Mathematics

We introduce new Elmendorf constructions for equivariant categories and posets, and we prove that they are compatible with the classical topological one. Our constructions are more concrete than their model-categorical counterparts, and they give rise to new proofs of the Elmendorf theorems for equivariant categories and posets.

We study the dependence of the embedding calculus Taylor tower on the smooth structures of the source and target. We prove that embedding calculus does not distinguish exotic smooth structures in dimension 4, implying a negative answer to a question of Viro. In contrast, we show that embedding calculus does distinguish certain exotic spheres in higher dimensions. As a technical tool of independent interest, we prove an isotopy extension theorem for the limit of the embedding calculus tower, which we use to investigate several further examples.

We prove convergence of Goodwillie--Weiss' embedding calculus for spaces of diffeomorphisms of surfaces and for spaces of arcs in surfaces. This exhibits the first nontrivial class of examples for which embedding calculus converges in handle codimension less than three and results in particular in a description of the mapping class group of a closed surface in terms of derived maps of modules over the framed E 2 -operad.

Given an m -dimensional CW complex K , we use a version of the Goodwillie-Weiss tower to formulate an obstruction theory for embeddings into a Euclidean space R d . For 2 -complexes in R 4 a geometric analogue is also introduced, based on intersections of Whitney disks and more generally on the intersection theory of Whitney towers developed by Schneiderman and Teichner. The focus in this paper is on the first obstruction beyond the classical embedding obstruction of van Kampen. In this case we show the two approaches give the same result, and also relate it to the Arnold class in the cohomology of configuration spaces. The obstructions are shown to be realized in a family of examples. Conjectures are formulated, relating higher versions of these homotopy-theoretic, geometric and cohomological theories.

Consider the set of all rectangular n×m matrices with entries in a field. Recall that unitriangular group T n consists of upper triangular matrices with 1's on the diagonal. The product T n × T m naturally acts on the aforementioned set: X↦AX B −1 . Our first observation is that each orbit of this action contains a unique matrix which has at most one non-zero entry in each row and in each column. Thus these non-zero numbers and their positions are invariants of a matrix under this action. This is a variation of a classical Bruhat decomposition for GL . When applied in the setting of Morse theory, this linear algebraic construction leads to invariants of a strong Morse function f . Namely, positions of non-zero entries correspond to the well-known Barannikov decomposition (also known as persistent homology) of f . The novelty is the values themselves, which correspond to numbers, carried by Barannikov pairs (also known as bars in the barcode). Considering further a complex, constructed from a strong Morse function, we interpret the product of all the numbers as a torsion of chain complex.

The complete enriched Lie algebras constitue the natural extension of graded Lie algebras for connected spaces. Each complete enriched Lie algebra is the rational homotopy Lie algebra of a connected space. This text is the first part of a general study of those Lie algebras

In this paper we present background results in enriched category theory and enriched model category theory necessary for developing model categories of enriched functors suitable for doing functor calculus.

The equivariant cohomology of the classical configuration space F( R d ,n) has been been of great interest and has been studied intensively starting with the classical papers by Artin (1925/1947) on the theory of braids, by Fox and Neuwirth (1962), Fadell and Neuwirth (1962), and Arnol'd (1969). We give a brief treatment of the subject from the beginnings to recent developments. However, we focus on the mod 2 equivariant cohomology algebras of the classical configuration space F( R d ,n) , as described in an influential paper by Hung (1990). We show with a new, detailed proof that his main result is correct, but that the arguments that were given by Hung on the way to his result are not, as are some of the intermediate results in his paper. This invalidates a paper by three of the present authors, Blagojević, Lück \& Ziegler (2016), who used a claimed intermediate result from Hung (1990) in order to derive lower bounds for the existence of k -regular and ℓ -skew embeddings. Using our new proof for Hung's main result, we get new lower bounds for existence of highly regular embeddings: Some of them agree with the previously claimed bounds, some are weaker.

Wiltshire-Gordon has introduced a homotopy model for ordered configuration spaces on a given simplicial complex. That author asserts that, after a suitable subdivision, his model also works for unordered configuration spaces. We supply details justifying Wiltshire-Gordon's assertion and, more importantly, uncover the equivariant properties of his more-general simplicial-difference model for the complement of a subcomplex inside a larger complex. This is achieved by proving an equivariant version of the Nerve Lemma. In addition, in the case of configuration spaces, we show that a slight variation of the model has better properties: it is regular and sits inside the configuration space as a strong and equivariant deformation retract. Our variant for the configuration-space model comes from a comparison, in the equivariant setting, between Wiltshire's simplicial difference and a well known model for the complement of a full subcomplex on a simplicial complex.

We present an Eilenberg--Steenrod-like axiomatic framework for equivariant coarse homology and cohomology theories. We also discuss a general construction of such coarse theories from topological ones and the associated transgression maps. A large part of this paper is devoted to showing how some well-established coarse (co-)homology theories with already existing or newly introduced equivariance fit into this setup. Furthermore, a new and more flexible notion of coarse homotopy is given which is more in the spirit of topological homotopies. Some, but not all, coarse (co-)homology theories are even invariant under these new homotopies. They also led us to a meaningful concept of topological actions of locally compact groups on coarse spaces.