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We describe compatible open books for the fiber connected sum along binding components and along multisections of open books. As an application, the first description provides simple ways of constructing open books supporting all tight contact structures on , recovering a result by Van Horn-Morris, as well as an open book supporting the result of a Lutz twist along a binding component of an open book, recovering a result by Ozbagci and Pamuk.
We prove a theorem that bounds the Heegaard genus from below under special kinds of toroidal amalgamations of –manifolds. As a consequence, we conclude that for any pair of knots , where denotes the tunnel number of .
We consider paths of Hamiltonian diffeomorphisms preserving a given compact monotone lagrangian in a symplectic manifold that extend to an –Hamiltonian action. We compute the leading term of the associated lagrangian Seidel elements. We show that such paths minimize the lagrangian Hofer length. Finally, we apply these computations to lagrangian uniruledness and to give a nice presentation of the quantum cohomology of real lagrangians in monotone symplectic toric manifolds.
For a closed, orientable hyperbolic –manifold and an onto homomorphism that is not induced by a fibration , we bound the ranks of the subgroups for , below, linearly in . The key new ingredient is the following result: if is a closed, orientable hyperbolic –manifold and is a connected, two-sided incompressible surface of genus that is not a fiber or semifiber, then a reduced homotopy in has length at most .
We first classify singular fibers of proper stable maps of –dimensional manifolds with boundary into surfaces. Then we compute the cohomology groups of the associated universal complex of singular fibers, and obtain certain cobordism invariants for Morse functions on compact surfaces with boundary.
We give a generalization of the concept of near-symplectic structures to dimensions. According to our definition, a closed –form on a –manifold is near-symplectic if it is symplectic outside a submanifold of codimension where vanishes. We depict how this notion relates to near-symplectic –manifolds and broken Lefschetz fibrations via some examples. We define a generalized broken Lefschetz fibration as a singular map with indefinite folds and Lefschetz-type singularities. We show that, given such a map on a –manifold over a symplectic base of codimension , the total space carries such a near-symplectic structure whose singular locus corresponds precisely to the singularity set of the fibration. A second part studies the geometry around the codimension- singular locus . We describe a splitting property of the normal bundle that is also present in dimension four. A tubular neighbourhood theorem for is provided, which has a Darboux-type theorem for near-symplectic forms as a corollary.
Let be a weakly reducible unstabilized genus three Heegaard splitting in an orientable, irreducible –manifold . In this article, we prove that either the disk complex is contractible or is critical. Hence, the topological index of is if is topologically minimal.
We modify transchromatic character maps of the second author to land in a faithfully flat extension of Morava –theory. Our construction makes use of the interaction between topological and algebraic localization and completion. As an application we prove that centralizers of tuples of commuting prime-power order elements in good groups are good and we compute a new example.
Although it is well known that the complex cobordism ring is isomorphic to the polynomial ring , an explicit description for convenient generators has proven to be quite elusive. The focus of the following is to construct complex cobordism polynomial generators in many dimensions using smooth projective toric varieties. These generators are very convenient objects since they are smooth connected algebraic varieties with an underlying combinatorial structure that aids in various computations. By applying certain torus-equivariant blow-ups to a special class of smooth projective toric varieties, such generators can be constructed in every complex dimension that is odd or one less than a prime power. A large amount of evidence suggests that smooth projective toric varieties can serve as polynomial generators in the remaining dimensions as well.
We give a natural construction and a direct proof of the Adams isomorphism for equivariant orthogonal spectra. More precisely, for any finite group , any normal subgroup of , and any orthogonal –spectrum , we construct a natural map of orthogonal –spectra from the homotopy –orbits of to the derived –fixed points of , and we show that is a stable weak equivalence if is cofibrant and –free. This recovers a theorem of Lewis, May and Steinberger in the equivariant stable homotopy category, which in the case of suspension spectra was originally proved by Adams. We emphasize that our Adams map is natural even before passing to the homotopy category. One of the tools we develop is a replacement-by-–spectra construction with good functorial properties, which we believe is of independent interest.
We explicitly construct pseudo-Anosov maps on the closed surface of genus with orientable foliations whose stretch factor is a Salem number with algebraic degree . Using this result, we show that there is a pseudo-Anosov map whose stretch factor has algebraic degree , for each positive even integer such that .
We define a new –dimensional symplectic cut and paste operation which is analogous to Fintushel and Stern’s rational blow-down. We use this operation to produce multiple constructions of symplectic smoothly exotic complex projective spaces blown up eight, seven, and six times. We also show how this operation can be used in conjunction with knot surgery to construct an infinite family of minimal exotic smooth structures on the complex projective space blown-up seven times.
We study the relationship between exotic ’s and Stein surfaces as it applies to smoothing theory on more general open –manifolds. In particular, we construct the first known examples of large exotic ’s that embed in Stein surfaces. This relies on an extension of Casson’s embedding theorem for locating Casson handles in closed –manifolds. Under sufficiently nice conditions, we show that using these ’s as end-summands produces uncountably many diffeomorphism types while maintaining independent control over the genus-rank function and the Taylor invariant.
We prove that, if a pair of semicosimplicial spaces arises from a colored operad, then the semitotalization has the homotopy type of a relative double loop space and the pair is weakly equivalent to an explicit algebra over the two dimensional Swiss-cheese operad .
A knot in the –sphere is called an L-space knot if it admits a nontrivial Dehn surgery yielding an L-space, ie a rational homology –sphere with the smallest possible Heegaard Floer homology. Given a knot , take an unknotted circle and twist times along to obtain a twist family . We give a sufficient condition for to contain infinitely many L-space knots. As an application we show that for each torus knot and each hyperbolic Berge knot , we can take so that the twist family contains infinitely many hyperbolic L-space knots. We also demonstrate that there is a twist family of hyperbolic L-space knots each member of which has tunnel number greater than one.
We study the exponents of metastable homotopy of mod Moore spaces. We prove that the double loop space of –dimensional mod Moore spaces has a multiplicative exponent below the range of times the connectivity. As a consequence, the homotopy groups of –dimensional mod Moore spaces have an exponent of below the range of times the connectivity.
Given any (open or closed) cover of a space , we associate certain homotopy classes of maps from to –spheres. These homotopy invariants can then be considered as obstructions for extending covers of a subspace to a cover of all of . We use these obstructions to obtain generalizations of the classic KKM (Knaster–Kuratowski–Mazurkiewicz) and Sperner lemmas. In particular, we show that in the case when is a –sphere and is a –disk there exist KKM-type lemmas for covers by sets if and only if the homotopy group is nontrivial.
Let be a compact connected simple Lie group. Any principal –bundle over is classified by an integer , and we denote the corresponding gauge group by . We prove that if and only if , and for any prime if and only if , where is the gcd of integers .