Related papers: On Poincar\'e Surgery
The main aim of this paper is to extend Bochner's technique to statistical structures. Other topics related to this technique are also introduced to the theory of statistical structures. It deals, in particular, with Hodge's theory,…
In this paper, we explain how the more general context of generalised equivariant bundles allows for a simple inductive proof of the ECHP. We also make clear the link between the ECHP and the theory of Hurewicz fibrations.
The paper is the survey of the modern results and applications of the theory of homotopes. The notion of a well-tempered element in an associative algebra is introduced and it is proven that the category of representations of the homotope…
We present a sheaf-theoretic construction of shape space -- the space of all shapes. We do this by describing a homotopy sheaf on the poset category of constructible sets, where each set is mapped to its Persistent Homology Transform (PHT).…
We establish a Quillen equivalence relating the homotopy theory of Segal operads and the homotopy theory of simplicial operads, from which we deduce that the homotopy coherent nerve functor is a right Quillen equivalence from the model…
We prove a "gluing" theorem for monotone homotopies; a monotone homotopy is a homotopy through simple contractible closed curves which themselves are pairwise disjoint. We show that two monotone homotopies which have appropriate overlap can…
We present a geometric proof of the Poincar\'e-Dulac Normalization Theorem for analytic vector fields with singularities of Poincar\'e type. Our approach allows us to relate the size of the convergence domain of the linearizing…
We survey several mathematical developments in the holonomy approach to gauge theory. A cornerstone of this approach is the introduction of group structures on spaces of based loops on a smooth manifold, relying on certain homotopy…
We show that the KZB connection on the punctured torus and on the configuration space of points of the punctured torus can be constructed via the homotopy transfer theorem.
In this paper, we initiate the generalisation of the operadic calculus which governs the properties of homotopy algebras to a properadic calculus which governs the properties of homotopy gebras over a properad. In this first article of a…
The classical Poincar{\'e} conjecture that every homotopy 3-sphere is diffeomorphic to the 3-sphere is confirmed by Perelman in arXiv papers solving Thurston's program on geometrizations of 3-manifolds. A new confirmation of this conjecture…
This is the second in a series of papers. Here we develop here an intersection theory for manifolds equipped with an action of a finite group. As in our previous paper, our approach will be homotopy theoretic, enabling us to circumvent the…
We study collections of curves in generic position on a closed surface whose complement consists of one disk only, up to orientation-preserving homeomorphism of the surface. We define a surgery operation on the set of such collections and…
Basic examples show that coincidence theory is intimately related to central subjects of differential topology and homotopy theory such as Kervaire invariants and divisibility properties of Whitehead products and of Hopf invariants. We…
We give a detailed exposition of the homotopy theory of equivalence relations, perhaps the simplest nontrivial example of a model structure.
For G an almost-connected Lie group, we study G-equivariant index theory for proper co-compact actions with various applications, including obstructions to and existence of G-invariant Riemannian metrics of positive scalar curvature. We…
We show that for a wide class of manifold pairs N, M satisfying dim(M) = dim(N) + 1, every \pi_1-injective map f : N --> M factorises up to homotopy as a finite cover of an embedding. This result, in the spirit of Waldhausen's torus…
Under certain homological hypotheses on a compact 4-manifold, we prove exactness of the topological surgery sequence at the stably smoothable normal invariants. The main examples are the class of finite connected sums of 4-manifolds with…
This paper is an introduction to the use of the cobordism of chain complexes with Poincar\'e duality in surgery theory. It is a companion to the author's paper "An introduction to algebraic surgery" math.AT/0008071 (to appear in Volume 2 of…
Final revision. To appear in the Journal of Differential Geometry. This paper studies knots that are transversal to the standard contact structure in $\reals^3$, bringing techniques from topological knot theory to bear on their transversal…