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Register a new userSmooth maps with singularities of bounded K-codimensions
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Authors
Yoshifumi Ando
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We will prove the relative homotopy principle for smooth maps with singularities of a given {cal K}-invariant class with a mild condition. We next study a filtration of the group of homotopy self-equivalences of a given manifold P by considering singularities of non-negative {cal K}-codimensions.
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Let $M$ be a smooth manifold and let $F$ be a codimension one, $C^infty$ foliation on $M$, with isolated singularities of Morse type. The study and classification of pairs $(M,F)$ is a challenging (and difficult) problem. In this setting, a classical result due to Reeb cite{Reeb} states that a manifold admitting a foliation with exactly two center-type singularities is a sphere. In particular this is true if the foliation is given by a function. Along these lines a result due to Eells and Kuiper cite{Ku-Ee} classify manifolds having a real-valued function admitting exactly three non-degenerate singular points. In the present paper, we prove a generalization of the above mentioned results. To do this, we first describe the possible arrangements of pairs of singularities and the corresponding codimension one invariant sets, and then we give an elimination procedure for suitable center-saddle and some saddle-saddle configurations (of consecutive indices). In the second part, we investigate if other classical results, such as Haefliger and Novikov (Compact Leaf) theorems, proved for regular foliations, still hold true in presence of singularities. At this purpose, in the singular set, $Sing(F)$ of the foliation $F$, we consider {em{weakly stable}} components, that we define as those components admitting a neighborhood where all leaves are compact. If $Sing(F)$ admits only weakly stable components, given by smoothly embedded curves diffeomorphic to $S^1$, we are able to extend Haefligers theorem. Finally, the existence of a closed curve, transverse to the foliation, leads us to state a Novikov-type result.
We construct a quasiconformally homogeneous hyperbolic Riemann surface-other than the hyperbolic plane-that does not admit a bounded pants decomposition. Also, given a connected orientable topological surface of infinite type with compact boundary components, we construct a complete hyperbolic metric on the surface that has bounded geometry but does not admit a bounded pants decomposition.
The local invariants of a meromorphic Abelian differential on a Riemann surface of genus $g$ are the orders of zeros and poles, and the residues at the poles. The main result of this paper is that with few exceptions, every pattern of orders and residues can be obtain by an Abelian differential. These exceptions are two families in genus zero when the orders of the poles are either all simple or all nonsimple. Moreover, we even show that the pattern can be realized in each connected component of strata. Finally we give consequences of these results in algebraic and flat geometry. The main ingredient of the proof is the flat representation of the Abelian differentials.
Let $W$ be a compact smooth $4$-manifold that deformation retract to a PL embedded closed surface. One can arrange the embedding to have at most one non-locally-flat point, and near the point the topology of the embedding is encoded in the singularity knot $K$. If $K$ is slice, then $W$ has a smooth spine, i.e., deformation retracts onto a smoothly embedded surface. Using the obstructions from the Heegaard Floer homology and the high-dimensional surgery theory, we show that $W$ has no smooth spines if $K$ is a knot with nonzero Arf invariant, a nontrivial L-space knot, the connected sum of nontrivial L-space knots, or an alternating knot of signature $<-4$. We also discuss examples where the interior of $W$ is negatively curved.
We give a version in characteristic $p>0$ of Mumfords theorem characterizing a smooth complex germ of surface $(X,x)$ by the triviality of the topological fundamental group of $U=Xsetminus {x}$. This note relies on discussions the authors had during the Christmas break 2009/10 in Ivry. They have been written down by Hel`ene in the night when Eckart died, as a despaired sign of love.
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