On fibering and splitting of 5-manifolds over the circle

Our main result is a generalization of Cappell's 5-dimensional splitting theorem. As an application, we analyze, up to internal s-cobordism, the smoothable splitting and fibering problems for certain 5-manifolds mapping to the circle. For example, these maps may have homotopy fibers which are in the class of finite connected sums of certain geometric 4-manifolds. Most of these homotopy fibers have non-vanishing second mod 2 homology and have fundamental groups of exponential growth, which are not known to be tractable by Freedman–Quinn topological surgery. Indeed, our key technique is topological cobordism, which may not be the trace of surgeries.     

Homotopy of gauge groups over non-simply-connected five-dimensional manifolds

Both the gauge groups and 5-manifolds are important in physics and mathematics. In this paper,we combine them to study the homotopy aspects of gauge groups over 5-manifolds. For principal bundles over non-simply connected oriented closed 5-manifolds of a certain type, we prove various homotopy decompositions of their gauge groups according to different geometric structures on the manifolds, and give the partial solution to the classification of the gauge groups. As applications, we estimate the homotopy exponents of their gauge groups, and show periodicity results of the homotopy groups of gauge groups analogous to the Bott periodicity.Our treatments here are also very effective for rational gauge groups in the general context, and applicable for higher dimensional manifolds.     

Dual decompositions of 4-manifolds III: s-cobordisms

The main result is that an s-cobordism (topological or smooth) of 4-manifolds has a product structure outside a ``core' sub-s-cobordism. These cores are arranged to have quite a bit of structure, for example they are smooth and abstractly (forgetting boundary structure) diffeomorphic to a standard neighborhood of a 1-complex. The decomposition is highly nonunique so cannot be used to define an invariant, but it shows that the topological s-cobordism question reduces to the core case. The simply-connected version of the decomposition (with 1-complex a point) is due to Curtis, Freedman, Hsiang and Stong. Controlled surgery is used to reduce topological triviality of core s-cobordisms to a question about controlled homotopy equivalence of 4-manifolds. There are speculations about further reductions. The decompositions on the ends of the s-cobordism are ``dual decompositions' with homotopically-controlled handle structures, and the main result is an application of earlier papers in the series.

     

Splitting and conjoining objects in monotopological categories

Power-sets are defined for any concrete category (over Set) with finite concrete products, and their structure described for monotopological categories. These sets are used to define the notions of splitting object and of conjoining object. Characterizations of the existence of these objects in monotopological categories are given. It is proved that no proper monotopological category can be concretely cartesian closed. Most well-known monotopological categories with splitting objects are topological or are c-categories, but it is shown that there are many proper monotopological categories which are not c-categories, and yet have splitting objects, and may even be cartesian closed. One of the characterizations of the existence of splitting objects is used to prove that a monotopological category with splitting objects is cartesian closed iff the largest initial completion in which it is epireflective is cartesian closed iff its MacNeille completion is cartesian closed.     

Almost Complex Structures Which Are Compatible with Kähler or Symplectic Structures

In this note we prove that half of all homotopy classes of almost complex structures on M is not compatible with any symplectic structure for a certain class of oriented compact 4-manifolds M. In particular, half of all homotopy classes of almost complex structures on an oriented 4-manifold is not compatible to any Kähler structure.     

Topological automorphic forms on U(1, 1)

The homotopy type and homotopy groups of some spectra TAF _GU of topological automorphic forms associated to a unitary similitude group GU of type (1, 1) are explicitly described in quasi-split cases. The spectrum TAF _GU is shown to be closely related to the spectrum TMF in these cases, and homotopy groups of some of these spectra are explicitly computed.     

Isotopy invariants for smooth tori in 4-manifolds

In this paper we define invariants under smooth isotopy for certain two-dimensional knots using some refined Cerf theory. One of the invariants is the knot type of some classical knot generalizing the string number of closed braids. The other invariant is a generalization of the unique invariant of degree 1 for classical knots in 3-manifolds. Possibly, these invariants can be used to distinguish smooth embeddings of tori in some 4-manifolds but which are equivalent as topological embeddings.     

A stable approach to the equivariant Hopf theorem

Let G be a finite group. For semi-free G-manifolds which are oriented in the sense of Waner [S. Waner, Equivariant RO(G)-graded bordism theories, Topology and its Applications 17 (1984) 1-26], the homotopy classes of G-equivariant maps into a G-sphere are described in terms of their degrees, and the degrees occurring are characterised in terms of congruences. This is first shown to be a stable problem, and then solved using methods of equivariant stable homotopy theory with respect to a semi-free G-universe.     

Combinatorics & Topology of Stratifications of The Space of Monic Polynomials With Real Coefficients

We study the stratification of the space of monic polynomials with real coefficients according to the number and multiplicities of real zeros. In the first part, for each of these strata we provide a purely combinatorial chain complex calculating (co)homology of its one-point compactification and describe the homotopy type by order complexes of a class of posets of compositions. In the second part, we determine the homotopy type of the one-point compactification of the space of monic polynomials of fixed degree which have only real roots (i.e., hyperbolic polynomials) and at least one root is of multiplicity k. More generally, we describe the homotopy type of the one-point compactification of strata in the boundary of the set of hyperbolic polynomials, that are defined via certain restrictions on root multiplicities, by order complexes of posets of compositions. In general, the methods are combinatorial and the topological problems are mostly reduced to the study of partially ordered sets.     

Seiberg-Witten invariants, the topological degree and wall crossing formula

Following S. Bauer and M. Furuta we investigate finite dimensional approximations of a monopole map in the case b ₁ = 0. We define a certain topological degree which is exactly equal to the Seiberg-Witten invariant. Using homotopy invariance of the topological degree a simple proof of the wall crossing formula is derived.     

An exotic involution of S4

CAPPELL and Shaneson [1] construct a family of smooth 4-manifolds which are simple homotopy equivalent to real projective 4-space RP⁴, but not even smoothly h- cobordant to RP⁴. (It is possible they are homeomorphic to RP⁴.) It is natural to ask whether their double covers are S⁴ or not.     

Triple Brackets and Lax Morphism Categories

Various aspects of the traditional homotopy theory of topological spaces may be developed in an arbitrary 2-category C with zeros. In particular certain secondary composition operations called box brackets recently have been defined for C; these are similar to, but extend, the familiar Toda brackets in the topological case. In this paper we introduce further the notion of a suspension functor in C and explore the ramifications of relativizing the theory in terms of the associated lax morphism category of C, denoted mC. Four operations associated to a 3-box diagram are introduced and relations among them are clarified. The results and insights obtained, while by nature somewhat technical, yield effective and efficient techniques for computing many operations of Toda bracket type. We illustrate by recording some computations from the homotopy groups of spheres. Also the properties of a new operation, the 2-sided matrix Toda bracket, are explored.     

Noncomplex smooth 4-manifolds with genus-2 Lefschetz fibrations

We construct noncomplex smooth 4-manifolds which admit genus-2 Lefschetz fibrations over . The fibrations are necessarily hyperelliptic, and the resulting 4-manifolds are not even homotopy equivalent to complex surfaces. Furthermore, these examples show that fiber sums of holomorphic Lefschetz fibrations do not necessarily admit complex structures.

     

Hausdorff Dimension and Limits of Kleinian Groups

In this paper we prove that if M is a compact, hyperbolizable 3-manifold, which is not a handlebody, then the Hausdorff dimension of the limit set is continuous in the strong topology on the space of marked hyperbolic 3-manifolds homotopy equivalent to M. We similarly observe that for any compact hyperbolizable 3-manifold M (including a handlebody), the bottom of the spectrum of the Laplacian gives a continuous function in the strong topology on the space of topologically tame hyperbolic 3-manifolds homotopy equivalent to M. Submitted: January 1998.     

Homotopy simplicial faces and the homology of realizations of simplicial topological spaces

The notion of a differential module with homotopy simplicial faces is introduced, which is a homotopy analog of the notion of a differential module with simplicial faces. The homotopy invariance of the structure of a differential module with homotopy simplicial faces is proved. Relationships between the construction of a differential module with homotopy simplicial faces and the theories of A _∞-algebras and D _∞-differential modules are found. Applications of the method of homotopy simplicial faces to describing the homology of realizations of simplicial topological spaces are presented.     

Bredon homology and ramified covering G-maps

Let G be a finite group. The objective of this paper is twofold. First we prove that the cellular Bredon homology groups with coefficients in an arbitrary coefficient system M are isomorphic to the homotopy groups of certain topological abelian group. And second, we study ramified covering G-maps of simplicial sets and of simplicial complexes. As an application, we construct a transfer for them in Bredon homology, when M is a Mackey functor. We also show that the Bredon-Illman homology with coefficients in M satisfies the equivariant weak homotopy equivalence axiom in the category of G-spaces.     

Spaces of Maps into Classifying Spaces for Equivariant Crossed Complexes, II: The General Topological Group Case

The results of a previous paper on the equivariant homotopy theory of crossed complexes are generalised from the case of a discrete group to general topological groups. The principal new ingredient necessary for this is an analysis of homotopy coherence theory for crossed complexes, using detailed results on the appropriate Eilenberg–Zilber theory, and of its relation to simplicial homotopy coherence. Again, our results give information not just on the homotopy classification of certain equivariant maps, but also on the weak equivariant homotopy type of the corresponding equivariant function spaces.     

Generalized Cn-Manifolds

This paper proves an important topological characterization of C-manifolds and especially of arbitrary finite dimensional Cⁿ-manifolds and arbitrary C∞-BANACH manifolds. Whereas differentiability structures in the usual sense may be proper classes this characterization always enables a definition of differentiability structures as sets. Further this characterization suggests a suitable generalization of the notion of differentiable manifold which we call Cⁿ-manifold. C ⁿ-manifolds have a lot of better properties than Cⁿ-manifolds and may be useful therefore in physics and technics. Some examples of C ⁿ-manifolds are given in the last two sections defined by means of certain geometric structures.     

Heegaard splittings of twisted torus knots

Little is known on the classification of Heegaard splittings for hyperbolic 3-manifolds. Although Kobayashi gave a complete classification of Heegaard splittings for the exteriors of 2-bridge knots, our knowledge of other classes is extremely limited. In particular, there are very few hyperbolic manifolds that are known to have a unique minimal genus splitting. Here we demonstrate that an infinite class of hyperbolic knot exteriors, namely exteriors of certain “twisted torus knots” originally studied by Morimoto, Sakuma and Yokota, have a unique minimal genus Heegaard splitting of genus two. We also conjecture that these manifolds possess irreducible yet weakly reducible splittings of genus three. There are no known examples of such Heegaard splittings.     

Homotopy types of locally linear representation forms

Summary The authors of [6] investigated certain locally linear actions of a cyclic groupG of odd order on homotopy spheres, the so-calledG-representation forms [16]. In particular, several conditions on a dimension function were described that made sure that it can be realized as the dimension function of aG-representation form. It remained unclear, whether all homotopy types with those dimension functions would support a locally linear structure. It is the aim of this note to show that this is not the case, i.e., to give examples of homotopy representations [17] with the same dimension functions some of which support a locally linear structure with stably trivial tangent bundle and others do not. The main tools are formulated as general splitting principles for fixed point and restriction functors that may have some interest in their own right, too. Part of the work with this paper was assembled while the authors were visiting Institut Mittag-Leffler at Djursholm, Sweden, whose support is gratefully acknowledged. This article was processed by the author using the LATEX style filecljour1 from Springer-Verlag.