Classifying Space BG Research Articles

The Morava K-theories of wreath products

In p-primary stable homotopy theory, recent developments have shown the importance of the Morava K-theory spectra K(n) for positive integers n. A current major problem concerns the behaviour of the K(n)-cohomologies on the classifying spaces of finite groups and on related spaces. In this paper we show how to compute the Morava K-theory of extended power constructions Here Xp is the p-fold product of some space X and Cp is the cyclic group of order p. In particular, if we take X as the classifying space BG for some group G, then Dp(X) forms the classifying space for , the wreath product of G by Cp.

Locally finite approximation of Lie groups. II

In an earlier paper [10], we constructed a ‘locally finite approximation away from a given prime p’ of the classifying space BG of a Lie group with finite component group. Such an approximation consists of a locally finite group g and a homotopy class of maps which in particular induces an isomorphism in cohomology with finite coefficients of order prime to p. The usefulness of such a construction is that it reduces various homotopy-theoretic questions concerning the space BG to the corresponding questions concerning Bπ for finite subgroups π. For example, we demonstrated in [10] how H. Miller's proof of the Sullivan conjecture concerning maps from , where π is a finite group and X is a finite-dimensional complex, can be extended to maps BG→X for G a Lie group with finite component group.

Squaring Operations in the 4-Connective Fibre Spaces over the Classifying Spaces of the Exceptional Lie Groups

In this paper we calculate the squaring operations in the 4connective fibre spaces over the classifying spaces of the exceptional Lie groups. Let G be a compact, 1 -connected and simple Lie group. As is well known, its classifying space BG is 3-connected and H*(BG', Z) ~Z. Choose a generator y^H^(BG',Z). Then the 4-connective fibre space BG over BG is, by definition, the homotopy fibre of y^:BG— > K(Z,ty. Note that BG is a classifying space of G, the 3-connective fibre space over G. Here we quote the results in [2] and [4]. Define the sets Jl (/ = 2, 4, 6, 7, 8) as follows:

Stable cohomotopy and cobordism of abelian groups

The object of this paper is to prove that for a finite abelian group G the natural map is injective, where Â(G) is the completion of the Burnside ring of G and σ0(BG) is the stable cohomotopy of the classifying space BG of G. The map â is detected by means of an M U* exponential characteristic class for permutation representations constructed in (11). The result is a generalization of a theorem of Laitinen (4) which treats elementary abelian groups using ordinary cohomology. One interesting feature of the present proof is that it makes explicit use of the universality of the formal group law of M U*. It also involves a computation of M U*(BG) in terms of the formal group law. This may be of independent interest. Since writing the paper the author has discovered that M U*(BG) has previously been calculated by Land-weber(5).

Simplicial de rham cohomology and characteristic classes of flat bundles

IT IS well-known that the classifying space BG of a Lie group G is the geometric realization of a simplicial manifold, i.e. a semi-simplicial set whose p-dimensional simplices constitute a C” manifold and whose boundary and degeneracy operators are C” maps (see e.g. Segal[231). In the study of characteristic classes in real cohomology it is therefore natural to look for a De Rham complex for a simplicial manifold X = {X,}. An obvious candidate is the total complex a*(X) of the double complex (aq(X,), 6, &) of C” q-forms on X,, where dx is the usual exterior differential and where 6 is the co-boundary of simplicial cochains. This is studied in a recent paper by Bott-Shulman-Stasheff [4], where one can find a proof of the fact that the homology H(&*(X)) is naturally isomorphic to the singular cohomology with real coefficients of the realization ]]X]]. However there is an even more natural De Rham complex A *(X) associated to a simplicial manifold where a “form” is roughly speaking a C” form on llX]l (see 02 for a precise definition). For X a discrete simplicial set the construction of A *(X) goes back to Whitney [26] and has recently been used by Sullivan[25] in his study of the rational homotopy type of a manifold. The advantage of A *(X) is (apart from the suggestive nature of the definition) that the multiplication is graded commutative as in the case of an ordinary manifold and so the usual Chern-Weil theory carries over word by word to the universal case X = NG, the nerve of a Lie group G. To a great extent this just leads to a reformulation of previous constructions by Bott and Shulman (see [2] and [3]) and by Kamber-Tondeur[l3]. However the present point of view gives rise to an interesting formula for the characteristic classes of flat bundles which we shall now describe. Let G be a connected semi-simple real Lie group with finite center and choose a maximal compact subgroup K 2 G. Let fi and I be the corresponding Lie algebras and let H*(g, f) be the relative Lie algebra cohomology. For a homomorphism f: I + G where I is a discrete group there is a well-known characteristic homomorphism jr: H*(g, f)+ H*(Br) whose definition via De Rham cohomology goes back to Matsushima[ 171 (see Kamber-Tondeur[ 131, $8 or $4 below) and which has recently been studied by Bore1 [l] for arithmetic subgroups I. Now H*(Br) is canonically isomorphic to H*(T) the Eilenberg-MacLane group cohomology of I with real coefficients (see e.g. MacLane [ 161, chapter 4, P5) and we want to express jr in terms of explicit cochains. Let g = p @ f be a Cartan decomposition (see e.g. Helgason[ 101, chapter 3, 57). Then a class in H’(fl, I) is represented by an alternating q-form cp on g/f = p. By left translation this gives a closed C” q-form 6 on G/K (the differential of the complex A*(p) is actually 0 since [p, p] c f). Endowed with a left invariant Riemannian metric G/K is a non-compact globally symmzric space and I acts via f as a group of isometries on G/K (so we shall write -+x instead of f(y)x for x E G/K and y E I). Now let (r,, . . . , yq) E (r)’ and let o = {K} E G/K be the base point. We define the geodesic simplex A(y,, . . . , yq) g G/K inductively as follows. A(-y,) is the geodesic arc from o to ylo and generally A(r,, . . . , yq) is the geodesic cone on yl . A(?*, . . . , yq) with top point o. The ordering of the vertices o, y,o, y,-yZo, . . . , ylyz . . . yqo, determines a natural orientation of A(?,, . , . , yq). In this notation we shall prove (54):

On the K-theory of the loop space of a Lie group

Let G be a simply connected, semi-simple, compact Lie group, let K* denote Z/2-graded, representable K-theory, and K* the corresponding homology theory. The K-theory of G and of its classifying space BG are well known, (8),(1). In contrast with ordinary cohomology, K*(G) and K*(BG) are torsion-free and have simple multiplicative structures. If ΩG denotes the space of loops on G, it seems natural to conjecture that K*(ΩG) should have, in some sense, a more simple structure than H*(ΩG).

A definition of exotic characteristic classes of spherical fibrations

The object of this paper is to define certain characteristic cohomology classes for spherical fibrations which are zero on vector bundles and to show that the classes defined are not always zero. For an introductory survey of characteristic classes and spherical fibrations the reader is referred to [12] and the references therein. Briefly there is a 'structure group' G and a classifying space BG for spherical fibrations and similar spaces (SG and BSG) for oriented spherical fibrations. SG has the homotopy type of lim,_~o~ (f2S)1, where (f2S)1 is the space of degree 1 basepoint preserving maps of to itself. The cohomology of all four spaces has recently been computed by Milgram ([I0]), May ([8]) and Tsuchiya ([18]). My object is to define certain classes e k e H r ~ I ( B S G ; Zp) for p an odd prime (where r = 2 p 2 ) and ekeH2~l l (BG; Z2) which I will refer to as exotic classes. In order to simplify notation I will only deal with the case o fp odd, but all of the theorems herein can be proved for p = 2 with the obvious changes in notation. All cohomology groups will have Zp coefficients unless otherwise indicated. The definition given here is similar to one given by Peterson in [13] and to a definition of e~ given by Gitler-Stasheff in [4]. In each case the exotic classes are defined in terms of twisted secondary cohomology operations (TSCO's) acting on the Thom class u e H * M S G , where MSG is the Thorn space of the universal bundle over BSG. TSCO's were introduced by Thomas ([16]) and axiomatized by McClendon ([9]). They are a generalization of ordinary secondary operations to the category of topological pairs (X, V) over a fixed space Y. The analogue of the Steenrod algebra in this category is A (Y) where A is the Steenrod algebra and A ( Y ) = H * Y| as a vector space with the multiplication appropriate to defining an A (Y) module structure on H* (X, V). TSCO's are derived from relations in A(Y) jus t as ordinary secondary operations are derived from relations in A. Indeed, ordinary secondary operations can be regarded as TSCO's for the special case Y=pt. Now the Thorn space of any oriented spherical fibration can be regarded as a pair over BSG, so relations in A (BSG) could be used to define characteristics classes on suitable spherical fibrations. In [13] Peterson defined an algebra injection O:A-~ A (BSG) with the property that O(a) annihilates the Thom class u of MSG

On spherical characteristic cohomology

The theory of characteristic classes, primarily a topological discipline, canl be (and, at its inception, was) treated geometrically by means of mappings into classifying spaces and invariant polynomials in the curvature forms of these spaces. The theory goes something like this. Suppose that n and p are positive integers, that G is a closed subgroup of the orthogonal group 0(n), and that S is an invariant symmetric q-linear functional on the Lie algebra of G. Let Q denote the curvature form of the canonical connection of the second kind on 0(n + p) as a bundle over the classifying space BG =O(n + p)/G x 0(p). The 2q-form

On equivariant Chern–Weil forms and determinant lines

A strong from of invariance under a group G is manifested in a family over the classifying space BG. We advocate a differential-geometric avatar of BG when G is a Lie group. Applied to G-equivariant connections on smooth principal or vector bundles, the equivariance-->families principle converts the G-equivariant extensions of curvature and Chern-Weil forms to the standard nonequivariant versions. An application of this technique yields the moment map of the determinant line of a G-equivariant Dirac operator, which in turn sheds light on some anomaly formulas in quantum field theory.