Mathematics

In this paper we prove the homotopy lifting property for symmetric products S P m (X) and F m (X) , with X a Hausdorff topological space. Furthermore, we introduce a new tool, the theory of topological puzzles, to get a useful decomposition of X m .

Curved algebras are algebras endowed with a predifferential, which is an endomorphism of degree -1 whose square is not necessarily 0. This makes the usual definition of quasi-isomorphism meaningless and therefore the homotopical study of curved algebras cannot follow the same path as differential graded algebras. In this article, we propose to study curved algebras by means of curved operads. We develop the theory of bar and cobar constructions adapted to this new notion as well as Koszul duality theory. To be able to provide meaningful definitions, we work in the context of objects which are filtered and complete and become differential graded after applying the associated graded functor. This setting brings its own difficulties but it nevertheless permits us to define a combinatorial model category structure that we can transfer to the category of curved operads and to the category of algebras over a curved operad using free-forgetful adjunctions. We address the case of curved associative algebras. We recover the notion of curved Aoo-algebras, and we show that the homotopy categories of curved associative algebras and of curved Aoo-algebras are Quillen equivalent.

Modern categories of spectra such as that of Elmendorf et al equipped with strictly symmetric monoidal smash products allows the introduction of symmetric monoids providing a new way to study highly coherent commutative ring spectra. These have categories of modules which are generalisations of the classical categories of spectra that correspond to modules over the sphere spectrum; passing to their derived or homotopy categories leads to new contexts in which homotopy theory can be explored. In this paper we describe some of the tools available for studying these `brave new homotopy theories' and demonstrate them by considering modules over the K -theory spectrum, closely related to Mahowald's theory of bo -resolutions. In a planned sequel we will apply these techniques to the much less familiar context of modules over the 2 -local connective spectrum of topological modular forms.

The index theory for the space of finite propagation unitary operators was developed by Gross, Nesme, Vogts and Werner from the viewpoint of quantum walks in mathematical physics. In particular, they proved that π 0 of the space is determined by the index. However, nothing is known about the higher homotopy groups. In this article, we describe the homotopy type of the space of finite propagation unitary operators on the Hilbert space of square summable C -valued Z -sequences, so we can determine its homotopy groups. We also study the space of (end-)periodic finite propagation unitary operators.

We study the homotopy type of the space of the unitary group U 1 ( C ??u (| Z n |)) of the uniform Roe algebra C ??u (| Z n |) of Z n . We show that the stabilizing map U 1 ( C ??u (| Z n |))??U ??( C ??u (| Z n |)) is a homotopy equivalence. Moreover, when n=1,2 , we determine the homotopy type of U 1 ( C ??u (| Z n |)) , which is the product of the unitary group U 1 ( C ??(| Z n |)) (having the homotopy type of U ??(C) or Z?B U ??(C) depending on the parity of n ) of the Roe algebra C ??(| Z n |) and rational Eilenberg--MacLane spaces.

We examine the relation between the gauge groups of SU(n) - and PU(n) -bundles over S 2i , with 2≤i≤n , particularly when n is a prime. As special cases, for PU(5) -bundles over S 4 , we show that there is a rational or p -local equivalence G 2,k ≃ (p) G 2,l for any prime p if, and only if, (120,k)=(120,l) , while for PU(3) -bundles over S 6 there is an integral equivalence G 3,k ≃ G 3,l if, and only if, (120,k)=(120,l) .

We compute a lower bound for the number of simplices that are needed to triangulate the Grassmann manifold G k ( R n ) . In particular, we show that the number of top-dimensional simplices grows exponentially with n . More precise estimates are given for k=2,3,4 . Our method can be used to estimate the minimal size of triangulations for other spaces, like Lie groups, flag manifolds, Stiefel manifolds etc.

We show that hypernetworks can be regarded as posets which, in their turn, have a natural interpretation as simplicial complexes and, as such, are endowed with an intrinsic notion of curvature, namely the Forman Ricci curvature, that strongly correlates with the Euler characteristic of the simplicial complex. This approach, inspired by the work of E. Bloch, allows us to canonically associate a simplicial complex structure to a hypernetwork, directed or undirected. In particular, this greatly simplifying the geometric Persistent Homology method we previously proposed.

The well-known Eckmann-Hilton Principle may be applied to prove that fundamental groups of H -spaces are commutative. In this paper, we identify an infinitary analogue of the Eckmann-Hilton Principle that applies to fundamental groups of all topological monoids and slightly more general objects called pre- Δ -monoids. In particular, we show that every pre- Δ -monoid M is "transfinitely π 1 -commutative" in the sense that permutation of the factors of any infinite loop-concatenation indexed by a countably infinite order and based at the identity e∈M is a homotopy invariant action. We also give a detailed account of fundamental groups of James reduced products and apply transfinite π 1 -commutativity to make several computations.

This is an introduction to the study of abstract homotopy theory by means of model categories and (∞,1) -categories. The only prerequisites are very basic general topology and abstract algebra. None categorical background is needed. The final objective is to show that classical homotopy theory for topological spaces can be more naturally understood in terms of categorical language.