Abstract:
Given a compact manifold X, the set of simple manifold structures on X x \Delta^k relative to the boundary can be viewed as the k-th homotopy group of a space \S^s (X). This space is called the block structure space of X. We study the block structure spaces of real projective spaces. Generalizing Wall's join construction we show that there is a functor from the category of finite dimensional real vector spaces with inner product to the category of pointed spaces which sends the vector space V to the block structure space of the projective space of V. We study this functor from the point of view of orthogonal calculus of functors; we show that it is polynomial of degree <= 1 in the sense of orthogonal calculus. This result suggests an attractive description of the block structure space of the infinite dimensional real projective space via the Taylor tower of orthogonal calculus. This space is defined as a colimit of block structure spaces of projective spaces of finite-dimensional real vector spaces and is closely related to some automorphisms spaces of real projective spaces.

Abstract:
In this paper, we continue the study of the category of functors Fquad, associated to F_2-vector spaces equipped with a nondegenerate quadratic form, initiated in two previous papers of the author. We define a filtration of the standard projective objects in Fquad; this refines to give a decomposition into indecomposable factors of the two first standard projective objects in Fquad. As an application of these two decompositions, we give a complete description of the polynomial functors of the category Fquad.

Abstract:
We restate the notion of orthogonal calculus in terms of model categories. This provides a cleaner set of results and makes the role of O(n)-equivariance clearer. Thus we develop model structures for the category of n-polynomial and n-homogeneous functors, along with Quillen pairs relating them. We then classify n-homogeneous functors, via a zig-zag of Quillen equivalences, in terms of spectra with an O(n)-action. This improves upon the classification of Weiss. As an application, we develop a variant of orthogonal calculus by replacing topological spaces with orthogonal spectra.

Abstract:
Manifold calculus of functors, due to M. Weiss, studies contravariant functors from the poset of open subsets of a smooth manifold to topological spaces. We introduce "multivariable" manifold calculus of functors which is a generalization of this theory to functors whose domain is a product of categories of open sets. We construct multivariable Taylor approximations to such functors, classify multivariable homogeneous functors, apply this classification to compute the derivatives of a functor, and show what this gives for the space of link maps. We also relate Taylor approximations in single variable calculus to our multivariable ones.

Abstract:
Let M be a smooth manifold and V a Euclidean space. Let Ebar(M,V) be the homotopy fiber of the map from Emb(M,V) to Imm(M,V). This paper is about the rational homology of Ebar(M,V). We study it by applying embedding calculus and orthogonal calculus to the bi-functor (M,V) |--> HQ /\Ebar(M,V)_+. Our main theorem states that if the dimension of V is more than twice the embedding dimension of M, the Taylor tower in the sense of orthogonal calculus (henceforward called ``the orthogonal tower'') of this functor splits as a product of its layers. Equivalently, the rational homology spectral sequence associated with the tower collapses at E^1. In the case of knot embeddings, this spectral sequence coincides with the Vassiliev spectral sequence. The main ingredients in the proof are embedding calculus and Kontsevich's theorem on the formality of the little balls operad. We write explicit formulas for the layers in the orthogonal tower of the functor HQ /\Ebar(M,V)_+. The formulas show, in particular, that the (rational) homotopy type of the layers of the orthogonal tower is determined by the (rational) homology type of M. This, together with our rational splitting theorem, implies that under the above assumption on codimension, the rational homology groups of Ebar(M,V) are determined by the rational homology type of M.

Abstract:
The category of small covariant functors from simplicial sets to simplicial sets supports the projective model structure. In this paper we construct various localizations of the projective model structure and also give a variant for functors from simplicial sets to spectra. We apply these model categories in the study of calculus of functors, namely for a classification of polynomial and homogeneous functors. In the $n$-homogeneous model structure, the $n$-th derivative is a Quillen functor to the category of spectra with $\Sigma_n$-action. After taking into account only finitary functors -- which may be done in two different ways -- the above Quillen map becomes a Quillen equivalence. This improves the classification of finitary homogeneous functors by T. G. Goodwillie.

Abstract:
Goodwillie's homotopy functor calculus constructs a Taylor tower of approximations to F, often a functor from spaces to spaces. Weiss's orthogonal calculus provides a Taylor tower for functors from vector spaces to spaces. In particular, there is a Weiss tower associated to the functor which sends a vector space V to F evaluated at the one-point compactification of V. In this paper, we give a comparison of these two towers and show that when F is analytic the towers agree up to weak equivalence. We include two main applications, one of which gives as a corollary the convergence of the Weiss Taylor tower of BO. We also lift the homotopy level tower comparison to a commutative diagram of Quillen functors, relating model categories for Goodwillie calculus and model categories for the orthogonal calculus.

Abstract:
We investigate covariant first order differential calculi on the quantum complex projective spaces CP_q^{N-1} which are quantum homogeneous spaces for the quantum group SU_q(N). Hereby, one more well-studied example of covariant first order differential calculus on a quantum homogeneous space is given. Since the complex projective spaces are subalgebras of the quantum spheres S_q^{2N-1} introduced by Vaksman and Soibelman, we get also an example of the relations between covariant differential calculus on two closely related quantum spaces. Two approaches are combined in obtaining covariant first order differential calculi on CP_q^{N-1}: 1. restriction of covariant first order differential calculi from S_q^{2N-1}; 2. classification of calculi under appropriate constraints, using methods from representation theory. The main result is that under three reasonable settings of dimension constraints, covariant first order differential calculi on CP_q^{N-1} exist and are (for N >= 6) uniquely determined. This is a clear difference as compared to the case of the quantum spheres where several parametrical series of calculi exist. For two of the constraint settings, the covariant first order calculi on CP_q^{N-1} are also obtained by restriction from calculi on S_q^{2N-1} as well as from calculi on the quantum group SU_q(N).

Abstract:
This is a (slightly edited) version of the PhD dissertation of the author, submitted to Brown University in July 2005. We construct a homotopy calculus of functors in the sense of Goodwillie for the categories of rational homotopy theory. More precisely, given a homotopy functor between any of the categories of differential graded vector spaces (DG), reduced differential graded vector spaces, differential graded Lie algebras (DGL), and differential graded coalgebras (DGC), we show that there is an associated approximating "rational Taylor tower" of excisive functors. The fibers in this tower are homogeneous functors which factor as homogeneous endomorphisms of the category of differential graded vector spaces. Furthermore, we develop very straightforward and simple models for all of the objects in this tower. Constructing these models entails first building very simple models for homotopy pushouts and pullbacks in the categories DG, DGL, and DGC. We end with a short example of the usefulness of our computationally simple models for rational Taylor towers, as well as a preview of some further results dealing with the structure of rational (and non-rational) Taylor towers.

Abstract:
In this article, we introduce symbol calculus on a projective scheme. Using holomorphic Poisson structures, we construct deformations of ring structures for structure sheaves on projective spaces.