Unstable atomicity and loop spaces on lie groups

Introduction. Let T be a simply connected CW complex of finite type. Its homotopy Lie is the rational Lie 7r*(9 T) 0 Q, equipped with the Samelson product. Let A be a local (noetherian) ring with residue field k (of arbitrary characteristic). Its homotopy Lie algebra, 7r*(A), is the naturally defined Lie whose enveloping is ExtA(k, k)-cf. [Av]. These Lie algebras have finite type (i.e. are finite dimensional in each degree) and are concentrated in strictly positive degrees. Thus, for the sake of simplicity, we shall use graded Lie algebra to mean graded Lie of finite type concentrated in strictly positive degrees throughout this paper. On the other hand these Lie algebras are usually nonzero in infinitely many degrees. Thus, while we may (and do) form the sum, R, of all the solvable ideals in such a Lie and call it the radical, it may well happen that R, itself, is not solvable. Indeed, this does happen in the topological context, since by Quillen's result [Q] every rational Lie of finite type occurs as the homotopy Lie of a space. By contrast, in the case of local rings, or of CW complexes with finite Lusternik-Schnirelmann (L.S.) category, serious restrictions apply to the homotopy Lie algebra. Indeed, our first main result is the remarkable

Every principal G-bundle over X is classified up to equivalence by a homotopy class X -> BG, where BG is the classifying space of G. On the other hand, for every nice topological space X Milnor constructed a strict model of its loop space (Omega) over tildeX, that is a group. Moreover, the morphisms of topological groups (Omega) over tildeX -> G generate all the G-bundles over X up to equivalence. In this paper, we show that the relation between Milnor's loop space and the classifying space functor is, in a precise sense, an adjoint pair between based spaces and topological groups in a homotopical context. This proof leads to a classification of principal bundles over a fixed space, that is dual to the classification of bundles with a fixed group. Such a result clarifies the deep relation that exists between the theory of bundles, the classifying space construction and the loop space, which are very important in topological K-theory, group cohomology, and homotopy theory.

We introduce fibred type-theoretic fibration categories which are fibred categories between categorical models of Martin-Lof type theory. Fibred type-theoretic fibration categories give a categorical description of logical predicates for identity types. As an application, we show a relational parametricity result for homotopy type theory. As a corollary, it follows that every closed term of type of polymorphic endofunctions on a loop space is homotopic to some iterated concatenation of a loop.

From the point of view of homotopy theory a compact Lie group G has the following remarkable combination of properties:(1) G can be given the structure of a finite CW-complex, and(2) there is a pointed space BG and a homotopy equivalence from G to the loop space QB G .Of course the space BG in (2) is the ordinary classifying space of G.In general, a finite complex X together with a chosen equivalence X -+ nBX for some BX is called a finite loop space.If p is a prime number and the geometric finiteness condition on X is replaced by the requirement that X be Fp-complete in the sense of [3] and have finite mod p cohomology, then X is called a p-adic finite loop space or a finite loop space at the prime p.A (p-adic) finite loop space is a strong homotopy-theoretic analogue of a compact Lie group.The study of these spaces is related to many classical questions in topology (for instance, to the problem of determining all spaces with polynomial cohomology rings).Call a p-adic finite loop space X exotic if it is not the Fp-completion of G for a compact Lie group G.There are many known examples of exotic p-adic finite loop spaces at odd primes p [5] and the classification of these spa~es is partially understood [8,9].However, until now there have been no known exotic 2-adic finite loop spaces.Recall [32] that the ring of rank 4 mod 2 Dickson invariants is the ring of invariants of the natural action of GL( 4, F 2) on the rank 4 polynomial algebra H*((BZ/2)4, F 2 ); this ring of invariants is a polynomial algebra on classes c s ' C 12 ' C 14 ' and C l5 with Sq 4 C s = c 12 ' Sq 2 c l2 = c 14 ' and Sql C l4 = C 15 • Our main theorem is the following one.1.1.Theorem.There exists an F 2-complete space B DI( 4) such that H*(BDI(4) , F 2 ) is isomorphic as an algebra over the Steenrod algebra to the ring of rank 4 mod 2 Dickson invariants.Let DI(4) = QBDI(4).Standard methods show that H*(DI(4) , F 2 ) is multiplicatively generated by elements x 7 , y 11' and z I3' with S q 4 X = y, Sq2y = z, Sql Z = x 2 =1= 0, and X4 = i = z2 = O.This space DI(4) is an exotic 2-adic finite loop space.

Let M be a compact connected Riemannian manifold whose sectional curvature values are all nonpositive. Let T denote the fundamental group of M. We prove that any homotopy equivalence : N -> M from a compact closed manifold TV is homotopic to a homeomorphism, provided that m > 5 where m = dim M . We show that the surgery L-group L k+m (T, w { ) is isomorphic to the set of homotopy classes of k k

Lie groups are very important objects in algebra, topology and geometry, as it is naturally endowed with two special structures: the algebraic structure of group and the geometric structure of differential manifold. Hence, it is quite meaningful to better know the structure of Lie groups and the actions, and the transformations between them. Two specific theorems about Lie groups and their homomorphism and Lie homomorphism are proved in this paper. First, the fact that there does not exist a surjective Lie group homomorphism between and is introduced. This is done by first having a trial of constructing Lie group surjective homomorphism but fails. Next, this result is further proved by applying topological method that to construct a covering map between and , followed by computing the fundamental group of . Finally, by applying that is a simple group, the fact that there does not exist a surjective group homomorphism between and is demonstrated, together with the statement that is a simple group.

D. H. Gottlieb proved in 1972 that the group of automorphisms of a numerable G G -bundle p : X → B p:X \to B is weakly homotopy equivalent to Ω Map ⁡ ( B , B G ; k ) \Omega \;\operatorname {Map}(B,{B_G};k) , where k : B → B G k:B \to {B_G} is a classifying map for p p . We refine this classical result by constructing a genuine homotopy equivalence between these two spaces which is natural with respect to numerable bundle morphisms, can be generalized to fibre bundles, and can be interpreted as a natural isomorphism between two suitably defined functors.

In this article we consider a space B_{com}G assembled from commuting\nelements in a Lie group G first defined in [Adem, Cohen, Torres-Giese 2012]. We\ndescribe homotopy-theoretic properties of these spaces using homotopy colimits,\nand their role as a classifying space for transitionally commutative bundles.\nWe prove that ZxB_{com}U is a loop space and define a notion of commutative\nK-theory for bundles over a finite complex X which is isomorphic to\n[X,ZxB_{com}U]. We compute the rational cohomology of B_{com}G for G equal to\nany of the classical groups U(n), SU(n) and Sp(n), and exhibit the rational\ncohomologies of B_{com}U, B_{com}SU and B_{com}Sp as explicit polynomial rings.\n

We survey recent results regarding embeddings of finite simple groups (and their nonsplit central extensions) in complex Lie groups, especially the Lie groups of exceptional type. Introduction Throughout this paper, L will be a finite group. Representation theory for L is usually understood to be the study of group morphisms L → GL(n, k) for distinguished collections of fields k (e.g., all overfields of a fixed field F ) and positive integers n . The topic of this survey is motivated by the question as to what happens if GL ( n ,·) is replaced by another algebraic group G (·). We shall mainly be concerned with the case where L is a finite simple group (that is, a finite nonabelian simple group) or a central extension thereof, and G(k) is a connected simple algebraic group over a field k . A further restriction of our discussion concerns the field k . It will mostly be taken to be the complex numbers, in which case we will mainly study group morphisms from L to the complex Lie group G (ℂ). (See below for some exceptions in §3 and §5.) For G (·) of classical type, the theory for representations L → G (ℂ) differs little from the usual one for GL ( n , ℂ). Indeed, a representation L → GL ( n , ℂ) decomposes into irreducible subrepresentations. The decomposition is well controlled by character theory.

Equivariant homotopy and deformations of diffeomorphisms

A cocycle Ω : P × G → H \Omega : P \times G \to H taking values in a Lie group H H for a free right action of G G on P P defines a principal bundle Q Q with the structure group H H over X = P / G . X=P/G. The Chern character of a vector bundle associated to Q Q then defines characteristic classes on X . X. This observation becomes useful in the case of infinite dimensional groups. It typically happens that a representation of G G is not given by linear operators which differ from the identity by a trace-class operator. For this reason the Chern character of a vector bundle associated to the principal fibration P → P / G P \to P/G is ill-defined. But it may happen that the Lie algebra representations of the group H H are given in terms of trace-class operators and therefore the Chern character is well-defined; this observation is useful especially if the map g ↦ Ω ( p ; g ) g\mapsto \Omega (p;g) is a homotopy equivalence on the image for any p ∈ P . p\in P. We apply this method to the case P = A , P= \mathcal {A}, the space of gauge connections in a finite-dimensional vector bundle, and G = G G= \mathcal {G} the group of (based) gauge transformations. The method for constructing the appropriate cocycle Ω \Omega comes from ideas in quantum field theory, used to define the renormalized gauge currents in a Fock space.

Detecting homotopy equivalences in base-point-free homotopy

The Weyl group, WG, of each exceptional simple Lie group G, is described in detail. Its structure is defined in terms of its coset decomposition with respect to the Weyl group, WH, of a classical semi-simple Lie group, H, embedded naturally in G. The concepts of G-dominance and G-equivalence are defined and used to determine, from the character formula of Weyl, the branching rule associated with the restriction of group elements from G to H. The Weyl group WG is used further to impose constraints on both the branching multiplicities for G to H and the weight multiplicities of G. These constraints are used to evaluate the weight multiplicities of F4, E6, E7 and E8 together with the branching multiplicities for E8 to SO(16).

We prove that if Γ is an icc irreducible lattice in a product of connected non-compact rank one simple Lie groups with finite center, then theII1{\mathrm{II}_{1}}factorL⁢(Γ){L(\Gamma)}is prime. In particular, we deduce that theII1{\mathrm{II}_{1}}factors associated to the arithmetic groupsPSL2⁢(ℤ⁢[d]){\mathrm{PSL}_{2}(\mathbb{Z}[\sqrt{d}])}andPSL2⁢(ℤ⁢[S-1]){\mathrm{PSL}_{2}(\mathbb{Z}[S^{-1}])}are prime for any square-free integerd≥2{d\geq 2}withd≢1⁢(mod⁡ 4){d\not\equiv 1~{}(\operatorname{mod}\,4)}and any finite non-empty set of primesS. This provides the first examples of primeII1{\mathrm{II}_{1}}factors arising from lattices in higher rank semisimple Lie groups. More generally, we describe all tensor product decompositions ofL⁢(Γ){L(\Gamma)}for icc countable groups Γ that are measure equivalent to a product of non-elementary hyperbolic groups. In particular, we show thatL⁢(Γ){L(\Gamma)}is prime, unless Γ is a product of infinite groups, in which case we prove a unique prime factorization result forL⁢(Γ){L(\Gamma)}.

We prove that irreducible unitary representations of the discrete series of simple Hermitian-symmetric Lie groups can be continuously deformed in the class of pure unitary pseudo-representations (note that these representations cannot be continuously deformed in the class of unitary representations). We also briefly recall the main definitions and facts related to the notions of quasi-symmetry and pseudo-symmetry and to the realization of representations of the holomorphic discrete series of simple hermitian-symmetric Lie groups.