On the Morse-Novikov number for 2-knots

Let $K\subset S^4$ be a 2-knot, that is, a smoothly embedded 2-sphere in $S^4$. The Morse-Novikov number $\mathcal M\mathcal N(K)$ is the minimal possible number of critical points of a Morse map $S^4\setminus K\to S^1$ belonging to the canonical class in $H^1(S^4\setminus K)$. We prove that for a classical knot $K\subset S^3$ the Morse-Novikov number of the spun knot $S(K)$ is $\leq 2\mathcal M\mathcal N(K)$. This enables us to compute $\mathcal M\mathcal N(S(K))$ for every classical knot $K$ with tunnel number 1.Comment: Latex, 14 page

Novikov homology, jump loci and Massey products

Let X be a finite CW-complex, denote its fundamental group by G. Let R be an n-dimensional complex repesentation of G. Any element A of the first cohomology group of X with complex coefficients gives rise to the exponential deformation of the representation R, which can be considered as a curve in the space of representations. We show that the cohomology of X with local coefficients corresponding to the generic point of this curve is computable from a spectral sequence starting from the cohomology of X with R-twisted coefficients. We compute the differentials of the spectral sequence in terms of Massey products. We show that the spectral sequence degenerates in case when X is a Kaehler manifold, and the representation R is semi-simple. If A is a real cohomology class, one associates to the triple (X,R,A) the twisted Novikov homology (a module over the Novikov ring). We show that the twisted Novikov Betti numbers equal the Betti numbers of X with coefficients in the above local system. We investigate the dependance of these numbers on A and prove that they are constant in the complement to a finite number of integral hyperplanes in the first cohomology group.Comment: Final version, references adde

Circle-valued Morse theory for complex hyperplane arrangements

Let A be an essential complex hyperplane arrangement in an n-dimensional complex vector space V. Let H denote the union of the hyperplanes, and M denote the complement to H in V. We develop the real-valued and circle-valued Morse theory for M and prove, in particular, that M has the homotopy type of a space obtained from a manifold fibered over a circle, by attaching cells of dimension n. We compute the Novikov homology of M for a large class of homomorphisms of the fundamental group of M to R.Comment: 15 pages, revised versio

On the tunnel number and the Morse-Novikov number of knots

We prove that the Morse-Novikov number of a link L in a 3-sphere is less than or equal to twice the tunnel number of L.Comment: 8 pages, revised and extende

On the fixed points of a Hamiltonian diffeomorphism in presence of fundamental group

Let M be a weakly monotone symplectic manifold, and H be a time-dependent Hamiltonian; we assume that the periodic orbits of the corresponding time-dependent Hamiltonian vector field are non-degenerate. We construct a refined version of the Floer chain complex associated to these data and any regular covering of M, and derive from it new lower bounds for the number of periodic orbits. We prove in particular that if the fundamental group of M is finite and solvable or simple, then the number of periodic orbits is not less than the minimal number of generators of the fundamental group. For a general closed symplectic manifold with infinite fundamental group, we show the existence of 1-periodic orbit of Conley-Zehnder index 1-n for any non-degenerate 1-periodic Hamiltonian system.Comment: revised and extended version; the estimates for periodic orbits of index 2-n adde

Morse-Novikov theory, Heegaard splittings and closed orbits of gradient flows

The works of Donaldson and Mark make the structure of the Seiberg-Witten invariant of 3-manifolds clear. It corresponds to certain torsion type invariants counting flow lines and closed orbits of a gradient flow of a circle-valued Morse map on a 3-manifold. We study these invariants using the Morse-Novikov theory and Heegaard splitting for sutured manifolds, and make detailed computations for knot complements.Comment: 27 pages, 12 figure

ON THE MORSE-NOVIKOV NUMBER FOR 2-KNOTS

International audienceLet K ⊂ S4 be a 2-knot. The Morse-Novikov number M N (K) is the minimal possible number of critical points of a Morse mapS4\K→S1belonging to the canonical class inH1(S4\K). We prove that for a classical knotK⊂S3the Morse-Novikov number of the spunknotS(K)is< 2 M N(K). This enables us to compute M N (S(K)) for every classical knot K with tunnel number