Edgar H. Brown

On the rational homotopy type of function spaces

The main result of this paper is the construction of a minimal model for the function space F(X,Y) of continuous functions from a finite type, finite dimensional space X to a finite type, nilpotent space Y in terms of minimal models for X and Y. For the component containing the constant map, 7r* (.F(X, Y)) 0 Q = 7r* (Y) 0 H-* (X; Q) in positive dimensions. When X is formal, there is a simple formula for the differential of the minimal model in terms of the differential of the minimal model for Y and the coproduct of H* (X; Q). We also give a version of the main result for the space of cross sections of a fibration.

On immersions of n-Manifolds

A classical conjecture in differential topology is that an n -manifold immerses in R 2 n - α ( n ) , where α ( n ) is the number of ones in the dyadic expansion of n . We prove a corollary of this conjecture, namely, that at the Thom space level, the map T(v) → MO factors through MO ( n - α ( n )).

SOME REMARKS ABOUT SYMMETRIC FUNCTIONS

A formula is proven which determines whether or not a symmet- ric function is decomposable. Some applications to topology are mentioned.

CHAPTER 17 – Real and Rational Homotopy Theory

This chapter discusses the real and rational homotopy theories. It gives an exposition of the Quillan, Sullivan rational homotopy theory and the extension of this theory to real homotopy theory. The treatment is via the Sullivan approach emphasizing differential forms. To fully deal with real homotopy theory, it is essential that one uses simplicial spaces and continuous cohomology. deRham cohomology with real coefficients and carry this as far as one can without continuous cohomology. To simplify the exposition and still capture the main ideas, then shifts to rational coefficients and nilpotent simplicial sets. In this context, the chapter develops four theorems which in view form the foundation of real and rational theory.

The Serre spectral sequence theorem for continuous and ordinary cohomology

Abstract We give a detailed proof of the Serre spectral sequence E 2 result for singular homology. The techniques then easily adapt to singular cohomology and to continuous cohomology for twisted Cartesian products of simplicial spaces. With a view to some particular applications, we include local coefficients for the homology and cohomology of the total space.

Some algebraic constructions in rational homotopy theory

Abstract In this note we describe constructions in the category of differential graded commutative algebras over the rational numbers Q which are analogs of the space F ( X , Y ) of continuous maps of X to Y , the component F(X, Y,ƒ) containing ƒ ϵ F(X, Y) , fibrations, induced fibrations, the space Γ(π) of sections of a fibration π : E → X , and the component Γ(π,σ) containing σ ϵ Γ (π). As a focus, we address the problem of expressing π ∗ (F(X, Y, ƒ)) = Hom (π ∗ (F(X,Y, ƒ)),Q) in terms of differential graded algebra models for X and Y .

Computation of the unoriented cobordism ring

This note gives a set of generators for the unoriented cobordism ring and it gives a simplification of the algebra involved in Thorns computa- tion of this ring. In this note we give a simplification of the algebra involved in Thorns computation of the unoriented cobordism ring (3) and, along the way. a recursive formula for the generators of this ring given by Liulevicius (4). Let A denote the mod two Steenrod algebra, A * its dual and ^ E A* the polynomial generators defined by Milnor (1). In the following all homology and cohomology is with Z2 coefficients. We identify HtiMO) with H^iBO) via the Thorn isomorphism. The A module structure of //*(A/0) defines an A* comodule structure on H^MO), V: H^MO) -> A* ® H^MO). The Whitney sum operation makes //+(A/0) into an algebra and V is an algebra map. Let F: UkBOx —> BOk be the classifying map of the product of the canonical line bundles. F embeds H*iBO) in H*iY°BOx) = Z2(tx, t2, . . . ) as the algebra of symmetric functions. For a partition w = {/,, i2, . . ., ik) let sw be the smallest symmetric function containing f,1 • • • tkk. Recall under the map H*iBO) -* H*iBO) H*iBO) induced by the Whitney sum,

KO-obstructions of non-abelian group action on spin manifolds

Abstract In this paper, we give some KO -obstructions of non-Abelian group action on spin manifolds. These are closely related to the existence of metrics of positive scalar curvature on spin manifolds.