The derived category of quasi-coherent sheaves and axiomatic stable homotopy

We prove in this paper that for a quasi-compact and semi-separated (nonnecessarily noetherian) scheme X, the derived category of quasi-coherent sheaves over X, D(Aqc(X)), is a stable homotopy category in the sense of Hovey, Palmieri and Strickland, answering a question posed by Strickland. Moreover we show that it is unital and algebraic. We also prove that for a noetherian semi-separated formal scheme X, its derived category of sheaves of modules with quasi-coherent torsion homologies Dqct(X) is a stable homotopy category. It is algebraic but if the formal scheme is not a usual scheme, it is not unital, therefore its abstract nature differs essentially from that of the derived category Dqc(X) (which is equivalent to D(Aqc(X))) in the case of a usual scheme.     

Fixed points on model solvmanifold pairs

In this paper we define model solvmanifold pairs and their diagonal type selfmaps in the tradition of Heath and Keppelmann. We derive an explicit formula for computing the relative Nielsen number N(F;X,A) on these spaces and selfmaps F:(X,A)→(X,A). We find that model solvmanifold pairs often exhibit interesting Schirmer theory, meaning N(F;X,A)>max{N(F),N(F|_A)}.     

Orbits and invariants of visible group actions

We consider actions G?×?X?→?X of the affine, algebraic group G on the irreducible, affine, variety X. If [k[X] ^G ]?=?[k[X]] ^G we call the action visible. Here [A] denotes the quotient field of the integral domain A. If the action is not visible we construct a G-invariant, birational morphism φ: Z?→?X such that G?×?Z?→?Z is a visible action. We use this to obtain visible open subsets U of X. We also discuss visibility in the presence of other desirable properties: What if G?×?X?→?X is stable? What if there is a semi-invariant f ∈ k[X] such that G?×?X _f ?→?X _f is visible? What if X is locally factorial? What if G is reductive?     

A discrete homotopy theory for binary reflexive structures

We present a simple combinatorial construction of a sequence of functors σ_k from the category of pointed binary reflexive structures to the category of groups. We prove that if the relational structure is a poset P then the groups are (naturally) isomorphic to the homotopy groups of P when viewed as a topological space with the topology of ideals, or equivalently, to the homotopy groups of the simplicial complex associated to P. We deduce that the group σ_k(X,x₀) of the pointed structure (X,x₀) is (naturally) isomorphic to the kth homotopy group of the simplicial complex of simplices of X, i.e. those subsets of X which are the homomorphic image of a finite totally ordered set.     

Fixed point property on symmetric products

For a metric continuum X, let Fn(X)={A⊂X:A is nonempty and has at most n points}. In this paper we show a continuum X such that F₂(X) has the fixed point property while X does not have it.     

Classifying Hilbert bundles

A Hilbert bundle (p, B, X) is a type of fibre space p:B → X such that each fibre p^?1(x) is a Hilbert space. However, p^?1(x) may vary in dimension as x varies in X. We generalize the classical homotopy classification theory of vector bundles to a “homotopy” classification of certain Hilbert bundles. An (m, n)-bundle over the pair (X, A) is a Hilbert bundle (p, B, X) such that the dimension of p^?1(x) is m for x in A and n otherwise. The main result here is that if A is a compact set lying in the “edge” of the metric space X (e.g. if X is a topological manifold and A is a compact subset of the boundary of X), then the problem of classifying (m, n)-bundles over (X, A) reduces to a problem in the classical theory of vector bundles. In particular, we show there is a one-to-one correspondence between the members of the orbit set, [A, G_m($\text{C}$ⁿ)]/[X, U(n)] ¦ A, and the isomorphism classes of (m, n)-bundles over (X, A) which are trivial over X, A.     

A counter-example to a conjecture of Félix

If X is a simply connected space of finite type, then the rational homotopy groups of the based loop space of X possess the structure of a graded Lie algebra, denoted L_X. The radical of L_X, which is an important rational homotopy invariant of X, is of finite total dimension if the Lusternik-Schnirelmann category of X is finite.Let X be a simply connected space with finite Lusternik-Schnirelmann category. If dimL_X<∞, i.e., if X is elliptic, then L_X is its own radical, and therefore the total dimension of the radical of L_X in odd degrees is less than or equal to its total dimension in even degrees (Friedlander and Halperin (1979) [8]). Félix conjectured that this inequality should hold for all simply connected spaces with finite Lusternik-Schnirelmann category.We prove Félix’s conjecture in some interesting special cases, then provide a counter-example to the general case.     

Existence and asymptotic behavior of a functional differential equation in Banach space

Existence and asymptotic behavior of solutions are given for the equation u′(t) = ?A(t)u(t) + F(t,u_t) (t ? 0) and u₀ = ? ? C([?r,0]; X)  C. The space X is a Banach space; the family $\text{{A(t) ¦ 0 ? t ? T}}$ of unbounded linear operators defined on D(A) ? X → X generates a linear evolution system and F: C → X is continuous with respect to a fractional power of A(t₀) for some t₀ ? [0, T].     

Ganea and Whitehead definitions for the tangential Lusternik-Schnirelmann category of foliations

This work solves the problem of elaborating Ganea and Whitehead definitions for the tangential category of a foliated manifold. We develop these two notions in the category S-Top of stratified spaces, that are topological spaces X endowed with a partition F and compare them to a third invariant defined by using open sets. More precisely, these definitions apply to an element (X,F) of S-Top together with a class A of subsets of X; they are similar to invariants introduced by M. Clapp and D. Puppe.If (X,F)∈S-Top, we define a transverse subset as a subspace A of X such that the intersection S∩A is at most countable for any S∈F. Then we define the Whitehead and Ganea LS-categories of the stratified space by taking the infimum along the transverse subsets. When we have a closed manifold, endowed with a C¹-foliation, the three previous definitions, with A the class of transverse subsets, coincide with the tangential category and are homotopical invariants.     

Realization of 2-solvable nilpotent groups as groups of classes of homotopy self-equivalences

Let X be a 1-connected CW-complex of finite type and ε_?(X) be the group of homotopy classes of self-equivalences of X which induce the identity on homotopy groups. In this paper, we prove that every finitely generated 2-solvable rational nilpotent group is realizable as ε_?(X) where X is the rationalization of a 1-connected CW-complex of finite type.     

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.     

The variety of the asymptotic values of a real polynomial etale map

A polynomial map F: R² → R² is said to satisfy the Jacobian condition if ∀(X, Y)ϵ R², J(F)(X, Y) ≠ 0. The real Jacobian conjecture was the assertion that such a map is a global diffeomorphism. Recently the conjecture was shown to be false by S. Pinchuk. According to a theorem of J. Hadamard any counterexample to the conjecture must have asymptotic values. We give the structure of the variety of all the asymptotic values of a polynomial map F: R² → R² that satisfies the Jacobian condition. We prove that the study of the asymptotic values of such maps can be reduced to those maps that have only X- or Y-finite asymptotic values. We prove that a Y-finite asymptotic value can be realized by F along a rational curve of the type (X^{− k}, A₀ + A₁ X + … + A_{N − 1} X^{N − 1} + YX^N), where X → 0, Y is fixed and K, N > 0 are integers. More precisely we prove that the coordinate polynomials P(U, V) of F(U, V) satisfy finitely many asymptotic identities, namely, identities of the following type, P(X^{− k}, A₀ + A₁ X + … + A_{N − 1} X^{N − 1} + YX^N) = A(X, Y)ϵ R[X, Y], which ‘capture’ the whole set of asymptotic values of F.     

The Cartesian product of a compactum and a space is a bifunctor in shape

In 2003 the author has associated with every cofinite inverse system of compact Hausdorff spaces X with limit X and every simplicial complex K (possibly infinite) with geometric realization P=|K| a resolution R(X,K) of X×P, which consists of paracompact spaces. If X consists of compact polyhedra, then R(X,K) consists of spaces having the homotopy type of polyhedra. In two subsequent papers the author proved that R(X,K) is a covariant functor in each of its variables X and K. In the present paper it is proved that R(X,K) is a bifunctor. Using this result, it is proved that the Cartesian product X×Z of a compact Hausdorff space X and a topological space Z is a bifunctor SSh(Cpt)×Sh(Top)→Sh(Top) from the product category of the strong shape category of compact Hausdorff spaces SSh(Cpt) and the shape category Sh(Top) of topological spaces to the category Sh(Top). This holds in spite of the fact that X×Z need not be a direct product in Sh(Top).     

On additive polynomials and certain maximal curves

We show that a maximal curve over F_q² given by an equation A(X)=F(Y), where A(X)∈F_q²[X] is additive and separable and where F(Y)∈F_q²[Y] has degree m prime to the characteristic p, is such that all roots of A(X) belong to F_q². In the particular case where F(Y)=Y^m, we show that the degree m is a divisor of q+1.     

A remark on “homotofp fibrations”

We define a category Top^F of homotopy fibrations with fibre F (or rather maps with homotopy fibre F) and show that this category is closed under certain colimits and homotopy colimits. It follows that the geometric realization of a semisimplicial object in Top^F is again in Top^F. As a corollary we show that for a homotopy everything H-space A_*(i.e. a (special) -space in the sense of G. Segal (s.[9],[10])) with homotopy inverse the loop space of the classifying space of A_* is homotopy equivalent (not only weakly s. [9],[iO])to A₁ even without assuming that all spaces involved have the homotopy type of CW-complexes (compare [8]).     

An operadic model for a mapping space and its associated spectral sequence

Let X and Y be simplicial sets and K a field. In [B. Fresse, Derived division functors and mapping spaces, 2002, Preprint arXiv:math.At/0208091], Fresse has constructed an algebra model over an E_∞K-operad E for the mapping space F(X,Y), whose source X is finite, provided the homotopy groups of the target Y are finite. In this paper, we show that if the underlying field K is the closure of the finite field F_p and the given mapping space is connected, then the finiteness assumption of the homotopy group of Y can be dropped in constructing the E-algebra model. Moreover, we give a spectral sequence converging to the cohomology of F(X,Y) with coefficients in , whose E₂-term is expressed via Lannes’ division functor in the category of unstable -algebra over the Steenrod algebra.     

On the derived category of a regular toric scheme

Let X be a quasi-compact scheme, equipped with an open covering by affine schemes U _σ = Spec A ^σ . A quasi-coherent sheaf on X gives rise, by taking sections over the U _σ , to a diagram of modules over the coordinate rings A ^σ , indexed by the intersection poset Σ of the covering. If X is a regular toric scheme over an arbitrary commutative ring, we prove that the unbounded derived category of quasi-coherent sheaves on X can be obtained from a category of Σ^op-diagrams of chain complexes of modules by inverting maps which induce homology isomorphisms on hyper-derived inverse limits. Moreover, we show that there is a finite set of weak generators, one for each cone in the fan Σ. The approach taken uses the machinery of Bousfield–Hirschhorn colocalisation of model categories. The first step is to characterise colocal objects; these turn out to be homotopy sheaves in the sense that chain complexes over different open sets U _σ agree on intersections up to quasi-isomorphism. In a second step it is shown that the homotopy category of homotopy sheaves is equivalent to the derived category of X.     

Splitting manifolds as M × R where M has a k-fold end structure

Let X be a locally finite simplicial complex of dimension n, n? 5, equipped with a k-fold end structure [4] and consider a piecewise linear (n + 1)-dimensional manifold M that is proper homotopy equivalent to X × R by F:M → X × R, where R is the set of real numbers. The question arises as to whether or not the manifold M can be split, i.e., written as M = N × R where N is a n-manifold and where there is a proper homotopy between F and (p₁ ° F₀) × id:N × R → X × R, preserving the natural (k+1)-fold end structure, where F₀ is F|_N and p₁ is the projection X × R → X. Of particular significance is the fact that X is noncompact. When the construction of such splittings is attempted, algebraic obstructions arise, which vanish if and only if the construction can be completed. This paper develops such an obstruction theory by utilizing methods of L.C. Siebenmann and the k-fold end structures of F. Waldhausen.     

Strongly e-movable convergence and spaces of ANR's

Let X be a metric space and let ANR(X) denote the hyperspace of all compact ANR's in X. This paper introduces a notion of a strongly e-movable convergence for sequences in ANR(X) and proves several characterizations of strongly e-movable convergence. For a (complete) separable metric space X we show that ANR(X) with the topology induced by strongly e-movable convergence can be metrized as a (complete) separable metric space. Moreover, if X is a finite-dimensional compactum, then strongly e-movable convergence induces on ANR(X) the same topology as that induced by Borsuk's homotopy metric.For a separable Q-manifold M, ANR(M) is locally arcwise connected and A, B ? ANR(M) can be joined by an arc in ANR(M) iff there is a simple homotopy equivalence ?: A → B homotopic to the inclusion of A into M.     

Realizing modules over the homology of a DGA

Let A be a DGA over a field and X a module over H_∗(A). Fix an A_∞-structure on H_∗(A) making it quasi-isomorphic to A. We construct an equivalence of categories between A_n+1-module structures on X and length n Postnikov systems in the derived category of A-modules based on the bar resolution of X. This implies that quasi-isomorphism classes of A_n-structures on X are in bijective correspondence with weak equivalence classes of rigidifications of the first n terms of the bar resolution of X to a complex of A-modules. The above equivalences of categories are compatible for different values of n. This implies that two obstruction theories for realizing X as the homology of an A-module coincide.