On the zeroth stable A1-homotopy group of a smooth curve

We provide a cohomological interpretation of the zeroth stable ${\mathbb{A}}^1$-homotopy group of a smooth curve over an infinite perfect field. We show that this group is isomorphic to the first Nisnevich (or Zariski) cohomology group of a certain sheaf closely related to the first Milnor–Witt K-theory sheaf. This cohomology group can be computed using an explicit Gersten-type complex. We show that if the base field is algebraically closed then the zeroth stable ${\mathbb{A}}^1$-homotopy group of a smooth curve coincides with the zeroth Suslin homology group that was identified by Suslin and Voevodsky with a relative Picard group. As a consequence we reobtain a version of Suslin's rigidity theorem.     

Many homotopy categories are homotopy categories

We show that any category that is enriched, tensored, and cotensored over the category of compactly generated weak Hausdorff spaces, and that satisfies an additional hypothesis concerning the behavior of colimits of sequences of cofibrations, admits a Quillen closed model structure in which the weak equivalences are the homotopy equivalences. The fibrations are the Hurewicz fibrations and the cofibrations are a subclass of the Hurewicz cofibrations. This result applies to various categories of spaces, unbased or based, categories of prespectra and spectra in the sense of Lewis and May, the categories of L-spectra and S-modules of Elmendorf, Kriz, Mandell and May, and the equivariant analogues of all the afore-mentioned categories.     

Mixing model structures

We prove that if a category has two Quillen closed model structures (W₁,F₁,C₁) and (W₂,F₂,C₂) that satisfy the inclusions W₁⊆W₂ and F₁⊆F₂, then there exists a “mixed model structure” (Wm,Fm,Cm) for which Wm=W₂ and Fm=F₁. This shows that there is a model structure for topological spaces (and other topological categories) for which Wm is the class of weak equivalences and Fm is the class of Hurewicz fibrations. The cofibrant spaces in this model structure are the spaces that have CW homotopy type.     

Splitting monoidal stable model categories

If C is a stable model category with a monoidal product then the set of homotopy classes of self-maps of the unit forms a commutative ring, [S,S]^C. An idempotent e of this ring will split the homotopy category: [X,Y]^C≅e[X,Y]^C⊕(1−e)[X,Y]^C. We prove that provided the localised model structures exist, this splitting of the homotopy category comes from a splitting of the model category, that is, C is Quillen equivalent to L_eSC×L_(1−e)SC and [X,Y]^L_eSC≅e[X,Y]^C. This Quillen equivalence is strong monoidal and is symmetric when the monoidal product of C is.     

Stable model categories are categories of modules

A stable model category is a setting for homotopy theory where the suspension functor is invertible. The prototypical examples are the category of spectra in the sense of stable homotopy theory and the category of unbounded chain complexes of modules over a ring. In this paper we develop methods for deciding when two stable model categories represent ‘the same homotopy theory’. We show that stable model categories with a single compact generator are equivalent to modules over a ring spectrum. More generally stable model categories with a set of generators are characterized as modules over a ‘ring spectrum with several objects’, i.e., as spectrum valued diagram categories. We also prove a Morita theorem which shows how equivalences between module categories over ring spectra can be realized by smashing with a pair of bimodules. Finally, we characterize stable model categories which represent the derived category of a ring. This is a slight generalization of Rickard's work on derived equivalent rings. We also include a proof of the model category equivalence of modules over the Eilenberg-Mac Lane spectrum HR and (unbounded) chain complexes of R-modules for a ring R.     

Iterated homotopy fixed points for the Lubin-Tate spectrum

When G is a profinite group and H and K are closed subgroups, with H normal in K, it is not known, in general, how to form the iterated homotopy fixed point spectrum (ZhH)^hK/H, where Z is a continuous G-spectrum and all group actions are to be continuous. However, we show that, if G=Gn, the extended Morava stabilizer group, and , where is Bousfield localization with respect to Morava K-theory, En is the Lubin-Tate spectrum, and X is any spectrum with trivial Gn-action, then the iterated homotopy fixed point spectrum can always be constructed. Also, we show that is just , extending a result of Devinatz and Hopkins.     

The Balmer spectrum of rational equivariant cohomology theories

The category of rational G-equivariant cohomology theories for a compact Lie group G is the homotopy category of rational G-spectra and therefore tensor-triangulated. We show that its Balmer spectrum is the set of conjugacy classes of closed subgroups of G, with the topology corresponding to the topological poset of [7]. This is used to classify the collections of subgroups arising as the geometric isotropy of finite G-spectra. The ingredients for this classification are (i) the algebraic model of free spectra of the author and B. Shipley [14], (ii) the Localization Theorem of Borel–Hsiang–Quillen [21] and (iii) tom Dieck's calculation of the rational Burnside ring [4].     

Strong cofibrations and fibrations in enriched categories

We define strong cofibrations and fibrations in suitably enriched categories using the relative homotopy extension resp. lifting property. We prove a general pairing result, which for topological spaces specializes to the well-known pushout-product theorem for cofibrations. Strong cofibrations and fibrations give rise to cofibration and fibration categories in the sense of homotopical algebra. We discuss various examples; in particular, we deduce that the category of chain complexes with chain equivalences and the category of categories with equivalences are symmetric monoidal proper closed model categories. Eine überarbeitete Fassung ging am 5. 12. 2001 ein     

Orbifold genera, product formulas and power operations

We generalize the definition of orbifold elliptic genus and introduce orbifold genera of chromatic level h, using h-tuples rather than pairs of commuting elements. We show that our genera are in fact orbifold invariants, and we prove integrality results for them. If the genus arises from an H_∞-map into the Morava-Lubin-Tate theory E_h, then we give a formula expressing the orbifold genus of the symmetric powers of a stably almost complex manifold M in terms of the genus of M itself. Our formula is the p-typical analogue of the Dijkgraaf-Moore-Verlinde-Verlinde formula for the orbifold elliptic genus [R. Dijkgraaf et al., Elliptic genera of symmetric products and second quantized strings Comm. Math. Phys. 185(1) (1997) 197-209]. It depends only on h and not on the genus.     

Rational torus-equivariant stable homotopy I: Calculating groups of stable maps

We construct an abelian category A(G) of sheaves over a category of closed subgroups of the r-torus G and show that it is of finite injective dimension. It can be used as a model for rational G-spectra in the sense that there is a homology theory

     

Stable splittings of surface mapping spaces

We study the homotopy type of mapping spaces from Riemann surfaces to spheres. Our main result is a stable splitting of these spaces into a bouquet of new finite spectra. From this and classical results, one may deduce splittings of the configuration spaces of surfaces.     

Modules over motivic cohomology

For fields of characteristic zero, we show that the homotopy category of modules over the motivic ring spectrum representing motivic cohomology is equivalent to Voevodsky's big category of motives. The proof makes use of some highly structured models for motivic stable homotopy theory, motivic Spanier-Whitehead duality, the homotopy theories of motivic functors and of motivic spaces with transfers as introduced from ground up in this paper. Working with rational coefficients, we extend the equivalence for fields of characteristic zero to all perfect fields by employing the techniques of alterations and homotopy purity in motivic homotopy theory.     

A generalized Conner-Floyd conjecture and the immersion problem for low 2-torsion lens spaces

Let α(d) denote the number of ones in the binary expansion of d. For 1?k?α(d) we prove that the 2(d+α(d)−k)+1-dimensional, 2^k-torsion lens space does not immerse in a Euclidian space of dimension 4d−2α(d) provided certain technical condition holds. The extra hypothesis is easily eliminated in the case k=1 recovering Davis’ strong non-immersion theorem for real projective spaces. For k>1 this is a deeper problem (solved only in part) that requires a close analysis of the interaction between the Brown-Peterson 2-series and its 2^k analogue. The methods are based on a partial generalization of the Brown-Peterson version for the Conner-Floyd conjecture used in this context to detect obstructions for the existence of Euclidian immersions.     

The extended Bloch groups of biquadratic and dihedral number fields

In this paper, we study the Galois action on the extended Bloch groups of biquadratic and dihedral number fields. We prove that if F is a biquadratic number field, then the index $Q_2(F)$ in Browkin and Gangl's formulas on the Brauer–Kuroda relation can only be 1 or 2. This is exactly what Browkin and Gangl predicted in their paper. Moreover we give the explicit criteria for $Q_2(F)=1$ or 2 in terms of the Tate kernels. We also prove that $Q_2(F)=1$ or p for any dihedral extension $F/\mathbb{Q}$ whose Galois group is the dihedral group of order 2p, where p is an odd prime.     

Quadrature formulas on the unit circle with prescribed nodes and maximal domain of validity

In this paper we investigate the Szeg?-Radau and Szeg?-Lobatto quadrature formulas on the unit circle. These are (n+m)-point formulas for which m nodes are fixed in advance, with m=1 and m=2 respectively, and which have a maximal domain of validity in the space of Laurent polynomials. This means that the free parameters (free nodes and positive weights) are chosen such that the quadrature formula is exact for all powers z^j, −p≤j≤p, with p=p(n,m) as large as possible.     

Minimal 2-spheres in a complex projective space

In this paper conformal minimal 2-spheres immersed in a complex projective space are studied by applying Lie theory and moving frames. We give differential equations of Kähler angle and square length of the second fundamental form. By applying these differential equations we give characteristics of conformal minimal 2-spheres of constant Kähler angle and obtain pinching theorems for curvature. We also discuss conformal minimal 2-spheres of constant normal curvature and prove that there does not exist any linearly full minimal 2-sphere immersed in a complex projective space CPn (n>2) with non-positive constant normal curvature. We also prove that a linearly full minimal 2-sphere immersed in a complex projective space CPn (n>2) with constant normal curvature and constant Kähler angle is of constant curvature.