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We show that in codimension at least 3, spaces of locally flat topological embeddings of manifolds are correctly modelled by derived spaces of maps between their configuration categories (under mild smoothability conditions). That general claim was reduced in an earlier paper to the special cases where the manifolds in question are euclidean spaces. We deal with these special cases by comparing to other special cases where the manifolds have the form "torus" and "torus times euclidean space" respectively, and by setting up a torus trick for configuration categories., Comment: 52 pages. Changes in v3: some minor changes to sections 4 and appendix A and bigger changes to sections 6 and 7

The configuration category of a manifold is a topological category which we view as a Segal space, via the nerve construction. Our main result is that the unordered configuration category, suitably truncated, admits a finite presentation as a complete Segal space if the manifold in question is the interior of a compact manifold., Comment: 15 pages

We investigate the relationship between the configuration category of a manifold and the configuration category of a covering space of that manifold., Comment: 18 pages

A construction related to the Boardman-Vogt tensor product of operads allows us to describe the configuration category of a product manifold $M\times N$ in terms of the configuration categories of the factors $M$ and $N$., Comment: 15 pages. v2: added details and theorem 2

The functor that takes a manifold to its configuration category exhibits a type of full faithfulness in some cases., Comment: 2 pages

The rational Pontryagin classes, evaluated on fiber bundles where the fiber is a 2n-dimensional euclidean space, can be nonzero in cohomology dimensions much greater than 4n. This makes a striking contrast with the Pontryagin classes of vector bundles., Comment: v2: corrections in appendix A, references added. v3: new appendix B. v4: bibliography updated, appendix A downsized, typos corrected. v5: abstract added. v6: details added. v7: section 2 simplified, minor mistakes corrected. v8: new fonts, erroneous crossrefs corrected. This will be published under another title, "Rational Pontryagin classes of Euclidean fiber bundles"

It was shown in a recent paper by Boavida de Brito and Weiss that a well-known construction which to a plain (=monochromatic) topological operad associates a topological category and a functor from it to the category of finite sets is homotopically fully faithful, under mild conditions on the operads. The main result here is a generalization of that statement to k-truncated plain topological operads. A k-truncated operad is a weaker version of operad where all operations have arity at most k., Comment: v2: just a few typos corrected. v3: corrections end of section 2

Let M be a smooth compact manifold with boundary. Under some geometric conditions on M, a homotopical model for the pair (M,boundary of M) can be recovered from the configuration category of the interior of M. The grouplike monoid of derived homotopy automorphisms of the configuration category of the interior of M then acts on the homotopical model of the pair (M,boundary of M), in a way which is compatible with a known homotopical action of the homeomorphism group of M minus boundary on (M,boundary of M)., Comment: v2: introduction added, appendix added. Some updates reflecting corrections and changes in the prequel v3: some typos corrected, references updated, many explanations added

Let K be a subset of a smooth manifold M. In some cases functor calculus methods lead to a homotopical formula for M minus K in terms of the subspaces M minus S, where S runs through the finite subsets of K., Comment: There is a sequel to this paper: title "Configuration categories and homotopy automorphisms". v2: some typos corrected, v3: some errors in section 1 corrected. v4: more errors corrected and therefore number of authors increased from 1 to 2. v5: more errors corrected especially in section 3. v6: just a few typos corrected and pointers added

In the homotopical study of spaces of smooth embeddings, the functor calculus method (Goodwillie-Klein-Weiss manifold calculus) has opened up important connections to operad theory. Using this and a few simplifying observations, we arrive at an operadic description of the obstructions to deforming smooth immersions into smooth embeddings. We give an application which in some respects improves on recent results of Arone-Turchin and Dwyer-Hess concerning high-dimensional variants of spaces of long knots., Comment: 84 pages. v2: minor errors in section 7 corrected, old section 8 improved and reorganized into three separate sections. v3: typos corrected

The structure space S(M) of a closed topological m-manifold M classifies bundles whose fibers are closed m-manifolds equipped with a homotopy equivalence to M. We construct a highly connected map from S(M) to a concoction of algebraic L-theory and algebraic K-theory spaces associated with M. The construction refines the well-known surgery theoretic analysis of the block structure space of M in terms of L-theory., Comment: approx 95 pages

Manifold calculus is a form of functor calculus concerned with functors from some category of manifolds to spaces. A weakness in the original formulation is that it is not continuous in the sense that it does not handle well the natural enrichments. In this paper, we correct this by defining an enriched version of manifold calculus which essentially extends the discrete setting. Along the way, we recast the Taylor tower as a tower of homotopy sheafifications. As a spin-off we obtain a natural connection to operads: the limit of the Taylor tower is a certain (derived) space of right module maps over the framed little discs operad., Comment: 19 pages; added minor corrections and improvements

The discriminant method is a tool for describing the cohomology, or the homotopy type, of certain spaces of smooth maps with uncomplicated singularities from a smooth compact manifold L to R^k. We recast some of it in the language of functor calculus. This reformulation allows us to use the discriminant method in a setting where we wish to impose conditions on the multilocal behavior of smooth maps f from L to R^k., Comment: Few changes in v3. Shorter title. Some promises removed

It is known that in the integral cohomology of BSO(2m), the square of the Euler class is the same as the Pontryagin class in degree 4m. Given that the Pontryagin classes extend rationally to the cohomology of BSTOP(2m), it is reasonable to ask whether the same relation between the Euler class and the Pontryagin class in degree 4m is still valid in the rational cohomology of BSTOP(2m). In this paper we use smoothing theory and tools from homotopy theory to reformulate the hypothesis, and variants, in a differential topology setting and in a functor calculus setting., Comment: Very minor changes in the last revision

We classify the most common local forms of smooth maps from a smooth manifold L to the plane. The word "local" can refer to locations in the source L, but also to locations in the target. The first point of view leads us to a classification of certain germs of maps, which we review here although it is very well known. The second point of view leads us to a classification of certain multigerms of maps., Comment: 25 pages

The main result of this paper amounts to a complete evaluation of the integral cohomological structure of the stable mapping class group. In particular it verifies the conjecture of D.Mumford about the rational cohomology of the stable mapping class group., Comment: Ca. 97 pages. For this version, many results have been reformulated and re-proved to be valid in all dimensions and with more general notions of orientation

This is a survey paper on spaces of automorphisms of manifolds and spaces of manifolds in a fixed homotopy type. It describes the main theorems of traditional surgery theory, but also the main theorems of pseudoisotopy theory, alias concordance theory, Waldhausen style. It culminates in (an outline of) a synthesis of these two theories, producing algebraic models, valid in a stable range, for spaces of manifolds in a fixed homotopy type. This is inspired by earlier work of Burghelea-Lashof and Hatcher. The algebraic models are a mix of algebraic L-theory and algebraic K-theory., Comment: 56 pages; survey paper; appears also in "Surveys on Surgery Theory: Volume 2" (eds. S.Cappell, A.Ranicki, J.Rosenberg), Princeton University Press, Princeton, New Jersey; January 2001