A looping–delooping adjunction for topological spaces

The cohomology of groups

The classifying space BG of a topological group G can be filtered by a sequence of subspaces B(q,G) using the descending central series of free groups. If G is finite, describing them as homotopy colimits is convenient when applying homotopy theoretic methods. In this thesis we introduce natural subspaces B(q,G)_p of B(q,G) defined for a fixed prime p. We show that B(q,G) is stably homotopy equivalent to a wedge sum of B(q,G)_p as p runs over the primes dividing the order of G. Colimits of abelian groups play an important role in understanding the homotopy type of these spaces. Extraspecial p-groups are key examples, for which these colimits turn out to be finite. We prove that for extraspecial p-groups of rank at least 4 the space B(2,G) does not have the homotopy type of a K(π,1) space. Furthermore, we give a group theoretic condition, applicable to symmetric groups and general linear groups, which implies the space B(2,G) not having the homotopy type of a K(π,1) space. For a finite group G, we compute the complex K-theory of B(2,G) modulo torsion.

It is a well-known result of the covering groups that a subgroup G of the fundamental group at the identity of a semilocally simply connected topological group determines a covering morphism of topological groups with characteristic group G. In this paper we generalize this result to a large class of algebraic objects called topological groups with operations, including topological groups. We also prove that the crossed modules and internal categories within topological groups with operations are equivalent. This equivalence enables us to introduce the cover of crossed modules within topological groups with operations. Finally, we draw relations between the coverings of an internal groupoid within topological groups with operations and those of the corresponding crossed module.

This chapter continues to study homology and cohomology theories through the concept of a spectrum and constructs its associated homology and cohomology theories, called spectral homology and cohomology theories. It also introduces the concept of generalized (or extraordinary) homology and cohomology theories. Moreover, this chapter conveys the concept of an \(\Omega \)-spectrum and constructs a new \(\Omega \)-spectrum \(\underline{A}\), generalizing the Eilenberg–MacLane spectrum K(G, n). It constructs a new generalized cohomology theory \(h^*(~ ~ ; \underline{A})\) associated with this spectrum \(\underline{A}\), which generalizes the ordinary cohomology theory of Eilenberg and Steenrod. This chapter works in the category \( {\mathcal {C}}\) whose objects are pairs of spaces having the homotopy type of finite CW-complex pairs and morphisms are continuous maps of such pairs. This is a full subcategory of the category of pairs of topological spaces and maps of pairs, and this admits the construction of mapping cones. Let \({\mathcal {C}_0}\) be the category whose objects are pointed topological spaces having the homotopy type of pointed finite CW-complexes and morphisms are continuous maps of such spaces. There exist the (reduced) suspension functor \(\Sigma : {\mathcal {C}_0} \rightarrow {\mathcal {C}_0}\) and its adjoint functor \(\Omega : {\mathcal {C}_0} \rightarrow {\mathcal {C}_0}\) which is the loop functor.

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.

This monograph introduces a framework for genuine proper equivariant stable homotopy theory for Lie groups. The adjective ‘proper’ alludes to the feature that equivalences are tested on compact subgroups, and that the objects are built from equivariant cells with compact isotropy groups; the adjective ‘genuine’ indicates that the theory comes with appropriate transfers and Wirthmüller isomorphisms, and the resulting equivariant cohomology theories support the analog of an R O ( G ) R O(G) -grading. Our model for genuine proper G G -equivariant stable homotopy theory is the category of orthogonal G G -spectra; the equivalences are those morphisms that induce isomorphisms of equivariant stable homotopy groups for all compact subgroups of G G . This class of π ∗ \pi _* -isomorphisms is part of a symmetric monoidal stable model structure, and the associated tensor triangulated homotopy category is compactly generated. Consequently, every orthogonal G G -spectrum represents an equivariant cohomology theory on the category of G G -spaces. These represented cohomology theories are designed to only depend on the ‘proper G G -homotopy type’, tested by fixed points under all compact subgroups. An important special case of our theory are infinite discrete groups. For these, our genuine equivariant theory is related to finiteness properties in the sense of geometric group theory; for example, the G G -sphere spectrum is a compact object in our triangulated equivariant homotopy category if the universal space for proper G G -actions has a finite G G -CW-model. For discrete groups, the represented equivariant cohomology theories on finite proper G G -CW-complexes admit a more explicit description in terms of parameterized equivariant homotopy theory, suitably stabilized by G G -vector bundles. Via this description, we can identify the previously defined G G -cohomology theories of equivariant stable cohomotopy and equivariant K-theory as cohomology theories represented by specific orthogonal G G -spectra.

In the setting of homotopy type theory, each type can be interpreted as a space. Moreover, given an element of a type, i.e. a point in the corresponding space, one can define another type which encodes the space of loops based at this point. In particular, when the type we started with is a groupoid, this loop space is always a group. Conversely, to every group we can associate a type (more precisely, a pointed connected groupoid) whose loop space is this group: this operation is called delooping. The generic procedures for constructing such deloopings of groups (based on torsors, or on descriptions of Eilenberg-MacLane spaces as higher inductive types) are unfortunately equipped with elimination principles which do not directly allow eliminating to untruncated types, and are thus difficult to work with in practice. Here, we construct deloopings of the cyclic groups Zm which are cellular, and thus do not suffer from this shortcoming. In order to do so, we provide type-theoretic implementations of lens spaces, which constitute an important family of spaces in algebraic topology. Our definition is based on the computation of an iterative join of suitable maps from the circle to an arbitrary delooping of Zm. In some sense, this work generalizes the construction of real projective spaces by Buchholtz and Rijke, which handles the case m = 2, although the general setting requires more involved tools. Finally, we use this construction to also provide cellular descriptions of dihedral groups, and explain how we can hope to use those to compute the cohomology and higher actions of such groups.

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.

A (k-1)-connected space X with Hk(X, Zp)~'~Z p has been defined to be patomic if any map f : X ~ X which induces an isomorphism f , : Hk(X, Zp) ~Hk(X , Zp) must induce an isomorphism f , : H,(X, Zp)->H,(X, Zp). We describe such X as being C.P. p-atomic (Cohen, Peterson) and consider some variants of the definition. C.P. p-atomicity is an indecomposability condition on a space implying for example that the space has no non trivial rood p homotopy retracts. We define a space X to be weakly p-atomic if any map f : X ~ X is either a mod p homotopy equivalence or for each n, f . : lq,(X, Zp) --* H,(X, Zp) is nilpotent under composition. If X is a (k-1)-connected CW complex of finite type, Hk(X, Zp)~--Zp, k > 1 and X is either an H-space or a co-H-space (in particular, a suspension), then X is C.P. p-atomic if and only if it is weakly p-atomic (see comment 2 following Proposition 5.1). The spaces considered satisfy a stronger condition. A space X is defined to be p-atomic if any map f : X ~ X is either a m o d p homotopy equivalence or f . : /~,(X, Zp)~ H. (X , Zv) is nilpotent, Not every weakly p-atomic space is p-atomic, for example, BSU is weakly 2-atomic but is not 2-atomic (see comment 2 following Lemma 3.3). A p-atomic space X is defined to be p-atomic of degree t if whenever f . : H. (X, Zv)--+ H, (X, Zv) is nilpotent, ( f , ) t= 0 but there exists some such f with (f.)t1 + 0, t > 0, The author does not know if every p-atomic space is p-atomic of finite degree. The significance of the degree t will become apparent below. We consider the 2-atomicity of O G where G is a 1-connected simple Lie group.

The core of classical homotopy theory is a body of ideas and theorems that emerged in the 1950s and was later largely codified in the notion of a model category. This core includes the notions of fibration and cofibration; CW complexes; long fiber and cofiber sequences; loop spaces and suspensions; and so on. Brown's representability theorems show that homology and cohomology are also contained in classical homotopy theory. This text develops classical homotopy theory from a modern point of view, meaning that the exposition is informed by the theory of model categories and that homotopy limits and colimits play central roles. The exposition is guided by the principle that it is generally preferable to prove topological results using topology (rather than algebra). The language and basic theory of homotopy limits and colimits make it possible to penetrate deep into the subject with just the rudiments of algebra. The text does reach advanced territory, including the Steenrod algebra, Bott periodicity, localization, the Exponent Theorem of Cohen, Moore, and Neisendorfer, and Miller's Theorem on the Sullivan Conjecture. Thus the reader is given the tools needed to understand and participate in research at (part of) the current frontier of homotopy theory. Proofs are not provided outright. Rather, they are presented in the form of directed problem sets. To the expert, these read as terse proofs; to novices they are challenges that draw them in and help them to thoroughly understand the arguments.

We define completion of the algebraic de Rham complex associated to the algebras of functionals smooth in the Chen–Souriau sense or in the Nualart–Pardoux sense over the loop space. We show that the stochastic algebraic de Rham cohomology groups are equal to the deterministic cohomology groups of the loop space.

Let M be a geometrically irreducible smooth projective variety, defined over a finite field k, such that M admits a k-rational point x0. Let (M,x0/ denote the corresponding fundamental group-scheme introduced by Nori. Let EG be a principal G-bundle over M, where G is a reduced reductive linear algebraic group defined over the field k. Fix a polarization ξ on M. We prove that the following three statements are equivalent:1. The principal G-bundle EG over M is given by a homomorphism (M,x0)→G.2. There are integers b > a ≥ 1, such that the principal G-bundle (FbM)* EG is isomorphic to (FaM) * EG where FM is the absolute Frobenius morphism of M.3. The principal G-bundle EG is strongly semistable, the degree(c2(ad(EG))c1 (ξ)d−2 = 0, where d = dimM, and the degree(c1(EG(χ))c1(ξ)d−1) = 0 for every character χ of G, where EG(χ) is the line bundle over M associated to EG for χ.In [16], the equivalence between the first statement and the third statement was proved under the extra assumption that dimM = 1 and G is semisimple.

A classification theorem for diagrams of simplicial sets

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.

§ i. General principal bundles i. F-structures Let P=(P,~,B,G) be a principal bundle where P and B are topological spaces and @ is a topological group. Let 0: F+G be a continuous homomorphism from a topological group F onto G with kernel K • 0 will be called central, if (i) K is discrete (ii) K is contained in the center of F Thus, in particular, K is abelian. It is easy to check that (ii) follows from (i) if F is connected. Thus every covering projection P: F÷G (F connected) satisfies the conditions above. Throughout this paper we shall assume that p is central. A r-structure on P is a F-principal bundle P=(P,~,B,F) together with a strong bundle map ~: P÷P which is equivariant under the right actions of the structure groups; that is