Mathematics

We study a natural generalization of covering projections defined in terms of unique lifting properties. A map p:E→X has the "continuous path-covering property" if all paths in X lift uniquely and continuously (rel. basepoint) with respect to the compact-open topology. We show that maps with this property are closely related to fibrations with totally path-disconnected fibers and to the natural quotient topology on the homotopy groups. In particular, the class of maps with the continuous path-covering property lies properly between Hurewicz fibrations and Serre fibrations with totally path-disconnected fibers. We extend the usual classification of covering projections to a classification of maps with the continuous path-covering property in terms of topological π 1 : for any path-connected Hausdorff space X , maps E→X with the continuous path-covering property are classified up to weak equivalence by subgroups H≤ π 1 (X, x 0 ) with totally path-disconnected coset space π 1 (X, x 0 )/H . Here, "weak equivalence" refers to an equivalence relation generated by formally inverting bijective weak homotopy equivalences.

This paper addresses two questions: (a) can we identify a sensible class of 2-parameter persistence modules on which the rank invariant is complete? (b) can we determine efficiently whether a given 2-parameter persistence module belongs to this class? We provide positive answers to both questions, and our class of interest is that of rectangle-decomposable modules. Our contributions include: on the one hand, a proof that the rank invariant is complete on rectangle-decomposable modules, together with an inclusion-exclusion formula for counting the multiplicities of the summands; on the other hand, algorithms to check whether a module induced in homology by a bifiltration is rectangle-decomposable, and to decompose it in the affirmative, with a better complexity than state-of-the-art decomposition methods for general 2-parameter persistence modules. Our algorithms are backed up by a new structure theorem, whereby a 2-parameter persistence module is rectangle-decomposable if, and only if, its restrictions to squares are. This local characterization is key to the efficiency of our algorithms, and it generalizes previous conditions derived for the smaller class of block-decomposable modules. It also admits an algebraic formulation that turns out to be a weaker version of the one for block-decomposability. By contrast, we show that general interval-decomposability does not admit such a local characterization, even when locality is understood in a broad sense. Our analysis focuses on the case of modules indexed over finite grids, the more general cases are left as future work.

We develop a theory of infinity properads enriched in a general symmetric monoidal infinity category. These are defined as presheaves, satisfying a Segal condition and a Rezk completeness condition, over certain categories of graphs. In particular, we introduce a new category of level graphs which also allow us to give a framework for algebras over an enriched infinity properad. We show that one can vary the category of graphs without changing the underlying theory. We also show that infinity properads cannot always be rectified, indicating that a conjecture of the second author and Robertson is unlikely to hold. This stands in stark contrast to the situation for infinity operads, and we further demarcate these situations by examining the cases of infinity dioperads and infinity output properads. In both cases, we provide a rectification theorem that says that each up-to-homotopy object is equivalent to a strict one.

This note is purely expository and is an extended version of math review to the paper [AP19]=arXiv:1901.07918v3 by S. Abramyan and T. Panov published in Proc. of Steklov Math. Inst. 305 (2019). The authors construct simplicial complexes for whose moment-angle complexes certain homotopy classes are non-trivial. I present in a shorter and clearer way the main definition and the statement of Theorem 5.1 from [AP19]. The clarification reveals that the main definition used in the statements of the main results is not given [AP19].

We study the orientability of vector bundles with respect to a family of cohomology theories called EO -theories. The EO -theories are higher height analogues of real K -theory KO . For each EO -theory, we prove that the direct sum of i copies of any vector bundle is EO -orientable for some specific integer i . Using a splitting principal, we reduce to the case of the canonical line bundle over CP ∞ . Our method involves understanding the action of an order p subgroup of the Morava stabilizer group on the Morava E -theory of CP ∞ . Our calculations have another application: We determine the homotopy type of the S 1 -Tate spectrum associated to the trivial action of S 1 on all EO -theories.

We completely calculate the RO(G) -graded coefficients of ordinary equivariant cohomology where G is the dihedral group of order 2p for a prime p>2 both with constant and Burnside ring coefficients. The authors first proved it for p=3 and then the second author generalized it to arbitrary p . These are the first such calculations for a non-abelian group.

For an odd prime p , we study the image of the Thom map from Brown-Peterson cohomology of BP U n to the ordinary cohomology in dimensions 0≤i≤2p+2 , where BP U n is the classifying space of the projective unitary group P U n . Also we show that a family of well understood p -torsion cohomology classes y p,k ∈ H 2 p k+1 +2 (BP U n ; Z (p) ) are in the image of the Thom map.

This note contains a generalization to p>2 of the authors' previous calculations of the coefficients of (Z/2 ) n -equivariant ordinary cohomology with coefficients in the constant Z/2 -Mackey functor. The algberaic results by S.Kriz allow us to calculate the coefficients of the geometric fixed point spectrum Φ (Z/p ) n HZ/p , and more generally, the Z -graded coefficients of the localization of HZ/ p (Z/p ) n by inverting any chosen set of embeddings S 0 → S α i where α i are non-trivial irreducible representations. We also calculate the RO(G ) + -graded coefficients of HZ/ p (Z/p ) n , which means the cohomology of a point indexed by an actual (not virtual) representation. (This is the "non-derived" part, which has a nice algebraic description.)

The aim of this paper is three-fold: (i) we construct a naturally occurring highly homotopy coherent operad structure on the derivatives of the identity functor on structured ring spectra which can be described as algebras over an operad O in spectra, (ii) we prove that every connected O -algebra has a naturally occurring left action of the derivatives of the identity, and (iii) we show that there is a naturally occurring weak equivalence of highly homotopy coherent operads between the derivatives of the identity on O -algebras and the operad O . Along the way, we introduce the notion of N -colored operads with levels which -- by construction -- provides a precise algebraic framework for working with and comparing highly homotopy coherent operads, operads, and their algebras.

We determine the Lusternik-Schnirelmann category of the projective product spaces introduced by D. Davis. We also obtained an upper bound for the topological complexity of these spaces, which improves the estimate given by J. González, M. Grant, E. Torres-Giese, and M. Xicoténcatl.