A spectral sequence for string cohomology

Let X be a 1-connected space with free-loop space ΛX. We introduce two spectral sequences converging towards H^*(ΛX;Z/p) and H^*((ΛX)_hT;Z/p). The E₂-terms are certain non-Abelian-derived functors applied to H^*(X;Z/p). When H^*(X;Z/p) is a polynomial algebra, the spectral sequences collapse for more or less trivial reasons. If X is a sphere it is a surprising fact that the spectral sequences collapse for p=2.     

Batalin-Vilkovisky algebras and the J-homomorphism

Let X be a topological space. The homology of the iterated loop space H_∗ΩnX is an algebra over the homology of the framed n-disks operad H_∗fDn [E. Getzler, Batalin-Vilkovisky algebras and two-dimensional topological field theories, Comm. Math. Phys. 159 (2) (1994) 265-285; P. Salvatore, N. Wahl, Framed discs operads and Batalin-Vilkovisky algebras, Q. J. Math. 54 (2) (2003) 213-231]. We explicitly determine this H_∗fDn-algebra structure on H_∗(ΩnX;Q). We show that the action of H_∗(SO(n)) on the iterated loop space H_∗ΩnX is related to the J-homomorphism and that the BV-operator on H_∗(Ω²X) vanishes on spherical classes only in characteristic other than 2.     

A completion theorem for pro-G-spectra

We give a very general completion theorem for pro-spectra. We show that, if G is a compact Lie group, M[∗] is a pro-G-spectrum, and F is a family of (closed) subgroups of G, then the mapping pro-spectrum F(EF₊,M[∗]) is the F-adic completion of M[∗], in the sense that the map M[∗]→F(EF₊,M[∗]) is the universal map into an algebraically F-adically complete pro-spectrum. Here, F(EF₊,M[∗]) denotes the pro-G-spectrum , where runs over the finite subcomplexes of EF₊.     

A fiberwise analogue of the Borsuk-Ulam theorem for sphere bundles over a 2-cell complex II

We describe a finite complex B as I-trivial if there does not exist a Z₂-map from Si−1 to S(α) for any vector bundle α over B and any integer i with i>dimα. We prove that the m-fold suspension of projective plane FP² is I-trivial if and only if m≠0,2,4 for F=C, m≠0,4 for F=H. In the case where F is the Cayley algebra, the m-fold suspension is shown to be I-trivial for every m>0.     

Primitives and central detection numbers in group cohomology

Fix a prime p. Given a finite group G, let H^∗(G) denote its mod p cohomology. In the early 1990s, Henn, Lannes, and Schwartz introduced two invariants d₀(G) and d₁(G) of H^∗(G) viewed as a module over the mod p Steenrod algebra. They showed that, in a precise sense, H^∗(G) is respectively detected and determined by Hd(CG(V)) for d?d₀(G) and d?d₁(G), with V running through the elementary abelian p-subgroups of G.The main goal of this paper is to study how to calculate these invariants. We find that a critical role is played by the image of the restriction of H^∗(G) to H^∗(C), where C is the maximal central elementary abelian p-subgroup of G. A measure of this is the top degree e(G) of the finite dimensional Hopf algebra H^∗(C)_⊗H^∗(G)Fp, a number that tends to be quite easy to calculate.Our results are complete when G has a p-Sylow subgroup P in which every element of order p is central. Using the Benson-Carlson duality, we show that in this case, d₀(G)=d₀(P)=e(P), and a similar exact formula holds for d₁. As a bonus, we learn that He(G)(P) contains nontrivial essential cohomology, reproving and sharpening a theorem of Adem and Karagueuzian.In general, we are able to show that d₀(G)?max{e(CG(V))|V<G} if certain cases of Benson's Regularity Conjecture hold. In particular, this inequality holds for all groups such that the difference between the p-rank of G and the depth of H^∗(G) is at most 2. When we look at examples with p=2, we learn that d₀(G)?14 for all groups with 2-Sylow subgroup of order up to 64, with equality realized when G=SU(3,4).En route we study two objects of independent interest. If C is any central elementary abelian p-subgroup of G, then H^∗(G) is an H^∗(C)-comodule, and we prove that the subalgebra of H^∗(C)-primitives is always Noetherian of Krull dimension equal to the p-rank of G minus the p-rank of C. If the depth of H^∗(G) equals the rank of Z(G), we show that the depth essential cohomology of G is nonzero (reproving and extending a theorem of Green), and Cohen-Macauley in a certain sense, and prove related structural results.     

The product of operators with closed range in Hilbert C-modules

Suppose T and S are bounded adjointable operators with close range between Hilbert C^∗-modules, then TS has closed range if and only if Ker(T)+Ran(S) is an orthogonal summand, if and only if Ker(S^∗)+Ran(T^∗) is an orthogonal summand. Moreover, if the Dixmier (or minimal) angle between Ran(S) and Ker(T)∩[Ker(T)∩Ran(S)]^⊥ is positive and is an orthogonal summand then TS has closed range.     

Lie theory and the Chern-Weil homomorphism

Let P→B be a principal G-bundle. For any connection θ on P, the Chern-Weil construction of characteristic classes defines an algebra homomorphism from the Weil algebra Wg=Sg^∗⊗∧g^∗ into the algebra of differential forms A=Ω(P). Invariant polynomials inv(Sg^∗)⊂Wg map to cocycles, and the induced map in cohomology inv(Sg^∗)→H(Abasic) is independent of the choice of θ. The algebra Ω(P) is an example of a commutativeg-differential algebra with connection, as introduced by H. Cartan in 1950. As observed by Cartan, the Chern-Weil construction generalizes to all such algebras.In this paper, we introduce a canonical Chern-Weil map Wg→A for possibly non-commutativeg-differential algebras with connection. Our main observation is that the generalized Chern-Weil map is an algebra homomorphism “up to g-homotopy”. Hence, the induced map inv(Sg^∗)→Hbasic(A) is an algebra homomorphism. As in the standard Chern-Weil theory, this map is independent of the choice of connection.Applications of our results include: a conceptually easy proof of the Duflo theorem for quadratic Lie algebras, a short proof of a conjecture of Vogan on Dirac cohomology, generalized Harish-Chandra projections for quadratic Lie algebras, an extension of Rouvière's theorem for symmetric pairs, and a new construction of universal characteristic forms in the Bott-Shulman complex.     

Simple homotopy types and finite spaces

We present a new approach to simple homotopy theory of polyhedra using finite topological spaces. We define the concept of collapse of a finite space and prove that this new notion corresponds exactly to the concept of a simplicial collapse. More precisely, we show that a collapse X↘Y of finite spaces induces a simplicial collapse K(X)↘K(Y) of their associated simplicial complexes. Moreover, a simplicial collapse K↘L induces a collapse X(K)↘X(L) of the associated finite spaces. This establishes a one-to-one correspondence between simple homotopy types of finite simplicial complexes and simple equivalence classes of finite spaces. We also prove a similar result for maps: We give a complete characterization of the class of maps between finite spaces which induce simple homotopy equivalences between the associated polyhedra. This class describes all maps coming from simple homotopy equivalences at the level of complexes. The advantage of this theory is that the elementary move of finite spaces is much simpler than the elementary move of simplicial complexes: It consists of removing (or adding) just a single point of the space.     

A concavity inequality for symmetric norms

We review some recent convexity results for Hermitian matrices and we add a new one to the list: Let A be semidefinite positive, let Z be expansive, Z^∗Z?I, and let f:[0,∞)→[0,∞) be a concave function. Then, for all symmetric norms

‖f(Z^∗AZ)‖?‖Z^∗f(A)Z‖.     

Epimorphisms of knot groups onto free products

Let k be a knot in S³. There is an epimorphism from π₁(S³−k) onto a free product of two nontrivial cyclic groups sending a meridian to an element of length two iff k has property Q (Topology of Manifolds, Markham, Chicago, IL, 1970, pp. 195-199) that is if there is a closed surface F in S³ containing k, such that k is imprimitive in H₁(X) and in H₁(Y) where X and Y are the closures of the components of S³−F. We give answers to questions of Simon (1970) about properties Q, Q∗ and Q∗∗. Epimorphisms from knot groups onto torus knot groups are also studied and some results on property P and surgery are included.     

On the direct image of intersections in exact homological categories

Given a regular epimorphism f:X?Y in an exact homological category C, and a pair (U,V) of kernel subobjects of X, we show that the quotient (f(U)∩f(V))/f(U∩V) is always abelian. When C is nonpointed, i.e. only exact protomodular, the translation of the previous result is that, given any pair (R,S) of equivalence relations on X, the difference mappingδ:Y/f(R∩S)?Y/(f(R)∩f(S)) has an abelian kernel relation. This last result actually holds true in any exact Mal'cev category. Setting Y=X/T, this result says that the difference mapping determined by the inclusion T∪(R∩S)?(T∪R)∩(T∪S) has an abelian kernel relation, which casts a new light on the congruence distributive property.     

Homology of iterated loop spaces of homogeneous spaces associated with exceptional Lie groups

We study the mod 2 homology of the double and triple loop spaces of homogeneous spaces associated with exceptional Lie groups. The main computational tools are the Serre spectral sequence for fibrations Ωⁿ⁺¹G→Ωⁿ⁺¹(G/H)→ΩⁿH for n=1,2, and the Eilenberg-Moore spectral sequence associated with related fiber squares, which both converge to the same destination space H_∗(Ωⁿ(G/H);F₂). We also develop the generalized Bockstein lemma to determine the higher Bockstein actions.     

The product of two dependent random variables with regularly varying or rapidly varying tails

Let X and Y be two nonnegative and dependent random variables following a generalized Farlie-Gumbel-Morgenstern distribution. In this short note, we study the impact of a dependence structure of X and Y on the tail behavior of XY. We quantify the impact as the limit, as x→∞, of the quotient of Pr(XY>x) and Pr(X^∗Y^∗>x), where X^∗ and Y^∗ are independent random variables identically distributed as X and Y, respectively. We obtain an explicit expression for this limit when X is regularly varying or rapidly varying tailed.     

Properties of the Holmes space

The following properties of the Holmes space H are established:

(i)

H has the Metric Approximation Property (MAP).

(ii)

The w^∗-closure of the set of extreme points of the unit ball BH^∗ of the dual space H^∗ is the whole ball BH^∗.

A family of compact subsets X⊂U of the Urysohn space is described such that the Lipschitz-free space F(X) has a finite-dimensional decomposition and is not complemented in H.     

An “almost” full embedding of the category of graphs into the category of groups

We construct a functor F:Graphs→Groups which is faithful and “almost” full, in the sense that every nontrivial group homomorphism FX→FY is a composition of an inner automorphism of FY and a homomorphism of the form Ff, for a unique map of graphs f:X→Y. When F is composed with the Eilenberg-Mac Lane space construction K(FX,1) we obtain an embedding of the category of graphs into the unpointed homotopy category which is full up to null-homotopic maps.We provide several applications of this construction to localizations (i.e. idempotent functors); we show that the questions:

(1)

Is every orthogonality class reflective?

(2)

Is every orthogonality class a small-orthogonality class?

have the same answers in the category of groups as in the category of graphs. In other words they depend on set theory: (1) is equivalent to weak Vopěnka's principle and (2) to Vopěnka's principle. Additionally, the second question, considered in the homotopy category, is also equivalent to Vopěnka's principle.     

Holomorphic spectral geometry of magnetic Schrödinger operators on Riemann surfaces

We study magnetic Schrödinger operators on line bundles over Riemann surfaces endowed with metrics of constant curvature. We show that for harmonic magnetic fields the spectral geometry of these operators is completely determined by the Bochner Laplacians of the line bundles. Therefore we are led to examine the spectral problem for the Bochner Laplacian ∇^∗∇ of a Hermitian line bundle L with connection ∇ over a Riemann surface S. This spectral problem is analyzed in terms of the natural holomorphic structure on L defined by the Cauchy-Riemann operator associated with ∇. By means of an elliptic chain of line bundles obtained by twisting L with the powers of the canonical bundle we prove that there exists a certain subset of the spectrum σhol(∇^∗∇) such that the eigensections associated with λ∈σhol(∇^∗∇) are given by the holomorphic sections of a certain line bundle of the elliptic chain. For genus p=0,1 we prove that σhol(∇^∗∇) is the whole spectrum, whereas for genus p>1 we get a finite number of eigenvalues.     

Submodule categories of wild representation type

Let Λ be a commutative local uniserial ring with radical factor field k. We consider the category S(Λ) of embeddings of all possible submodules of finitely generated Λ-modules. In case Λ=Z/〈pⁿ〉, where p is a prime, the problem of classifying the objects in S(Λ), up to isomorphism, has been posed by Garrett Birkhoff in 1934. In this paper we assume that Λ has Loewy length at least seven. We show that S(Λ) is controlled k-wild with a single control object I∈S(Λ). It follows that each finite dimensional k-algebra can be realized as a quotient End(X)/End(X)_I of the endomorphism ring of some object X∈S(Λ) modulo the ideal End(X)_I of all maps which factor through a finite direct sum of copies of I.     

Some inverse problems involving conditional expectations

Let (Ω, F, P) be a probability space, let H be a sub-σ-algebra of F, and let Y be positive and H-measurable with E[Y] = 1. We discuss the structure of the convex set CE(Y; H) = {X ∈ pF: Y = E[X|H]} of random variables whose conditional expectation given H is the prescribed Y. Several characterizations of extreme points of CE(Y; H) are obtained. A necessary and sufficient condition is given in order that CE(Y; H) be the closed, convex hull of its extreme points. For the case of finite F we explicitly calculate the extreme points of CE(Y; H), identify pairs of adjacent extreme points, and characterize extreme points of CE(Y; H) ? CE(Z; G), where G is a second sub-σ-algebra of F and Z ∈ pG. When H = σ(Y) and appropriate topological hypotheses hold, extreme points of CE(Y; H) are shown to be in explicit one-to-one correspondence with certain left inverses of Y. Finally, it is shown how the same approach can be applied to the problem of extremal random measures on $\text{R}$₊ with a prescribed compensator, to deduce that the number of extreme points is zero or one.     

Tychonoff expansions with prescribed resolvability properties

The recent literature offers examples, specific and hand-crafted, of Tychonoff spaces (in ZFC) which respond negatively to these questions, due respectively to Ceder and Pearson (1967) [3] and to Comfort and García-Ferreira (2001) [5]: (1) Is every ω-resolvable space maximally resolvable? (2) Is every maximally resolvable space extraresolvable? Now using the method of KID expansion, the authors show that every suitably restricted Tychonoff topological space (X,T) admits a larger Tychonoff topology (that is, an “expansion”) witnessing such failure. Specifically the authors show in ZFC that if (X,T) is a maximally resolvable Tychonoff space with S(X,T)?Δ(X,T)=κ, then (X,T) has Tychonoff expansions U=Ui (1?i?5), with Δ(X,Ui)=Δ(X,T) and S(X,Ui)?Δ(X,Ui), such that (X,Ui) is: (i=1) ω-resolvable but not maximally resolvable; (i=2) [if κ^′ is regular, with S(X,T)?κ^′?κ] τ-resolvable for all τ<κ^′, but not κ^′-resolvable; (i=3) maximally resolvable, but not extraresolvable; (i=4) extraresolvable, but not maximally resolvable; (i=5) maximally resolvable and extraresolvable, but not strongly extraresolvable.