Closed Orbits of Gradient Flows and Logarithms of Non-Abelian Witt Vectors

We consider the flows generated by generic gradients of Morse maps f: M S ¹. To each such flow we associate an invariant counting the closed orbits of the flow. Each closed orbit is counted with the weight derived from its index and homotopy class. The resulting invariant is called the eta function, and lies in a suitable quotient of the Novikov completion of the group ring of the fundamental group of M. Its Abelianization coincides with the logarithm of the twisted Lefschetz zeta function of the flow. For C ⁰-generic gradients we obtain a formula expressing the eta function in terms of the torsion of a special homotopy equivalence between the Novikov complex of the gradient flow and the completed simplicial chain complex of the universal cover.     

Rigidity and gluing for Morse and Novikov complexes

We obtain rigidity and gluing results for the Morse complex of a real-valued Morse function as well as for the Novikov complex of a circle-valued Morse function. A rigidity result is also proved for the Floer complex of a hamiltonian defined on a closed symplectic manifold (M,) with c₁|_2(M)=[]|_2(M)=0. The rigidity results for these complexes show that the complex of a fixed generic function/hamiltonian is a retract of the Morse (respectively Novikov or Floer) complex of any other sufficiently C⁰ close generic function/hamiltonian. The gluing result is a type of Mayer-Vietoris formula for the Morse complex. It is used to express algebraically the Novikov complex up to isomorphism in terms of the Morse complex of a fundamental domain. Morse cobordisms are used to compare various Morse-type complexes without the need of bifurcation theory.     

On the Whitehead group of Novikov rings associated to irrational homomorphisms

Given a homomorphism ξ:G→R we show that the natural map from the Whitehead group of G to the Whitehead group of the Novikov ring is surjective. The group is of interest for the simple chain homotopy type of the Novikov complex. It also contains the Latour obstruction for the existence of a nonsingular closed 1-form within a fixed cohomology class ξ∈H¹(M;R), where M is a closed connected smooth manifold.     

Lusternik—Schnirelman theory for closed 1-forms

S. P. Novikov developed an analog of the Morse theory for closed 1-forms. In this paper we suggest an analog of the Lusternik - Schnirelman theory for closed 1-forms. For any cohomology class x ? H¹(M,\R) \xi\in H^1(M,\R) we define an integer \cl(x) \cl(\xi) (the cup-length associated with x \xi ); we prove that any closed 1-form representing x \xi has at least \cl(x)-1 \cl(\xi)-1 critical points. The number \cl(x) \cl(\xi) is defined using cup-products in cohomology of some flat line bundles, such that their monodromy is described by complex numbers, which are not Dirichlet units.     

Morse Novikov theory and cohomology with forward supports

We present a new approach to Morse and Novikov theories, based on the deRham Federer theory of currents, using the finite volume flow technique of Harvey and Lawson [HL]. In the Morse case, we construct a noncompact analogue of the Morse complex, relating a Morse function to the cohomology with compact forward supports of the manifold. This complex is then used in Novikov theory, to obtain a geometric realization of the Novikov Complex as a complex of currents and a new characterization of Novikov Homology as cohomology with compact forward supports. Two natural ``backward-forward' dualities are also established: a Lambda duality over the Novikov Ring and a Topological Vector Space duality over the reals.     

The algebraic construction of the Novikov complex of a circle-valued Morse function

The Novikov complex of a circle-valued Morse function is constructed algebraically from the Morse-Smale complex of the restriction of the real-valued Morse function to a fundamental domain of the pullback infinite cyclic cover of M. Received: 23 November 2000 / Revised version: 3 May 2001 / Published online: 28 February 2002     

Surgery on the Novikov complex

Let M be a closed connected smooth manifold with dim M=n6, and : ₁(M) Z be an epimorphism. Denote by the group ring of ₁(M) and let ^– be its Novikov completion. Let D _* be a free-based finitely generated chain complex over ^– . Assume that D _ii=0 for i1 and in–1 and that D _* has the same simple homotopy type as the Novikov-completed simplicial chain complex of the universal covering M. Let N be an integer. We prove that D _* can be realized, up to the terms of ^– of degree N as the Novikov complex of a Morse map : M S ¹, belonging to . Applications to Arnold's conjectures and to the theory of fibering of M over S ¹ are given.     

Deformations of trianalytic subvarieties of hyperkähler manifolds

Let M be a compact complex manifold equipped with a hyperk?hler metric, and X be a closed complex analytic subvariety of M. In alg-geom 9403006, we proved that X is trianalytic (i.e., complex analytic with respect to all complex structures induced by the hyperk?hler structure), provided that M is generic in its deformation class. Here we study the complex analytic deformations of trianalytic subvarieties. We prove that all deformations of X are trianalytic and naturally isomorphic to X as complex analytic varieties. We show that this isomorphism is compatible with the metric induced from M. Also, we prove that the Douady space of complex analytic deformations of X in M is equipped with a natural hyperk?hler structure.     

On the Compactness Theorem for Differential Forms

Kichenassamy found conditions under which the space W _p ^k of differential forms on a closed manifold M embeds compactly in the space F _p ^k of currents on M. We give a version of Kichenassamy's theorem for an arbitrary Banach complex and, in particular, for an elliptic differential complex on a closed manifold.     

Superconnections and higher index theory

LetM be a smooth closed spin manifold. The higher index theorem, as given for example in Proposition 6.3 of [CM], computes the pairing between the group cohomology of ₁(M) and the Chern character of the higher index of a Dirac-type operator on M. Using superconnections, we give a heat equation proof of this theorem on the level of differential forms on a noncommutative base space. As a consequence, we obtain a new proof of the Novikov conjecture for hyperbolic groups.     

The Role of the Legendre Transform in the Study of the Floer Complex of Cotangent Bundles

Consider a classical Hamiltonian H on the cotangent bundle T*M of a closed orientable manifold M, and let L:TM → R be its Legendre‐dual Lagrangian. In a previous paper we constructed an isomorphism Φ from the Morse complex of the Lagrangian action functional that is associated to L to the Floer complex that is determined by H. In this paper we give an explicit construction of a homotopy inverse Ψ of Φ. Contrary to other previously defined maps going in the same direction, Ψ is an isomorphism at the chain level and preserves the action filtration. Its definition is based on counting Floer trajectories on the negative half‐cylinder that on the boundary satisfy half of the Hamilton equations. Albeit not of Lagrangian type, such a boundary condition defines Fredholm operators with good compactness properties. We also present a heuristic argument which, independently of any Fredholm and compactness analysis, explains why the spaces of maps that are used in the definition of Φ and Ψ are the natural ones. The Legendre transform plays a crucial role both in our rigorous and in our heuristic arguments. We treat with some detail the delicate issue of orientations and show that the homology of the Floer complex is isomorphic to the singular homology of the loop space of M with a system of local coefficients, which is defined by the pullback of the second Stiefel‐Whitney class of TM on 2‐tori in M.© 2015 Wiley Periodicals, Inc.     

Weakly Integrally Closed Numerical Monoids and Forbidden Patterns

An integral domain D is weakly integrally closed if whenever x is in the quotient field of D, and J is a nonzero finitely generated ideal of D such that xJ ? J ², then x is in D. We define weakly integrally closed (WIC) numerical monoids similarly. If a monoid algebra is weakly integrally closed, then so is the monoid. The characteristic function of a numerical monoid M can be thought of as an infinite binary string s(M). A pattern of finitely many 0's and 1's is called forbidden if whenever s(M) contains it, then M is not weakly integrally closed. The pattern 11011 is forbidden. We show that a numerical monoid M is WIC if and only if s(M) contains no forbidden patterns. We also show that for every finite set S of forbidden patterns, there exists a numerical monoid M that is not WIC and for which s(M) contains no stretch (in a natural sense) of a pattern in S.     

The Morse–Witten complex via dynamical systems

Given a smooth closed manifold M, the Morse–Witten complex associated to a Morse function f and a Riemannian metric g on M consists of chain groups generated by the critical points of f and a boundary operator counting isolated flow lines of the negative gradient flow. Its homology reproduces singular homology of M. The geometric approach presented here was developed in Weber [Der Morse–Witten Komplex, Diploma Thesis, TU Berlin, 1993] and is based on tools from hyperbolic dynamical systems. For instance, we apply the Grobman–Hartman theorem and the λ-lemma (Inclination Lemma) to analyze compactness and define gluing for the moduli space of flow lines.     

Branched immersions and braids

Branch points of a real 2-surface Σ in a 4-manifold M generalize branch points of complex curves in complex surfaces: for example, they can occur as singularities of minimal surfaces. We investigate such a branch point p when Σ is topologically embedded. It defines a link L(p), the components of which are closed braids with the same axis up to orientation. If Σ is closed without boundary, the contribution of p to the degree of the normal bundle of Σ in M can be computed on the link L(p), in terms of the algebraic crossing numbers of its components and of their linking numbers with one another.      

Counterexamples to the Kneser conjecture in dimension four

We construct a connected closed orientable smooth four-manifold whose fundamental group is the free product of two non-trivial groups such that it is not homotopy equivalent toM ₀#M ₁ unlessM ₀ orM ₁ is homeomorphic toS ⁴. LetN be the nucleus of the minimal elliptic Enrique surfaceV ₁(2, 2) and putM=N∪ _∂NN. The fundamental group ofM splits as ℤ/2 * ℤ/2. We prove thatM#k(S ²×S²) is diffeomorphic toM ₀#M ₁ for non-simply connected closed smooth four-manifoldsM ₀ andM ₁ if and only ifk≥8. On the other hand we show thatM is homeomorphic toM ₀#M ₁ for closed topological four-manifoldsM ₀ andM ₁ withπ ₁(M_i)=ℤ/2.     

Aubin’s Lemma for the Yamabe constants of infinite coverings and a positive mass theorem

Aubin’s Lemma says that, if the Yamabe constant of a closed conformal manifold (M, C) is positive, then it is strictly less than the Yamabe constant of any of its non-trivial finite conformal coverings. We generalize this lemma to the one for the Yamabe constant of any (M _∞, C _∞) of its infinite conformal coverings, provided that π ₁(M) has a descending chain of finite index subgroups tending to π ₁(M _∞). Moreover, if the covering M _∞ is normal, the limit of the Yamabe constants of the finite conformal coverings (associated to the descending chain) is equal to that of (M _∞, C _∞). For the proof of this, we also establish a version of positive mass theorem for a specific class of asymptotically flat manifolds with singularities.     

Profinite properties of graph manifolds

Let M be a closed, orientable, irreducible, geometrizable 3-manifold. We prove that the profinite topology on the fundamental group of π ₁(M) is efficient with respect to the JSJ decomposition of M. We go on to prove that π ₁(M) is good, in the sense of Serre, if all the pieces of the JSJ decomposition are. We also prove that if M is a graph manifold then π ₁(M) is conjugacy separable.     

Manifolds with Poly-Surface Fundamental Groups

We study closed topological 2n-dimensional manifolds M with poly-surface fundamental groups. We prove that if M is simple homotopy equivalent to the total space E of a Y-bundle over a closed aspherical surface, where Y is a closed aspherical n-manifold, then M is s-cobordant to E. This extends a well-known 4-dimensional result of Hillman in [14] to higher dimensions. Our proof is different from that of the quoted paper: we use Mayer-Vietoris techniques and the properties of the -theory assembly maps for such bundles.     

Classical r-Matrices and Novikov Algebras

We study the existence problem for Novikov algebra structures on finite-dimensional Lie algebras. We show that a Lie algebra admitting a Novikov algebra is necessarily solvable. Conversely we present a 2-step solvable Lie algebra without any Novikov structure. We use extensions and classical r-matrices to construct Novikov structures on certain classes of solvable Lie algebras.     

Strict quasicomplements and the operators of dense imbedding

A quasicomplementM to a subspaceN of a Banach spaceX is called strict ifM does not contain an infinite-dimensional subspaceM ₁ such that the linear manifoldN+M ₁ is closed. It is proved that ifX is separable, thenN always admits a strict quasicomplement. We study the properties of the restrictions of the operators of dense imbedding to infinite-dimensional closed subspaces of a space where these operators are defined. Zaporozhye University, Zaporozhye. Translated from Ukrainskii Matematicheskii Zhurmal, Vol. 46, No. 6, pp. 789–792, June, 1994