Poisson Traces and D-Modules on Poisson Varieties

To every Poisson algebraic variety X over an algebraically closed field of characteristic zero, we canonically attach a right D-module M(X) on X. If X is affine, solutions of M(X) in the space of algebraic distributions on X are Poisson traces on X, i.e. distributions invariant under Hamiltonian flow. When X has finitely many symplectic leaves, we prove that M(X) is holonomic. Thus, when X is affine and has finitely many symplectic leaves, the space of Poisson traces on X is finite-dimensional. More generally, to any morphism ${\phi : X \to Y}To every Poisson algebraic variety X over an algebraically closed field of characteristic zero, we canonically attach a right D-module M(X) on X. If X is affine, solutions of M(X) in the space of algebraic distributions on X are Poisson traces on X, i.e. distributions invariant under Hamiltonian flow. When X has finitely many symplectic leaves, we prove that M(X) is holonomic. Thus, when X is affine and has finitely many symplectic leaves, the space of Poisson traces on X is finite-dimensional. More generally, to any morphism f: X ? Y{\phi : X \to Y} and any quasicoherent sheaf of Poisson modules N on X, we attach a right D-module M_f(X,N){M_\phi(X,N)} on X, and prove that it is holonomic if X has finitely many symplectic leaves, f{\phi} is finite, and N is coherent.     

A function space model approach to the rational evaluation subgroups

Let f : U → X be a map from a connected nilpotent space U to a connected rational space X. The evaluation subgroup G _*(U, X; f), which is a generalization of the Gottlieb group of X, is investigated. The key device for the study is an explicit Sullivan model for the connected component containing f of the function space of maps from U to X, which is derived from the general theory of such a model due to Brown and Szczarba (Trans Am Math Soc 349, 4931–4951, 1997). In particular, we show that non Gottlieb elements are detected by analyzing a Sullivan model for the map f and by looking at non-triviality of higher order Whitehead products in the homotopy group of X. The Gottlieb triviality of a fibration in the sense of Lupton and Smith (The evaluation subgroup of a fibre inclusion, 2006) is also discussed from the function space model point of view. Moreover, we proceed to consideration of the evaluation subgroup of the fundamental group of a nilpotent space. In consequence, the first Gottlieb group of the total space of each S ¹-bundle over the n-dimensional torus is determined explicitly in the non-rational case.      

The classifying space of a categorical crossed module

Any pointed CW‐complex X has associated a categorical crossed module WX whose homotopy groups coincide with those of the space up to dimension 3. Here we associate WX more closely with the homotopy 3‐type of X. We introduce the nerve of a categorical crossed module L and define its classifying space BL as the geometrical realization of the nerve. Then we prove that there is a map X → BWX inducing isomorphism of the homotopy groups π_i for i ≤ 3. Finally, comparison with other algebraic models of 3‐types is achieved (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Extending local homotopy equivalences

We give an elementary proof of one of tom Dieck’s theorems. The theorem says that iff:X → Y is a local homotopy equivalence in a strong enough sense, thenf is a homotopy equivalence globally. Applications, 1. The base space of any numerable principalG-bundle is of the sme homotopy type as the Borel space of the bundle. 2. The nerve of a numerable coveringU ofX for which all finite intersections are contractible is of the same homotopy type asX.     

The dualizing spectrum of a topological group

To a topological group G, we assign a naive G-spectrum , called the dualizing spectrum of G. When the classifying space BG is finitely dominated, we show that detects Poincaré duality in the sense that BG is a Poincaré duality space if and only if is a homotopy finite spectrum. Secondly, we show that the dualizing spectrum behaves multiplicatively on certain topological group extensions. In proving these results we introduce a new tool: a norm map which is defined for any G and for any naive G-spectrum E. Applications of the dualizing spectrum come in two flavors: (i) applications in the theory of Poincaré duality spaces, and (ii) applications in the theory of group cohomology. On the Poincaré duality space side, we derive a homotopy theoretic solution to a problem posed by Wall which says that in a fibration sequence of fini the total space satisfies Poincaré duality if and only if the base and fiber do. The dualizing spectrum can also be used to give an entirely homotopy theoretic construction of the Spivak fibration of a finitely dominated Poincaré duality space. We also include a new proof of Browder's theorem that every finite H-space satisfies Poincaré duality. In connection with group cohomology, we show how to define a variant of Farrell-Tate cohomology for any topological or discrete group G, with coefficients in any naive equivariant cohomology theory E. When E is connective, and when G admits a subgroup H of finite index such that BH is finitely dominated, we show that this cohomology coincides with the ordinary cohomology of G with coefficients in E in degrees greater than the cohomological dimension of H. In an appendix, we identify the homotopy type of for certain kinds of groups. The class includes all compact Lie groups, torsion free arithmetic groups and Bieri-Eckmann duality groups. Received July 14, 1999 / Revised May 17, 2000 / Published online February 5, 2001     

Rationalization and the Gray index of phantom maps

In this paper we study a homotopy invariant of phantom maps called the Gray index. We give a new interpretation of the Gray index of a phantom map f:X→Y, in terms of the rationalization of X. We use this interpretation, in order to detect phantom maps of a specific Gray index. Finally, we examine the set of phantom maps with infinite Gray index in a tower theoretic way.     

Resolving Extensions of Finitely Presented Systems

In this paper we extend certain central results of zero dimensional systems to higher dimensions. The first main result shows that if (Y,f) is a finitely presented system, then there exists a Smale space (X,F) and a u-resolving factor map π ₊:X→Y. If the finitely presented system is transitive, then we show there is a canonical minimal u-resolving Smale space extension. Additionally, we show that any finite-to-one factor map between transitive finitely presented systems lifts through u-resolving maps to an s-resolving map.     

On homotopy groups of quandle spaces and the quandle homotopy invariant of links

For a quandle X, the quandle space BX is defined, modifying the rack space of Fenn, Rourke and Sanderson (1995) [13], and the quandle homotopy invariant of links is defined in Z[π₂(BX)], modifying the rack homotopy invariant of Fenn, Rourke and Sanderson (1995) [13]. It is known that the cocycle invariants introduced in Carter et al. (2005) [3], Carter et al. (2003) [5], Carter et al. (2001) [6] can be derived from the quandle homotopy invariant.In this paper, we show that, for a finite quandle X, π₂(BX) is finitely generated, and that, for a connected finite quandle X, π₂(BX) is finite. It follows that the space spanned by cocycle invariants for a finite quandle is finitely generated. Further, we calculate π₂(BX) for some concrete quandles. From the calculation, all cocycle invariants for those quandles are concretely presented. Moreover, we show formulas of the quandle homotopy invariant for connected sum of knots and for the mirror image of links.     

Free Group Actions on Spaces Homotopy Equivalent to a Sphere

For infinite-dimensional homotopy space forms X/G with G nontrivial finite cyclic groups, we study the homotopy type of X/G. We show that the Euler class of X/G is not zero if and only if X/G is finitely dominated.     

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.     

Existence of finitely dominatedCW-complexes withG 1(X) = π1(X) and non-vanishing finiteness obstruction

We show for a finite abelian groupG and any element in the image of the Swan homomorphism sw: that it can be realized as the finiteness obstruction of a finitely dominated connectedCW-complexX with fundamental group π₁(X) =G such that π₁(X) is equal to the subgroupG ₁(X) defined by Gottlieb. This is motivated by the observation that anyH-spaceX satisfies π₁(X) =G ₁(X) and still the problem is open whether any finitely dominatedH-space is up to homotopy finite.     

Algebraic geometry of topological spaces I

We use techniques from both real and complex algebraic geometry to study K-theoretic and related invariants of the algebra C(X) of continuous complex-valued functions on a compact Hausdorff topological space X. For example, we prove a parameterized version of a theorem by Joseph Gubeladze; we show that if M is a countable, abelian, cancellative, torsion-free, semi-normal monoid, and X is contractible, then every finitely generated projective module over C(X)[M] is free. The particular case gives a parameterized version of the celebrated theorem proved independently by Daniel Quillen and Andrei Suslin that finitely generated projective modules over a polynomial ring over a field are free. The conjecture of Jonathan Rosenberg which predicts the homotopy invariance of the negative algebraic K-theory of C(X) follows from the particular case . We also give algebraic conditions for a functor from commutative algebras to abelian groups to be homotopy invariant on C ^*-algebras, and for a homology theory of commutative algebras to vanish on C ^*-algebras. These criteria have numerous applications. For example, the vanishing criterion applied to nil K-theory implies that commutative C ^*-algebras are K-regular. As another application, we show that the familiar formulas of Hochschild–Kostant–Rosenberg and Loday–Quillen for the algebraic Hochschild and cyclic homology of the coordinate ring of a smooth algebraic variety remain valid for the algebraic Hochschild and cyclic homology of C(X). Applications to the conjectures of Beĭlinson-Soulé and Farrell–Jones are also given.     

Calculus I: The first derivative of pseudoisotopy theory

Let P be the (stable, smooth) pseudoisotopy space of the space X. For any map Y: YX of spaces, we identify the homotopy type of the fiber of P(f): P(f) P(f) in a stable range, roughly twice the connectivity of the map YX. We establish some language for discussing and manipulating such stable-range relative calculations for any homotopy functor. The theorem about P has a corollary about Waldhausen's A. Dedicated to Alexander GrothendieckResearch supported in part by NSF grant DMS-8806444 and by a Sloan Fellowship. Research at MSRI supported in part by NSF grant DMS-8505550.     

Algebraicity of analytic maps to a hyperbolic variety

Let X be a complex algebraic variety. We say that X is Borel hyperbolic if, for every finite type reduced scheme S over the complex numbers, every holomorphic map from S to X is algebraic. We use a transcendental specialization technique to prove that X is Borel hyperbolic if and only if, for every smooth affine complex algebraic curve C, every holomorphic map from C to X is algebraic. We use the latter result to prove that Borel hyperbolicity shares many common features with other notions of hyperbolicity such as Kobayashi hyperbolicity.     

Estimation of the number of fixed points of map extensions

Letf:(X,A)→(X,A) be an extension of a given map ψ:A→A, where (X,A) is a pair of compact polyhedra. We shall introduce a special Nielsen number,SN(f|ψ), which is a lower bound for the number of fixed points onX-A for all extensions in the homotopy class off. It is shown that for many space pairs this lower bound is the best possible one, and that it can be realized without the by-passing condition.     

On the homotopy type of p-completions of infra-nilmanifolds

Infra-nilmanifolds are compact K(G,1)-manifolds with G a torsion-free, finitely generated, virtually nilpotent group. Motivated by previous results of various authors on p-completions of K(G,1)-spaces with G a finite or a nilpotent group, we study the homotopy type of p-completions of infra-nilmanifolds, for any prime p. We prove that the p-completion of an infra-nilmanifold is a virtually nilpotent space which is either aspherical or has infinitely many nonzero homotopy groups. The same is true for p-localization. Moreover, we show by means of examples that rationalizations of infra-nilmanifolds may be elliptic or hyperbolic. Received: 12 December 2001 / Published online: 5 September 2002     

Polyhedra dominating finitely many different homotopy types

In 1968 K. Borsuk asked: Is it true that every finite polyhedron dominates only finitely many different shapes? In this question the notions of shape and shape domination can be replaced by the notions of homotopy type and homotopy domination.We obtained earlier a negative answer to the Borsuk question and next results that the examples of such polyhedra are not rare. In particular, there exist polyhedra with nilpotent fundamental groups dominating infinitely many different homotopy types. On the other hand, we proved that every polyhedron with finite fundamental group dominates only finitely many different homotopy types. Here we obtain next positive results that the same is true for some classes of polyhedra with Abelian fundamental groups and for nilpotent polyhedra. Therefore we also get that every finitely generated, nilpotent torsion-free group has only finitely many r-images up to isomorphism.     

<Emphasis Type="Italic">A</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript> theory,L.S. category,and strong category

Relations between category and strong category are studied. The notion of a homotopy coalgebra of order r over the Ganea comonad is introduced. It is shown that cat(X) =Cat(X) holds if a finite 1-connected complex X carries such a structure with r sufficiently large.     

On the location of fixed points on pairs of spaces

Let f: (X, A)→(X, A) be an admissible selfmap of a pair of metrizable ANR's. A Nielsen number of the complement Ñ(f; X, A) and a Nielsen number of the boundary ñ(f; X, A) are defined. Ñ(f; X, A) is a lower bound for the number of fixed points on C1(X - A) for all maps in the homotopy class of f. It is usually possible to homotope f to a map which is fixed point free on Bd A, but maps in the homotopy class of f which have a minimal fixed point set on X must have at least ñ(f; X, A) fixed points on Bd A. It is shown that for many pairs of compact polyhedra these lower bounds are the best possible ones, as there exists a map homotopic to f with a minimal fixed point set on X which has exactly Ñ(f; X - A) fixed points on C1(X−A) and ñ(f; X, A) fixed points on Bd A. These results, which make the location of fixed points on pairs of spaces more precise, sharpen previous ones which show that the relative Nielsen number N(f; X, A) is the minimum number of fixed points on all of X for selfmaps of (X, A), as well as results which use Lefschetz fixed point theory to find sufficient conditions for the existence of one fixed point on C1(X−A).