Simply-connected Case Research Articles

On the Grothendieck-Serre conjecture about principal bundles and its generalizations

Let $U$ be a regular connected affine semi-local scheme over a field $k$. Let $G$ be a reductive group scheme over $U$. Assuming that $G$ has an appropriate parabolic subgroup scheme, we prove the following statement. Given an affine $k$-scheme $W$, a principal $G$-bundle over $W\times_kU$ is trivial if it is trivial over the generic fiber of the projection $W\times_kU\to U$. We also simplify the proof of the Grothendieck-Serre conjecture: let $U$ be a regular connected affine semi-local scheme over a field $k$. Let $G$ be a reductive group scheme over $U$. A principal $G$-bundle over $U$ is trivial if it is trivial over the generic point of $U$. We generalize some other related results from the simple simply-connected case to the case of arbitrary reductive group schemes.

Concordance of surfaces in 4‐manifolds and the Freedman–Quinn invariant

We prove a concordance version of the 4-dimensional light bulb theorem for $\pi_1$-negligible compact orientable surfaces, where there is a framed but not necessarily embedded dual sphere. That is, we show that if $F_0$ and $F_1$ are such surfaces in a 4-manifold $X$ that are homotopic and there exists an immersed framed 2-sphere $G$ in $X$ intersecting $F_0$ geometrically once, then $F_0$ and $F_1$ are concordant if and only if their Freedman-Quinn invariant $\mathop{fq}$ vanishes. The proof of the main result involves computing $\mathop{fq}$ in terms of intersections in the universal covering space and then applying work of Sunukjian in the simply-connected case.

Two purity theorems and the Grothendieck-Serre conjecture concerning principal -bundles

The main results of the paper are two purity theorems for reductive group schemes over regular local rings containing a field. Using these two theorems a well-known Grothendieck-Serre conjecture on principal bundles is reduced to the simply-connected case. We point out that the mentioned reduction is one of the major steps in the proof of the conjecture that the author published in another work. Bibliography: 25 titles.

Conformal invariants associated with quadratic differentials

We associate a functional of pairs of simply-connected regions D2 ⊆ D1 to any quadratic differential on D1 with specified singularities. This functional is conformally invariant, monotonic, and negative. Equality holds if and only if the inner domain is the outer domain minus trajectories of the quadratic differential. This generalizes the simply-connected case of results of Z. Nehari [20], who developed a general technique for obtaining inequalities for conformal maps and domain functions from contour integrals and the Dirichlet principle for harmonic functions. Nehari’s method corresponds to the special case that the quadratic differential is of the form (∂q)2 for a singular harmonic function q on D1. As an application we give a one-parameter family of monotonic, conformally invariant functionals which correspond to growth theorems for bounded univalent functions. These generalize and interpolate the Pick growth theorems, which appear in a conformally invariant form equivalent to a two-point distortion theorem of W. Ma and D. Minda [16].

The Carathéodory Reflection Principle and Osgood–Carathéodory Theorem on Riemann Surfaces

AbstractThe Osgood–Carathéodory theorem asserts that conformal mappings between Jordan domains extend to homeomorphisms between their closures. For multiply-connected domains on Riemann surfaces, similar results can be reduced to the simply-connected case, but we find it simpler to deduce such results using a direct analogue of the Carathéodory reflection principle.

Equivariant basic cohomology of Riemannian foliations

Abstract The basic cohomology of a complete Riemannian foliation with all leaves closed is the cohomology of the leaf space. In this paper we introduce various methods to compute the basic cohomology in the presence of both closed and non-closed leaves in the simply-connected case (or more generally for Killing foliations): We show that the total basic Betti number of the union C of the closed leaves is smaller than or equal to the total basic Betti number of the foliated manifold, and we give sufficient conditions for equality. If there is a basic Morse–Bott function with critical set equal to C, we can compute the basic cohomology explicitly. Another case in which the basic cohomology can be determined is if the space of leaf closures is a simple, convex polytope. Our results are based on Molino’s observation that the existence of non-closed leaves yields a distinguished transverse action on the foliated manifold with fixed point set C. We introduce equivariant basic cohomology of transverse actions in analogy to equivariant cohomology of Lie group actions enabling us to transfer many results from the theory of Lie group actions to Riemannian foliations. The prominent role of the fixed point set in the theory of torus actions explains the relevance of the set C in the basic setting.

Surgery stable curvature conditions

We give a simple criterion for a pointwise curvature condition to be stable under surgery. Namely, a curvature condition C, which is understood to be an open, convex, $${{\mathrm{O}}}(n)$$ -invariant cone in the space of algebraic curvature operators, is stable under surgeries of codimension at least c provided it contains the curvature operator corresponding to $$S^{c-1} \times \mathbb {R}^{n-c+1}$$ , $$c \ge 3$$ . This is used to generalize the well-known classification result of positive scalar curvature in the simply-connected case in the following way: Any simply-connected manifold $$M^n$$ , $$n \ge 5$$ , which is either spin with vanishing $$\alpha $$ -invariant or else is non-spin admits for any $$\epsilon > 0$$ a metric such that the curvature operator satisfies $$R > - \epsilon \left\| R\right\| $$ .

Strongly solvable spaces

This work builds on the foundation laid by Gordon and Wilson in the study of isometry groups of solvmanifolds, i.e. Riemannian manifolds admitting a transitive solvable group of isometries. We restrict ourselves to a natural class of solvable Lie groups called almost completely solvable; this class includes the completely solvable Lie groups. When the commutator subalgebra contains the center, we have a complete description of the isometry group of any left-invariant metric using only metric Lie algebra information. Using our work on the isometry group of such spaces, we study quotients of solvmanifolds. Our first application is to the classification of homogeneous Ricci soliton metrics. We show that the verification of the Generalized Alekseevsky Conjecture reduces to the simply-connected case. Our second application is a generalization of a result of Heintze on the rigidity of existence of compact quotients for certain homogeneous spaces. Heintze's result applies to spaces with negative curvature. We remove all the geometric requirements, replacing them with algebraic requirements on the homogeneous structure.

Formality and the Lefschetz property in symplectic and cosymplectic geometry

Abstract We review topological properties of Kähler and symplectic manifolds, and of their odd-dimensional counterparts, coKähler and cosymplectic manifolds. We focus on formality, Lefschetz property and parity of Betti numbers, also distinguishing the simply-connected case (in the Kähler/symplectic situation) and the b1 = 1 case (in the coKähler/cosymplectic situation).

An algebraic chain model of string topology

A chain complex model for the free loop space of a connected, closed and oriented manifold is presented, and on its homology, the Gerstenhaber and Batalin-Vilkovisky algebra structures are defined and identified with the string topology structures. The gravity algebra on the equivariant homology of the free loop space is also modeled. The construction includes non simply-connected case, and therefore gives an algebraic and chain level model of Chas-Sullivan's String Topology.

Lie algebras over operads and their applications in homotopy theory

This paper is devoted to the development of algebraic devices that are necessary for describing the homotopy groups of topological spaces. As shown in the note [1], the notion of a Lie algebra over the operad must be used for this purpose. Here we consider this notion in more detail, prove its most important properties and clarify the question about the algebraic structure on the homotopy groups of topological spaces. In particular, we show that the homotopy groups of a topological space possess the structure of a Lie -algebra which determines the homotopy type of the space in the simply-connected case. Since the notion of an operad is analogous to that of algebra, we begin with recalling the notions of algebra and coalgebra and reviewing the main constructions over them. Then we transfer these constructions to the operad case and use them to investigate the structure of Lie algebras and coalgebras over an operad.

Surgery and the Generalized Kervaire Invariant, I

(i) Synopsis. The discovery, around 1960, of the 'Kervaire Invariant' for almost framed manifolds of dimension 4k+ 2 (see [12]) was an important stimulant for the development of surgery theory; but it also led to the theory of the 'generalized Kervaire Invariant' of Browder and Brown [2, 3]. The present paper is an attempt at uniting these two theories, by constructing a non-simply-connected and in other respects updated version of the generalized Kervaire Invariant. The construction has three surprising aspects. Firstly, it is conceptually satisfying and, in the simply-connected case, clarifies Brown's original theory; for instance, the 'product formula problem' (see [4]) evaporates. Most of the new concepts are borrowed from the 'algebraic theory of surgery'; see [15]. Secondly, it is computationally satisfying. Thirdly, it has applications to classical surgery theory, especially to the calculation of the symmetric L-groups of [13] and [15]; and therefore to anything which involves product formulae for surgery obstructions. A black box description of the theory has been given in [22]; in this introduction I shall concentrate on the concepts inside the box.

Fitted diffeomorphisms of non-simply connected manifolds

TOPOLOGICAL ENTROPY roughly speaking measures the complexity of a dynamical system. In “Homology Theory and Dynamical Systems”[ l] Shub and Sullivan defined a class of structurally stable diffeomorphisms called fitted which are Co dense in oil?‘(M). They show how to relate the dynamics of a fitted diffeomorphism to the induced chain map in homology theory, and using this determine the minimum topological entropy of fitted diffeomorphisms in each component of D%‘(M), for simply connected manifolds of dimension greater than five. In this paper we extend these results to the non simply-connected case. The main tools are a sharpened statement of the Whitney canceIling lemma for handles of index 2 and (n 2) using relative homotopy groups, and classical results of Whitelead concerning these groups[2]. Our methods also lead to an algebraic characterization of the chain complexes over Z[II,] which arise from handle decompositions of high dimensional manifolds, extending a theorem of Smale’s in the simply connected case. It is a pleasure to acknowledge many helpful conversations with Michael Shub and Dennis Sullivan.

Some New Four-Manifolds

The quotient space Q = X'/Z, of the involution is a smooth manifold of the (simple) homotopy type of real projective 4-space P', but not diffeomorphic or even piecewise linear (PL) homeomorphic to P'. As a corollary of the theorem and the vanishing of L,(Z2, -), there are precisely two s-cobordism classes of homotopy 4-dimensional real projective spaces. The manifold Q is obtained from Pi by removing the complement of a tube around P2 c P', a non-orientable 3-disk bundle over S', and replacing it by a certain bundle over S' with fibre the complement of an open cell in the 3-torus T3. In [6] we gave a classification theory of 4-manifolds, modulo connected sum with copies of S2 x S2, extending the results of Wall for the simplyconnected case. In particular, the connected sum of Q with many copies of S2 x SI is diffeomorphic to the manifold constructed in [6, 2.4], and the connected sum of Q with arbitrarily many copies of S2 x S2 is not PL homeomorphic or even h-cobordant to the connected sum of Pi with arbitrarily many copies of S2 x S2. Let V be a manifold with dim V > 1, and dim V > 2 if Vhas non-empty boundary. Then one can show, using topological surgery and the open h-cobordism theorem, that Pi x V and Q x V are topologically homeomorphic. It is not known if any of the manifolds Q constructed below are actually homeomorphic to P'. It is also not known if their universal covering spaces are diffeomorphic, PL homeomorphic, or even just homeomorphic to the 4-sphere S. 1 Partially supported by NSF.