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The category of –spaces is the diagram category of spaces indexed by finite sets and injections. This is a symmetric monoidal category whose commutative monoids model all –spaces. Working in the category of –spaces enables us to simplify and strengthen previous work on group completion and units of –spaces. As an application we clarify the relation to –spaces and show how the spectrum of units associated with a commutative symmetric ring spectrum arises through a chain of Quillen adjunctions.
Applying mod 2 homology to the Goodwillie tower of the functor sending a spectrum to the suspension spectrum of its space, leads to a spectral sequence for computing , which converges strongly when is 0–connected. The term is the homology of the extended powers of , and thus is a well known functor of , including structure as a bigraded Hopf algebra, a right module over the mod 2 Steenrod algebra , and a left module over the Dyer–Lashof operations. This paper is an investigation of how this structure is transformed through the spectral sequence.
Hopf algebra considerations show that all pages of the spectral sequence are primitively generated, with primitives equal to a subquotient of the primitives in .
We use an operad action on the tower, and the Tate construction, to determine how Dyer–Lashof operations act on the spectral sequence. In particular, has Dyer–Lashof operations induced from those on .
We use our spectral sequence Dyer–Lashof operations to determine differentials that hold for any spectrum . The formulae for these universal differentials then lead us to construct an algebraic spectral sequence depending functorially on an –module . The topological spectral sequence for agrees with the algebraic spectral sequence for for many spectra , including suspension spectra and almost all Eilenberg–Mac Lane spectra. The term of the algebraic spectral sequence has form and structure similar to , but now the right –module structure is unstable. Our explicit formula involves the derived functors of destabilization as studied in the 1980’s by W Singer, J Lannes and S Zarati, and P Goerss.
We derive a version of the Rothenberg–Steenrod, fiber-to-base, spectral sequence for cohomology theories represented in model categories of simplicial presheaves. We then apply this spectral sequence to calculate the equivariant motivic cohomology of with a general –action; this coincides with the equivariant higher Chow groups. The motivic cohomology of and some of the equivariant motivic cohomology of a Stiefel variety, , with a general –action is deduced as a corollary.
Let be the open unit disc in the Euclidean plane and let be the group of smooth compactly supported area-preserving diffeomorphisms of . For every natural number we construct an injective homomorphism , which is bi-Lipschitz with respect to the word metric on and the autonomous metric on . We also show that the space of homogeneous quasimorphisms vanishing on all autonomous diffeomorphisms in the above group is infinite-dimensional.
We generalize the notion of a sheaf of sets over a space to define the notion of a small stack of groupoids over an étale stack. We then provide a construction analogous to the étalé space construction in this context, establishing an equivalence of –categories between small stacks over an étale stack and local homeomorphisms over it. These results hold for a wide variety of types of spaces, for example, topological spaces, locales, various types of manifolds, and schemes over a fixed base (where stacks are taken with respect to the Zariski topology). Along the way, we also prove that the –category of topoi is fully reflective in the –category of localic stacks.
A closed topological –manifold is of –category if it can be covered by open subsets such that for each path-component of the subsets the image of its fundamental group is an amenable group. is the smallest number such that admits such a covering. For , has –category . We characterize all closed –manifolds of –category , and .
We view closed orientable –manifolds as covers of branched over hyperbolic links. To a cover , of degree and branched over a hyperbolic link , we assign the complexity . We define an invariant of –manifolds, called the link volume and denoted by , that assigns to a 3-manifold the infimum of the complexities of all possible covers , where the only constraint is that the branch set is a hyperbolic link. Thus the link volume measures how efficiently can be represented as a cover of .
We study the basic properties of the link volume and related invariants, in particular observing that for any hyperbolic manifold , is less than . We prove a structure theorem that is similar to (and uses) the celebrated theorem of Jørgensen and Thurston. This leads us to conjecture that, generically, the link volume of a hyperbolic –manifold is much bigger than its volume.
Finally we prove that the link volumes of the manifolds obtained by Dehn filling a manifold with boundary tori are linearly bounded above in terms of the length of the continued fraction expansion of the filling curves.
We restate the notion of orthogonal calculus in terms of model categories. This provides a cleaner set of results and makes the role of –equivariance clearer. Thus we develop model structures for the category of –polynomial and –homogeneous functors, along with Quillen pairs relating them. We then classify –homogeneous functors, via a zig-zag of Quillen equivalences, in terms of spectra with an –action. This improves upon the classification theorem of Weiss. As an application, we develop a variant of orthogonal calculus by replacing topological spaces with orthogonal spectra.
By a result of R Meyerhoff, it is known that among all cusped hyperbolic 3–orbifolds the quotient of by the tetrahedral Coxeter group has minimal volume. We prove that the group has smallest growth rate among all non-cocompact cofinite hyperbolic Coxeter groups, and that it is as such unique. This result extends to three dimensions some work of W Floyd who showed that the Coxeter triangle group has minimal growth rate among all non-cocompact cofinite planar hyperbolic Coxeter groups. In contrast to Floyd’s result, the growth rate of the tetrahedral group is not a Pisot number.
We study Farber’s topological complexity (TC) of Davis’ projective product spaces (PPS’s). We show that, in many nontrivial instances, the TC of PPS’s coming from at least two sphere factors is (much) lower than the dimension of the manifold. This is in marked contrast with the known situation for (usual) real projective spaces for which, in fact, the Euclidean immersion dimension and TC are two facets of the same problem. Low TC-values have been observed for infinite families of nonsimply connected spaces only for H-spaces, for finite complexes whose fundamental group has cohomological dimension at most , and now in this work for infinite families of PPS’s. We discuss general bounds for the TC (and the Lusternik–Schnirelmann category) of PPS’s, and compute these invariants for specific families of such manifolds. Some of our methods involve the use of an equivariant version of TC. We also give a characterization of the Euclidean immersion dimension of PPS’s through a generalized concept of axial maps or, alternatively (in an appendix), nonsingular maps. This gives an explicit explanation of the known relationship between the generalized vector field problem and the Euclidean immersion problem for PPS’s.
We show that the unstable splittings of the spaces in the Omega spectra representing , and from [Amer. J. Math. 97 (1975) 101–123] may be obtained for the real analogs of these spectra using techniques similar to those in [Progr. Math. 196 (2001) 35–45]. Explicit calculations for are given.
We extend to dihedral sets Drinfeld’s filtered-colimit expressions of the geometric realization of simplicial and cyclic sets. We prove that the group of homeomorphisms of the circle continuously act on the geometric realization of a dihedral set. We also see how these expressions of geometric realization clarify subdivision operations on simplicial, cyclic and dihedral sets defined by Bökstedt, Hsiang and Madsen, and Spaliński.
We present general techniques for constructing functorial factorizations appropriate for model structures that are not known to be cofibrantly generated. Our methods use “algebraic” characterizations of fibrations to produce factorizations that have the desired lifting properties in a completely categorical fashion. We illustrate these methods in the case of categories enriched, tensored and cotensored in spaces, proving the existence of Hurewicz-type model structures, thereby correcting an error in earlier attempts by others. Examples include the categories of (based) spaces, (based) –spaces and diagram spectra among others.
We show that essential punctured spheres in the complement of links with distance three or greater bridge spheres have bounded complexity. We define the operation of tangle product, a generalization of both connected sum and Conway product. Finally, we use the bounded complexity of essential punctured spheres to show that the bridge number of a tangle product is at least the sum of the bridge numbers of the two factor links up to a constant error.
If a –manifold contains a nonseparating sphere, then some twisted Heegaard Floer homology of is zero. This simple fact allows us to prove several results about Dehn surgery on knots in such manifolds. Similar results have been proved for knots in –spaces.
We call attention to the intermediate constructions in Goodwillie’s Calculus of homotopy functors, giving a new model which naturally gives rise to a family of towers filtering the Taylor tower of a functor. We also establish a surprising equivalence between the homotopy inverse limits of these towers and the homotopy inverse limits of certain cosimplicial resolutions. This equivalence gives a greatly simplified construction for the homotopy inverse limit of the Taylor tower of a functor under general assumptions.
We study the topology of moduli spaces of closed linkages in depending on a length vector . In particular, we use equivariant Morse theory to obtain information on the homology groups of these spaces, which works best for odd . In the case we calculate the Poincaré polynomial in terms of combinatorial information encoded in the length vector.
The Künneth Theorem for equivariant (complex) –theory , in the form developed by Hodgkin and others, fails dramatically when is a finite group, and even when is cyclic of order . We remedy this situation in this very simplest case by using the power of –graded equivariant –theory.