Samuel B. Smith

CONTINUOUS TRACE C*-ALGEBRAS, GAUGE GROUPS AND RATIONALIZATION

Let ζ be an n-dimensional complex matrix bundle over a compact metric space X and let Aζ denote the C*-algebra of sections of this bundle. We determine the rational homotopy type as an H-space of UAζ, the group of unitaries of Aζ. The answer turns out to be independent of the bundle ζ and depends only upon n and the rational cohomology of X. We prove analogous results for the gauge group and the projective gauge group of a principal bundle over a compact metric space X.

Banach algebras and rational homotopy theory

Let A be a unital commutative Banach algebra with maximal ideal space Max(A). We determine the rational H-type of GLn(A), the group of invertible n × n matrices with coefficients in A, in terms of the rational cohomology of Max(A). We also address an old problem of J. L. Taylor. Let Lcn(A) denote the space of “last columns” of GLn(A). We construct a natural isomorphism Ȟ(Max(A);Q) ∼= π2n−1−s(Lcn(A))⊗ Q for n > 1 2 s+1 which shows that the rational cohomology groups of Max(A) are determined by a topological invariant associated to A. As part of our analysis, we determine the rational H-type of certain gauge groups F (X,G) for G a Lie group or, more generally, a rational H-space.

RATIONAL HOMOTOPY OF THE SPACE OF SELF-MAPS OF COMPLEXES WITH FINITELY MANY HOMOTOPY GROUPS

For simply connected CW complexes X with finitely many, finitely generated homotopy groups,1 the path components of the function space M(X, X) of free self-maps of X are all of the same rational homotopy type if and only if all the ^-invariants of X are of finite order. In case X is rationally a two-stage Postnikov system the space Mq(X , X) of inessential self-maps of X has the structure of rational //-space if and only if the fc-invariants of X are of finite order. Given spaces X and Y let M(X, Y) denote the space of free, continuous maps from X to Y with the compact-open topology. Given a map / : X —* Y, we let Mf(X, Y) denote the component of / in M(X, Y). In particular, we denote by Mq(X , Y) the space of inessential maps from X to Y and by Mx (X, X) the space of self-maps of X which are homotopic to the identity. If X and Y are spaces with base points then we have the subspace M(X, T), of M(X, Y) consisting of all based maps between X and Y. Given a based map / : X —► Y, we write Mf(X, 7), for the component of / in M(X, Y%. In his 1956 paper (20), R. Thorn used a Postnikov decomposition of Y to show that the homotopy groups of M(X, Y) are determined up to a series of group extensions involving the cohomology of X with coefficients in the homo- topy of Y. To illustrate his technique, Thorn computed the rational homotopy groups of certain function spaces. He also showed how by analyzing the eval- uation map it is possible, in certain cases, to determine the rational homotopy type of components of M(X, Y). While predating by many years Sullivans development of the theory of minimal models, Thorns work established the techniques for studying function spaces within the framework of rational ho- motopy theory. In (7), Haefliger completed a program initiated by Sullivan ( 18) with the construction of a rational homotopy-theoretic model for the space of sections of a nilpotent bundle. The construction embodied the two aspects

Criteria for Components of A Function Space to Be Homotopy Equivalent

We give a general method that may be effectively applied to the question of whether two components of a function space map(X, Y) have the same homotopy type. We describe certain group-like actions on map(X, Y). Our basic results assert that if maps f, g: X ? Y are in the same orbit under such an action, then the components of map(X, Y) that contain f and g have the same homotopy type.

Rank of the fundamental group of any component of a function space

We compute the rank of the fundamental group of any connected component of the space map(X, Y) for X and Y connected, nilpotent CW complexes of finite type with X finite. For the component corresponding to a general homotopy class f : X → Y, we give a formula directly computable from the Sullivan model for f. For the component of the constant map, our formula retrieves a known expression for the rank in terms of classical invariants of X and Y. When both X and Y are rationally elliptic spaces with positive Euler characteristic, we use our formula to determine the rank of the fundamental group of any component of map(X, Y) explicitly in terms of the homomorphism induced by f on rational cohomology.

The rational homotopy Lie algebra of classifying spaces for formal two-stage spaces

We compute the center and nilpotency of the graded Lie algebra π∗(ΩBaut1(X))⊗Q for a large class of formal spaces X. The latter calculation determines the rational homotopical nilpotency of the space of self-equivalences aut1(X) for these X. Our results apply, in particular, when X is a complex or symplectic flag manifold.

Rational homotopy type of the classifying space for fibrewise self-equivalences

Let p be a fibration of simply connected CW complexes with finite base B and fibre F. Let aut_1(p) denote the identity component of the space of all fibre-homotopy self-equivalences of p and Baut_1(p) the classifying space for this topological monoid. We give a differential graded Lie algebra model for Baut_1(p). We use this model to give classification results for the rational homotopy types represented by Baut_1(p) and also to obtain conditions under which the monoid aut_1(p) is a double loop-space after rationalization.

Postnikov sections of formal and hyperformal spaces

Let X be a simply connected CW complex and Xn its n th Postnikov section. We prove that X is formal provided H q (Xn;I Q) is additively generated by decomposables for all q and n with q > n: Recall from [4] that a space X is said to be hyperformal if its rational cohomology algebra is the quotient of a free graded algebra by an ideal generated by a regular sequence. Using the main result of [4] we show our sucien t condition for formality is actually equivalent to hyperformality.