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Michael Hutchings

Publications and source records attributed to Michael Hutchings.

44 records · Page 3Linked to original sources
arXiv2007

Floer homology of families I

In principle, Floer theory can be extended to define homotopy invariants of families of equivalent objects (e.g. Hamiltonian isotopic symplectomorphisms, 3-manifolds, Legendrian knots, etc.) parametrized by a smooth manifold B. The invariant of a family consists of a filtered chain homotopy type, which gives rise to a spectral sequence whose E^2 term is the homology of B with twisted coefficients in the Floer homology of the fibers. The filtered chain homotopy type also gives rise to a "family Floer homology" to which the spectral sequence converges. For any particular version of Floer theory, some analysis needs to be carried out in order to turn this principle into a theorem. This paper constructs the invariant in detail for the model case of finite dimensional Morse homology, and shows that it recovers the Leray-Serre spectral sequence of a smooth fiber bundle. We also generalize from Morse homology to Novikov homology, which involves some additional subtleties.

math.SG↗
arXiv2007

Gluing pseudoholomorphic curves along branched covered cylinders I

This paper and its sequel prove a generalization of the usual gluing theorem for two index 1 pseudoholomorphic curves u_+ and u_- in the symplectization of a contact 3-manifold. We assume that for each embedded Reeb orbit gamma, the total multiplicity of the negative ends of u_+ at covers of gamma agrees with the total multiplicity of the positive ends of u_- at covers of gamma. However, unlike in the usual gluing story, here the individual multiplicities are allowed to differ. In this situation, one can often glue u_+ and u_- to an index 2 curve by inserting genus zero branched covers of R-invariant cylinders between them. We establish a combinatorial formula for the signed count of such gluings. As an application, we deduce that the differential d in embedded contact homology satisfies d^2=0. This paper explains the more algebraic aspects of the story, and proves the above formulas using some analytical results from part II.

math.SG↗
arXiv2005

The periodic Floer homology of a Dehn twist

The periodic Floer homology of a surface symplectomorphism, defined by the first author and M. Thaddeus, is the homology of a chain complex which is generated by certain unions of periodic orbits, and whose differential counts certain embedded pseudoholomorphic curves in R cross the mapping torus. It is conjectured to recover the Seiberg-Witten Floer homology of the mapping torus for most spin-c structures, and is related to a variant of contact homology. In this paper we compute the periodic Floer homology of some Dehn twists.

math.SG↗
arXiv2001

An index inequality for embedded pseudoholomorphic curves in symplectizations

Let $Σ$ be a surface with a symplectic form, let $ϕ$ be a symplectomorphism of $Σ$, and let $Y$ be the mapping torus of $ϕ$. We show that the dimensions of moduli spaces of embedded pseudoholomorphic curves in $\R\times Y$, with cylindrical ends asymptotic to periodic orbits of $ϕ$ or multiple covers thereof, are bounded from above by an additive relative index. We deduce some compactness results for these moduli spaces. This paper establishes some of the foundations for a program with Michael Thaddeus, to understand the Seiberg-Witten Floer homology of $Y$ in terms of such pseudoholomorphic curves. Analogues of our results should also hold in three dimensional contact homology.

math.SG↗
arXiv2000

Reidemeister torsion in generalized Morse theory

In two previous papers with Yi-Jen Lee, we defined and computed a notion of Reidemeister torsion for the Morse theory of closed 1-forms on a finite dimensional manifold. The present paper gives an a priori proof that this Morse theory invariant is a topological invariant. It is hoped that this will provide a model for possible generalizations to Floer theory.

math.DG↗
arXiv1999

Circle-valued Morse theory and Reidemeister torsion

Let X be a closed manifold with zero Euler characteristic, and let f: X --> S^1 be a circle-valued Morse function. We define an invariant I which counts closed orbits of the gradient of f, together with flow lines between the critical points. We show that our invariant equals a form of topological Reidemeister torsion defined by Turaev [Math. Res. Lett. 4 (1997) 679-695]. We proved a similar result in our previous paper [Topology, 38 (1999) 861-888], but the present paper refines this by separating closed orbits and flow lines according to their homology classes. (Previously we only considered their intersection numbers with a fixed level set.) The proof here is independent of the previous proof and also simpler. Aside from its Morse-theoretic interest, this work is motivated by the fact that when X is three-dimensional and b_1(X)>0, the invariant I equals a counting invariant I_3(X) which was conjectured in our previous paper to equal the Seiberg-Witten invariant of X. Our result, together with this conjecture, implies that the Seiberg-Witten invariant equals the Turaev torsion. This was conjectured by Turaev [Math. Res. Lett. 4 (1997) 679-695] and refines the theorem of Meng and Taubes [Math. Res. Lett. 3 (1996) 661-674].

dg-ga↗
arXiv1996

Circle-valued Morse theory, Reidemeister torsion, and Seiberg-Witten invariants of 3-manifolds

Let X be a compact oriented Riemannian manifold and let $ϕ:X\to S^1$ be a circle-valued Morse function. Under some mild assumptions on $ϕ$, we prove a formula relating: (a) the number of closed orbits of the gradient flow of $ϕ$ of any given degree; (b) the torsion of a ``Morse complex'', which counts gradient flow lines between critical points of $ϕ$; and (c) a kind of Reidemeister torsion of X determined by the homotopy class of $ϕ$. When $\dim(X)=3$ and $b_1(X)>0$, we state a conjecture analogous to Taubes's ``SW=Gromov'' theorem, and we use it to deduce (for closed manifolds, modulo signs) the Meng-Taubes relation between the Seiberg- Witten invariants and the ``Milnor torsion'' of X.

dg-ga↗