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     

on the o-minimal LS-category

We introduce the o-minimal LS-category of definable sets in o-minimal expansions of ordered fields and we establish a relation with the semialgebraic and the classical one. We also study the o-minimal LS-category of definable groups. Along the way, we show that two definably connected definably compact definable groups G and H are definable homotopy equivalent if and only if L(G) and L(H) are homotopy equivalent, where L is the functor which associates to each definable group its corresponding Lie group via Pillay’s conjecture.     

On equivariant Kasparov theory and Spanier-Whitehead duality

Suppose thatG is a second countable compact Lie group and thatA andB are commutativeG-C*-algebras. Then the Kasparov groupKK _* ^G (A, B) is a bifunctor onG-spaces. It is computed here in terms of equivariant stable homotopy theory. This result is a consequence of a more general study of equivariant Spanier-Whitehead duality and uses in an essential way the extension of the Kasparov machinery to the setting of -G-C*-algebras. As a consequence, we show that if (X, x ₀) is a based separable compact metricG-ENR (such as a smooth compactG-manifold) and (Y, y ₀) is a based countableG-CW-complex then there is a natural isomorphism

Spherical orbit closures in simple projective spaces and their normalizations

Let G be a simply connected semisimple algebraic group over an algebraically closed field k of characteristic 0 and let V be a rational simple G-module. If G/H ⊂ P(V) is a spherical orbit and if X = [`(G/H)] X = \overline {G/H} is its closure, then we describe the orbits of X and those of its normalization [(X)\tilde] \tilde{X} . If, moreover, the wonderful completion of G/H is strict, then we give necessary and sufficient combinatorial conditions so that the normalization morphism [(X)\tilde] ? X \tilde{X} \to X is a homeomorphism. Such conditions are trivially fulfilled if G is simply laced or if H is a symmetric subgroup.     

Higher homotopy of groups definable in o-minimal structures

It is known that a definably compact group G is an extension of a compact Lie group L by a divisible torsion-free normal subgroup. We show that the o-minimal higher homotopy groups of G are isomorphic to the corresponding higher homotopy groups of L. As a consequence, we obtain that all abelian definably compact groups of a given dimension are definably homotopy equivalent, and that their universal covers are contractible.     

Soluble minimal non-(finite-by-Baer)-groups

Let \mathfrakX{\mathfrak{X}} be a class of groups. A group G is called a minimal non- \mathfrakX{\mathfrak{X}}-group if it is not an \mathfrakX{\mathfrak{X}}-group but all of whose proper subgroups are \mathfrakX{\mathfrak{X}}-groups. In [16], Xu proved that if G is a soluble minimal non-Baer-group, then G/G ^′′ is a minimal non-nilpotent-group which possesses a maximal subgroup. In the present note, we prove that if G is a soluble minimal non-(finite-by-Baer)-group, then for all integer n ≥ 2, G/γ _n (G′) is a minimal non-(finite-by-abelian)-group.     

Disconnected Equivariant Rational Homotopy Theory

We present a variant of the disconnected equivariant rational homotopy theory to complete the result shown in [8]. For a finite group G let O(G) be the category of its canonical orbits. We prove that the category O(G)-DGA _Q of O(G)S-complete differential graded algebras over the rationals is a closed model category, where S runs over all O(G)-sets. Then, by means of the equivariant KS-minimal models, we show that the homotopy category of O(G)-DGA _Q is equivalent to the rational homotopy category of G-nilpotent disconnected simplicial sets provided G is a finite Hamiltonian group.     

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     

Upper bounds on the linear chromatic number of a graph

A proper vertex coloring of a graph G is linear if the graph induced by the vertices of any two color classes is a union of vertex-disjoint paths. The linear chromatic number of G is the smallest number of colors in a linear coloring of G.Let G be a graph with maximum degree Δ(G). In this paper we prove the following results: (1) ; (2) if Δ(G)≤4; (3) if Δ(G)≤5; (4) if G is planar and Δ(G)≥52.     

Product of nilpotent subgroups

We will say that a subgroup X of G satisfies property C in G if C_G(X?X^g)\leqq X?X^g{\rm C}_{G}(X\cap X^{{g}})\leqq X\cap X^{{g}} for all g ? G{g}\in G. We obtain that if X is a nilpotent subgroup satisfying property C in G, then XF(G) is nilpotent. As consequence it follows that if N\triangleleft GN\triangleleft G is nilpotent and X is a nilpotent subgroup of G then C_G(N?X)\leqq XC_G(N\cap X)\leqq X implies that NX is nilpotent.¶We investigate the relationship between the maximal nilpotent subgroups satisfying property C and the nilpotent injectors in a finite group.     

Duality of Fourier type with respect to locally compact Abelian groups

It is shown that a Banach space X has Fourier type p with respect to a locally compact abelian group G if and only if the dual space X′ has Fourier type p with respect to G if and only if X has Fourier type p with respect to the dual group of G. This extends previously known results for the classical groups and the Cantor group to the setting of general locally compact abelian groups. Supported by DFG grant Hi 584/2-2. Partially supported by a DAAD-grant A/02/46571.     

Ohba’s conjecture for graphs with independence number five

Ohba has conjectured that if G is a k-chromatic graph with at most 2k+1 vertices, then the list chromatic number or choosability of G is equal to its chromatic number χ(G), which is k. It is known that this holds if G has independence number at most three. It is proved here that it holds if G has independence number at most five. In particular, and equivalently, it holds if G is a complete k-partite graph and each part has at most five vertices.     

Fibred sites and stack cohomology

The usual notion of the site associated to a stack is expanded to a definition to a site fibred over a presheaf of categories A on a site . If the presheaf of categories is a presheaf of groupoids G, then the associated homotopy theory is Quillen equivalant to the homotopy theory of simplicial presheaves over BG, and so the homotopy theory for the fibred site is an invariant of the homotopy type of G. Similar homotopy invariance results obtain for presheaves of spectra and presheaves of symmetric spectra on . In particular, stack cohomology can be calculated on the fibred site for any representing presheaf of groupoids within a fixed homotopy type.     

Triangulated functors,homological functors,tilts, and lifts

 Let T be a triangulated category and let X be an object of T. This paper studies the questions: Does there exist a triangulated functor G : D(ℤ)  T with G(ℤ)≌X? Does there exist a triangulated functor H : T  D(ℤ) with h⁰ ⊚ H ⋍ Hom_T (X, −)? To what extent are G and H unique? One spin off is a proof that the homotopy category of spectra is not the stable category of any Frobenius category with set indexed coproducts. Received: 8 March 2002 / Revised version: 18 October 2002 Published online: 14 February 2003 Mathematics Subject Classification (2000): 18E30, 55U35     

Homotopy equivalences of p-completed classifying spaces of finite groups

We study homotopy equivalences of p-completions of classifying spaces of finite groups. To each finite group G and each prime p, we associate a finite category ℒ _p ^c (G) with the following properties. Two p-completed classifying spaces BG _p ^∧ and BG’ _p ^∧ have the same homotopy type if and only if the associated categories ℒ _p ^c (G) and ℒ _p ^c (G’) are equivalent. Furthermore, the topological monoid Aut(BG _p ^∧) of self equivalences is determined by the self equivalences of the associated category ℒ _p ^c (G). Oblatum 5-VII-2001 & 28-VIII-2002?Published online: 8 November 2002 RID="*" ID="*"C. Broto is partially supported by DGICYT grant PB97–0203. RID="**" ID="**"R. Levi is partially supported by EPSRC grant GR/M7831. RID="***" ID="***"B. Oliver is partially supported by UMR 7539 of the CNRS.     

On a conjecture of ōshima

The set of homotopy classes of self maps of a compact, connected Lie group G is a group by the pointwise multiplication which we denote by H(G), and it is known to be nilpotent. ōshima [H. ōshima, Self homotopy group of the exceptional Lie group G₂, J. Math. Kyoto Univ. 40 (1) (2000) 177-184] conjectured: if G is simple, then H(G) is nilpotent of class ?rankG. We show this is true for PU(p) which is the first high rank example.     

Counterexamples to a fiber theorem

We exhibit a counterexample to a fiber theorem stated by F. Fumagalli in [Francesco Fumagalli, On the homotopy type of the Quillen complex of finite soluble groups, J. Algebra 283 (2) (2005) 639–654] and show how it affects the rest of Fumagalli's paper. As a consequence, whether the poset A_p(G) is homotopy equivalent to a wedge of spheres for any finite solvable group G seems to remain an open question.     

‐subgroups in the space Cremona group

We prove that if X is a rationally connected threefold and G is a p‐subgroup in the group of birational selfmaps of X, then G is an abelian group generated by at most 3 elements provided that . We also prove a similar result for under an assumption that G acts on a (Gorenstein) G‐Fano threefold, and show that the same holds for under an assumption that G acts on a G‐Mori fiber space.     

Equivariant Kasparov Theory and Generalized Homomorphisms

Let G be a locally compact group. We describe elements of KK ^G (A, B) by equivariant homomorphisms, following Cuntz's treatment in the non-equivariant case. This yields another proof for the universal property of KK ^G : It is the universal split exact stable homotopy functor. To describe a Kasparov triple (, , F) for A, B by an equivariant homomorphism, we have to arrange for the Fredholm operator F to be equivariant. This can be done if A is of the form ; and more generally if the group action on A is proper in the sense of Exel and Rieffel.