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Register a new userClassical Morse theory revisited I -- Backward $lambda$-Lemma and homotopy type
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Authors
Joa Weber
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We introduce two tools, dynamical thickening and flow selectors, to overcome the infamous discontinuity of the gradient flow endpoint map near non-degenerate critical points. More precisely, we interpret the stable fibrations of certain Conley pairs $(N,L)$, established in [2,3], as a dynamical thickening of the stable manifold. As a first application and to illustrate efficiency of the concept we reprove a fundamental theorem of classical Morse theory, Milnors homotopical cell attachment theorem [1]. Dynamical thickening leads to a conceptually simple and short proof.
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We reprove the $lambda$-Lemma for finite dimensional gradient flows by generalizing the well-known contraction method proof of the local (un)stable manifold theorem. This only relies on the forward Cauchy problem. We obtain a rather quantitative description of (un)stable foliations which allows to equip each leaf with a copy of the flow on the central leaf -- the local (un)stable manifold. These dynamical thickenings are key tools in our recent work [Web]. The present paper provides their construction.
The paper is devoted to finding conditions to the existence of a self-indexing energy function for Morse-Smale diffeomorphisms on a 3-manifold. These conditions involve how the stable and unstable manifolds of saddle points are embedded in the ambient manifold. We also show that the existence of a self-indexing energy function is equivalent to the existence of a Heegaard splitting of a special type with respect to the considered diffeomorphism.
We prove that that the homotopy type of the path connected component of the identity in the contactomorphism group is characterized by the homotopy type of the diffeomorphism group plus some data provided by the topology of the formal contactomorphism space. As a consequence, we show that every connected component of the space of Legendrian long knots in $R^3$ has the homotopy type of the corresponding smooth long knot space. This implies that any connected component of the space of Legendrian embeddings in $NS^3$ is homotopy equivalent to the space $K(G,1)timesU(2)$, with $G$ computed by A. Hatcher and R. Budney. Similar statements are proven for Legendrian embeddings in $R^3$ and for transverse embeddings in $NS^3$. Finally, we compute the homotopy type of the contactomorphisms of several tight $3$-folds: $NS^1 times NS^2$, Legendrian fibrations over compact orientable surfaces and finite quotients of the standard $3$-sphere. In fact, the computations show that the method works whenever we have knowledge of the topology of the diffeomorphism group. We prove several statements on the way that have interest by themselves: the computation of the homotopy groups of the space of non-parametrized Legendrians, a multiparametric convex surface theory, a characterization of formal Legendrian simplicity in terms of the space of tight contact structures on the complement of a Legendrian, the existence of common trivializations for multi-parametric families of tight $R^3$, etc.
We use the homotopy Brouwer theory of Handel to define a Poincar{e} index between two orbits for an orientation preserving fixed point free homeomorphism of the plane. Furthermore, we prove that this index is almost additive.
Homotopy Brouwer theory is a tool to study the dynamics of surface homeomorphisms. We introduce and illustrate the main objects of homotopy Brouwer theory, and provide a proof of Handels fixed point theorem. These are the notes of a mini-course held during the workshop Superficies en Montevideo in March 2012.
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