Mathematics

We study the cohomology of the complement M(A) of a manifold arrangement A in a smooth manifold M without boundary. We first give the concept of monoidal cosheaf on a locally geometric poset L , and then define the generalized Orlik--Solomon algebra A ∗ (L,C) over a commutative ring with unit, which is built by the classical Orlik--Solomon algebra and a monoidal cosheaf C as coefficients. Furthermore, we construct a monoidal cosheaf C ^ (A) associated with A , so that the generalized Orlik--Solomon algebra A ∗ (L, C ^ (A)) becomes a double complex with suitable multiplication structure and the associated total complex Tot( A ∗ (L, C ^ (A))) is a differential algebra. Our main result is that H ∗ (Tot( A ∗ (L, C ^ (A)))) is isomorphic to H ∗ (M(A)) as algebras. Our argument is of topological with the use of a spectral sequence induced by a geometric filtration associated with A . In particular, we also discuss the mixed Hodge complex structure on our model if M and all elements in A are complex smooth varieties, and show that it induces the canonical mixed Hodge structure of M(A) . As an application, we calculate the cohomology of chromatic configuration spaces, which agrees with many known results in some special cases. In addition, some explicit formulas with respect to Poincaré polynomial and chromatic polynomial are also given.

For any natural k?�n , the rational cohomology ring of the space of continuous maps S 2k?? ??S 2n?? (respectively, S 4k?? ??S 4n?? ) equivariant under the Hopf action of the circle (respectively, of the group S 3 of unit quaternions) is naturally isomorphic to that of the Stiefel manifold V k ( C n ) (respectively, V k ( H n ) ).

This paper studies combinatorial properties of the 'complex of planar injective words', a chain complex of modules over the Temperley-Lieb algebra that arose in our work on homological stability. Despite being a linear rather than a discrete object, our chain complex nevertheless exhibits interesting combinatorial properties. We show that the Euler characteristic of this complex is the n-th Fine number. We obtain an alternating sum formula for the representation given by its top-dimensional homology module and, under further restrictions on the ground ring, we decompose this module in terms of certain standard Young tableaux. This trio of results - inspired by results of Reiner and Webb for the complex of injective words - can be viewed as an interpretation of the n-th Fine number as the 'planar' or 'Dyck path' analogue of the number of derangements of n letters. This interpretation has precursors in the literature, but here emerges naturally from considerations in homological stability. Our final result shows a surprising connection between the boundary maps of our complex and the Jacobsthal numbers.

Commutative d -torsion K -theory is a variant of topological K -theory constructed from commuting unitary matrices of order dividing d . Such matrices appear as solutions of linear constraint systems that play a role in the study of quantum contextuality and in applications to operator-theoretic problems motivated by quantum information theory. Using methods from stable homotopy theory we modify commutative d -torsion K -theory into a cohomology theory which can be used for studying operator solutions of linear constraint systems. This provides an interesting connection between stable homotopy theory and quantum information theory.

In this paper we introduce a simplicial analogue of principal bundles with commutativity structure and their classifying spaces defined for topological groups. Our construction is a variation of the W ¯ ¯ ¯ ¯ ¯ -construction for simplicial groups. We show that the commutative W ¯ ¯ ¯ ¯ ¯ -construction is a classifying space for simplicial principal bundles with a commutativity structure and geometric realization relates our constructions to the topological version.

This paper provides conditions for Morava K-theory to commute with certain homotopy limits. These conditions extend previous work on this question by allowing for homotopy limits of sequences of spectra that are not uniformly bounded below. As an application, we prove the K(n) -local triviality (for sufficiently large n ) of the algebraic K-theory of algebras over truncated Brown-Peterson spectra, building on work of Bruner and Rognes and extending a classical theorem of Mitchell on K(n) -local triviality of the algebraic K-theory spectrum of the integers for large enough n.

The orthogonal and unitary calculi give a method to study functors from the category of real or complex inner product spaces to the category of based topological spaces. We construct functors between the calculi from the complexification-realification adjunction between real and complex inner product spaces. These allow for movement between the versions of calculi, and comparisons between the Taylor towers produced by both calculi. We show that when the inputted orthogonal functor is weakly polynomial, the Taylor tower of the functor restricted through realification and the restricted Taylor tower of the functor agree up to weak equivalence. We further lift the homotopy level comparison of the towers to a commutative diagram of Quillen functors relating the model categories for orthogonal calculus and the model categories for unitary calculus.

We analyze a model for the homotopy theory of complete filtered Z -graded L ∞ -algebras, which lends itself well to computations in deformation theory and homotopical algebra. We first give an explicit proof of an unpublished result of E. Getzler which states that the category Lie ˆ [1 ] ∞ of such L ∞ -algebras and filtration-preserving ∞ -morphisms admits the structure of a category of fibrant objects (CFO) for a homotopy theory. As a novel application, we use this result to show that, under some mild conditions, every L ∞ -quasi-isomorphism between L ∞ -algebras in Lie ˆ [1 ] ∞ has a filtration preserving homotopy inverse. Finally, building on previous joint work with V. Dolgushev, we prove that the simplicial Maurer--Cartan functor, which assigns a Kan simplicial set to each complete filtered L ∞ -algebra, is an exact functor between the respective categories of fibrant objects. We interpret this as an optimal homotopy-theoretic generalization of the classical Goldman--Millson theorem from deformation theory. One immediate application is the `` ∞ -categorical'' uniqueness theorem for homotopy transferred structures previously sketched by the author in arXiv:1612.07868.

Complete digraphs are referred to in the combinatorics literature as tournaments. We consider a family of semi-simplicial complexes, that we refer to as "tournaplexes", whose simplices are tournaments. In particular, given a digraph G , we associate with it a "flag tournaplex" which is a tournaplex containing the directed flag complex of G , but also the geometric realisation of cliques that are not directed. We define several types of filtrations on tournaplexes, and exploiting persistent homology, we observe that flag tournaplexes provide finer means of distinguishing graph dynamics than the directed flag complex. We then demonstrate the power of these ideas by applying them to graph data arising from the Blue Brain Project's digital reconstruction of a rat's neocortex.

Let f 1 ,..., f k :M→N be maps between closed manifolds, N( f 1 ,..., f k ) and R( f 1 ,..., f k ) be the Nielsen and the Reideimeister coincidence numbers respectively. In this note, we relate R( f 1 ,..., f k ) with R( f 1 , f 2 ),...,R( f 1 , f k ) . When N is a torus or a nilmanifold, we compute R( f 1 ,..., f k ) which, in these cases, is equal to N( f 1 ,..., f k ) .