Four-manifolds with π<Subscript>1</Subscript>–free second homotopy

 We study the homotopy type of closed connected orientable topological 4–manifolds M with Λ–free second homotopy group, where Λ is the integral group ring of π₁(M). This is related with problem N.4.53 of [23], and extends some results proved for the class of closed 4–manifolds with free fundamental group [10][11]. Other applications on special classes of closed topological manifolds complete the paper. Received: 27 November 2001 / Revised version: 28 October 2002 Published online: 19 May 2003 Work performed under the auspices of the GNSAGA of the CNR of Italy and partially supported by Ministero per la Ricerca Scientifica e Tecnologica of Italy within the project Proprietà Geometriche delle Varietà Reali e Complesse, and by a research grant of the University of Modena and Reggio Emilia. Mathematics Subject Classification (2000): Primary: 57 N 65, 57 R 67; Secondary: 57 Q 10, 57 R 80     

Splitting Obstruction Groups in Codimension 2

The splitting obstruction groups depend functorially on the square of fundamental groups. In the paper the problem of splitting along a submanifold of codimension two under some restrictions on the square of fundamental groups is considered. New exact sequences and commutative diagrams containing Wall groups, splitting obstruction groups, and surgery obstruction groups for manifold pairs are obtained. Examples of computation of splitting obstruction groups and natural maps are considered.     

The π-π-theorem for manifold pairs with boundaries

The surgery obstruction of a normal map to a simple Poincaré pair (X, Y) lies in the relative surgery obstruction group L _*(π ₁(Y) → π ₁(X)). A well-known result of Wall, the so-called π-π-theorem, states that in higher dimensions a normal map of a manifold with boundary to a simple Poincaré pair with π ₁(X) ? π ₁(Y) is normally bordant to a simple homotopy equivalence of pairs. In order to study normal maps to a manifold with a submanifold, Wall introduced the surgery obstruction groups LP _* for manifold pairs and splitting obstruction groups LS _*. In the present paper, we formulate and prove for manifold pairs with boundary results similar to the π-π-theorem. We give direct geometric proofs, which are based on the original statements of Wall’s results and apply obtained results to investigate surgery on filtered manifolds.     

Transfer maps for triples of manifolds

Transfer maps are closely related to the problem of splitting a homotopy equivalence along a submanifold and with the problem of surgery on a pair of manifolds. In the present paper, we describe relations between various transfer maps for a triple of embedded manifolds.     

Tangent maps and tangent groupoid for Carnot manifolds

This paper studies the infinitesimal structure of Carnot manifolds. By a Carnot manifold we mean a manifold together with a subbundle filtration of its tangent bundle which is compatible with the Lie bracket of vector fields. We introduce a notion of differential, called Carnot differential, for Carnot manifolds maps (i.e., maps that are compatible with the Carnot manifold structure). This differential is obtained as a group map between the corresponding tangent groups. We prove that, at every point, a Carnot manifold map is osculated in a very precise way by its Carnot differential at the point. We also show that, in the case of maps between nilpotent graded groups, the Carnot differential is given by the Pansu derivative. Therefore, the Carnot differential is the natural generalization of the Pansu derivative to maps between general Carnot manifolds. Another main result is a construction of an analogue for Carnot manifolds of Connes' tangent groupoid. Given any Carnot manifold $(M,H)$ we get a smooth groupoid that encodes the smooth deformation of the pair $M\times M$ to the tangent group bundle GM. This shows that, at every point, the tangent group is the tangent space in a true differential-geometric fashion. Moreover, the very fact that we have a groupoid accounts for the group structure of the tangent group. Incidentally, this answers a well-known question of Bellaïche [11].     

Sectional curvatures in nonlinear optimization

The aim of the paper is to show how to explicitly express the function of sectional curvature with the first and second derivatives of the problem’s functions in the case of submanifolds determined by equality constraints in the n-dimensional Euclidean space endowed with the induced Riemannian metric, which is followed by the formulation of the minimization problem of sectional curvature at an arbitrary point of the given submanifold as a global minimization one on a Stiefel manifold. Based on the results, the sectional curvatures of Stiefel manifolds are analysed and the maximal and minimal sectional curvatures on an ellipsoid are determined. This research was supported in part by the Hungarian Scientific Research Fund, Grant No. OTKA-T043276 and OTKA-K60480.     

Induction of Geometric Actions

We prove that, in some situations, an induced action from a normal subgroup preserves a geometric structure. Combined with known geometric rigidity results, this result implies certain rigidity statements concerning the full diffeomorphism group of a manifold. It also provides many examples of actions on Lorentz manifolds. Combining these with a small number of well-known actions, we get the full list of connected, simply connected Lie groups admitting a locally faithful, orbit nonproper action by isometries of a connected Lorentz manifold. We give an example of a connected nilpotent Lie group with no complicated action on a Lorentz manifold. We show that, if a connected Lie group has a normal closed subgroup isomorphic to a (two-dimensional) cylinder, then it admits a locally faithful, orbit nonproper action by isometries of a connected Lorentz manifold.     

Volumes of hyperbolic Löbell 3-manifolds

In 1931 F. Löbell constructed the first example of a closed orientable three-dimensional hyperbolic manifold. In the present paper we study properties of closed hyperbolic 3-manifolds generalizing Löbell's classical example. Explicit formulas for the volumes of these manifolds in terms of the Lobachevski function are obtained.Translated fromMatematicheskie Zametki, Vol. 64, No. 1, pp. 17–23, July, 1998.This research was partially supported by GARC-KOSEF (Global Analysis Research Center of National Seoul University) and by the Russian Foundation for Basic Research under grant No. 95-01-01410.     

Vector fields with a given set of singular points

Theorems on the existence of vector fields with given sets of indexes of isolated singular points are proved for the cases of closed manifolds, pairs of manifolds, manifolds with boundary, and gradient fields. It is proved that, on a two-dimensional manifold, an index of an isolated singular point of the gradient field is not greater than one. Kiev University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 10, pp. 1373–1384, October, 1997.     

Surgery of closed manifolds with dihedral fundamental group

In the paper the obstruction groups to obtaining simple homotopy equivalence by surgery from normal degree 1 maps of closed manifolds with dihedral fundamental group are computed. The cases of trivial orientation for the dihedral group and nontrivial orientation for the order 2 cyclic subgroup are considered. New results concerning the Browder-Livesey groups and natural maps ofL-groups arising in index 2 inclusions of the cyclic group into the dihedral group are obtained.Translated fromMatematicheskie Zametki, Vol. 64, No. 2, pp. 238–250, August, 1998.The work of the first-named author was partially supported by of the Grant of the President of the Russian Federation, grant No. 96-15-96841. The work of the second-named author was partially suported the Ministry of Science and Technology of the Republic of Sloveniya, grant No. J1-7039-0101-95.     

On the splitting problem for manifold pairs with boundaries

The problem of splitting a homotopy equivalence along a submanifold is closely related to the surgery exact sequence and to the problem of surgery of manifold pairs. In classical surgery theory there exist two approaches to surgery in the category of manifolds with boundaries. In the rel ∂ case the surgery on a manifold pair is considered with the given fixed manifold structure on the boundary. In the relative case the surgery on the manifold with boundary is considered without fixing maps on the boundary. Consider a normal map to a manifold pair (Y, ∂Y) ⊂ (X, ∂X) with boundary which is a simple homotopy equivalence on the boundary∂X. This map defines a mixed structure on the manifold with the boundary in the sense of Wall. We introduce and study groups of obstructions to splitting of such mixed structures along submanifold with boundary (Y, ∂Y). We describe relations of these groups to classical surgery and splitting obstruction groups. We also consider several geometric examples.     

Stein structures and holomorphic mappings

We prove that every continuous map from a Stein manifold X to a complex manifold Y can be made holomorphic by a homotopic deformation of both the map and the Stein structure on X. In the absence of topological obstructions, the holomorphic map may be chosen to have pointwise maximal rank. The analogous result holds for any compact Hausdorff family of maps, but it fails in general for a noncompact family. Our main results are actually proved for smooth almost complex source manifolds (X,J) with the correct handlebody structure. The paper contains another proof of Eliashberg’s (Int J Math 1:29–46, 1990) homotopy characterization of Stein manifolds and a slightly different explanation of the construction of exotic Stein surfaces due to Gompf (Ann Math 148(2): 619–693, 1998; J Symplectic Geom 3:565–587, 2005).      

Residual properties of graph manifold groups

Let f:M→N be a continuous map between closed irreducible graph manifolds with infinite fundamental group. Perron and Shalen (1999) [16] showed that if f induces a homology equivalence on all finite covers, then f is in fact homotopic to a homeomorphism. Their proof used the statement that every graph manifold is finitely covered by a 3-manifold whose fundamental group is residually p for every prime p. We will show that this statement regarding graph manifold groups is not true in general, but we will show how to modify the argument of Perron and Shalen to recover their main result. As a by-product we will determine all semidirect products Z?Zd which are residually p for every prime p.     

On Lorentz dynamics: From group actions to warped products via homogeneous spaces

We show a geometric rigidity of isometric actions of non-compact (semisimple) Lie groups on Lorentz manifolds. Namely, we show that the manifold has a warped product structure of a Lorentz manifold with constant curvature by a Riemannian manifold.

     

Computation of non-smooth local centre manifolds

^** Email: msjolly{at}indiana.edu^*** Email: rrosa{at}ufrj.br An iterative Lyapunov–Perron algorithm for the computationof inertial manifolds is adapted for centre manifolds and appliedto two test problems. The first application is to compute aknown non-smooth manifold (once, but not twice differentiable),where a Taylor expansion is not possible. The second is to asmooth manifold arising in a porous medium problem, where rigorouserror estimates are compared to both the correction at eachiteration and the addition of each coefficient in a Taylor expansion.While in each case the manifold is 1D, the algorithm is well-suitedfor higher dimensional manifolds. In fact, the computationalcomplexity of the algorithm is independent of the dimension,as it computes individual points on the manifold independentlyby discretising the solution through them. Summations in thealgorithm are reformulated to be recursive. This accelerationapplies to the special case of inertial manifolds as well.     

Euclidean 4-Simplices and Invariants of Four-Dimensional Manifolds: II. An Algebraic Complex and Moves 2↔4

We present sequences of linear maps of vector spaces with fixed bases. Each term of a sequence is a linear space of differentials of metric values ascribed to the elements of a simplicial complex determining a triangulation of a manifold. If a sequence is an acyclic complex, then we can construct a manifold invariant using its torsion. We demonstrate this first for three-dimensional manifolds and then construct the part of this program for four-dimensional manifolds pertaining to moves 24.     

Path and quasi-homotopy for Sobolev maps between manifolds

We study the relationship between quasi-homotopy and path homotopy for Sobolev maps between manifolds. By employing singular integrals on manifolds we show that, in the critical exponent case, path homotopic maps are quasi-homotopic – and observe the rather surprising fact that quasi-homotopic maps need not be path homotopic. We also study the case where the target is an aspherical manifold, e.g. a manifold with non-positive sectional curvature, and the contrasting case of the target being a sphere.     

Stable ℱ-harmonic maps between Finsler manifolds

In this paper, we derive the first and second variation formulas for JC-harmonic maps between Finsler manifolds, and when F″≤ 0 and n ≥ 3, we prove that there is no nondegenerate stable F-harmonic map between a Riemannian unit sphere Sn and any compact Finsler manifold.     

On Embeddings of Tori in Euclidean Spaces

Using the relation between the set of embeddings of tori into Euclidean spaces modulo ambient isotopies and the homotopy groups of Stiefel manifolds, we prove new results on embeddings of tori into Euclidean spaces.