tower, and the crucial lemmas apply to the sphere theorem as well as Dehns lemma. However, if one is content with Dehns lemma, the proof can be simplified. This chapter presents a simplified proof of Dehns lemma and also presents a proof of an analogous theorem for surfaces with more than one boundary curve. By surface of type (p,r) it is meant that a connected, compact, orientable surface of genus p with a boundary consists of r 1-spheres. Thus, the Euler characteristic of such a surface is 2 (1–p)–r.

On analytical complexes

1. In his Colloquium Lecturesf one of us outlined a proof of an important theorem regarding the covering of analytic loci by complexes. A proof for algebraic varieties had previously been given by B. van der WaerdenJ and B. 0. Koopman and A. B. Brown§ have recently proved the theorem for analytic loci. The object of this paper is to give a detailed proof along the lines indicated in Topology. 2. We begin with certain general observations! I concerning the nature of a configuration £ (at first complex) represented by an analytic system

On linear connections

With the introduction of infinitesimal parallelism, by T. Levi-Civitat in 1917, and independently by J. A. Schouten4 in 1918, tangent spaces began to play a leading role in differential geometry. The tangent space at a point, x, is the totality of all contravariant vectors, or differentials, associated with that point. By means of an affine connection? the tangent spaces at any two points on a curve are related by an affine transformation, which will in general depend on the curve. Linear connections of another kind were defined by R. K6nig,1| who associated with each point of a given n-dimensional manifold a space of m dimensions. A linear connection arises in differential equations of the form?

NOTE ON COHOMOLOGY SYSTEMS

Chang introduced certain new numerical invariants of a polyhedron P , assuming that dim P ≤ n+2 and π r ( P ) = 0 ( r = 1,…, n —1), where n > 2. They are defined in terms of the Steenrod homomorphism, Sq n– 2 : H n (2)→ H n +2 (2). This chapter discusses similar invariants of any finite polyhedron. They are defined in terms of Sq n – k , operating on H n (2), for any even value of k such that 0 k ≤ n . The chapter presents the definition of a cohomology spectrum. The definition consists of a 2-index family of additive, Abelian groups H n ( m ) ( n = 0,1,…) related by certain homomorphisms μ,Δ .