Mathematics

We compute the Lusternik-Schnirelmann category and the topological complexity of no k -equal manifolds M (k) d (n) for certain values of d , k and n . This includes instances where M (k) d (n) is known to be rationally non-formal. The key ingredient in our computations is the knowledge of the cohomology ring H ∗ ( M (k) d (n)) as described by Dobrinskaya and Turchin. A fine tuning comes from the use of obstruction theory techniques.

In the present work, we realize the space of string 2-links L as a free algebra over a colored operad denoted SCL (for "Swiss-Cheese for links"). This result extends works of Burke and Koytcheff about the quotient of L by its center and is compatible with Budney's freeness theorem for long knots. From an algebraic point of view, our main result refines Blaire, Burke and Koytcheff's theorem on the monoid of isotopy classes of string links. Topologically, it expresses the homotopy type of the isotopy class of a string 2-link in terms of the homotopy types of the classes of its prime factors.

Let B be a Borel subgroup of GL n (C) and T a maximal torus contained in B . Then T acts on GL n (C)/B and every Schubert variety is T -invariant. We say that a Schubert variety is of complexity k if a maximal T -orbit in X w has codimension k . In this paper, we discuss topology, geometry, and combinatorics related to Schubert varieties of complexity one.

The aim of this paper is to show the importance of the Steenrod construction of homology theories for the disassembly process in surgery on a generalized n -manifold X n , in order to produce an element of generalized homology theory, which is basic for calculations. In particular, we show how to construct an element of the n -th Steenrod homology group H st n ( X n , L + ), where L + is the connected covering spectrum of the periodic surgery spectrum L , avoiding the use of the geometric splitting procedure, which is standardly used in surgery on topological manifolds.

In this note we present a new homology theory, we call it geometric homology theory (or GHT for brevity). We prove that the homology groups of GHT are isomorphic to the singular homology groups, which solves a Conjecture of Voronov. GHT has several nice properties compared with singular homology, which makes itself more suitable than singular homology in some situations, especially in chain-level theories. We will develop further of this theory in our sequel paper.

For a Kan complex with a vertex, we have the notion of its simplicial homotopy groups. In this paper, for a weak complicial set in the sense of Verity with a vertex, we construct monoids which are a generalization of simplicial homotopy groups.

We prove the height two case of a conjecture of Hovey and Strickland that provides a K(n) -local analogue of the Hopkins--Smith thick subcategory theorem. Our approach first reduces the general conjecture to a problem in arithmetic geometry posed by Chai. We then use the Gross--Hopkins period map to verify Chai's Hope at height two and all primes. Along the way, we show that the graded commutative ring of completed cooperations for Morava E -theory is coherent, and that every finitely generated Morava module can be realized by a K(n) -local spectrum as long as 2p−2> n 2 +n . Finally, we deduce consequences of our results for descent of Balmer spectra.

We determine π ∗ (BDif f ∂ ( D 2n ))⊗Q for 2n≥6 completely in degrees ∗≤4n−10 , far beyond the pseudoisotopy stable range. Furthermore, above these degrees we discover a systematic structure in these homotopy groups: we determine them outside of certain "bands" of degrees.

Following ideas of Lurie, we give in this article a general construction of equivariant elliptic cohomology without restriction to characteristic zero. Specializing to the universal elliptic curve we obtain in particular equivariant spectra of topological modular forms. We compute the fixed points of these spectra for the circle group and more generally for tori.

In a triangulated category, cofibre fill-ins always exist. Neeman showed that there is always at least one "good" fill-in, i.e., one whose mapping cone is exact. Verdier constructed a fill-in of a particular form in his proof of the 4×4 lemma, which we call "Verdier good". We show that for several classes of morphisms of exact triangles, the notions of good and Verdier good agree. We prove a lifting criterion for commutative squares in terms of (Verdier) good fill-ins. Using our results on good fill-ins, we also prove a pasting lemma for homotopy cartesian squares.