Exotic Spheres Research Articles

Curvature and Symmetry of Milnor Spheres

Since Milnor’s discovery of exotic spheres [Mi], one of the most intriguing problems in Riemannian geometry has been whether there are exotic spheres with positive curvature. It is well known that there are exotic spheres that do not even admit metrics with positive scalar curvature [Hi]. On the other hand, there are many examples of exotic spheres with positive Ricci curvature (cf. [Ch1], [He], [Po], and [Na]) and this work recently culminated in [Wr] where it is shown that every exotic sphere that bounds a parallelizable manifold has a metric of positive Ricci curvature. This includes all exotic spheres in dimension 7. So far, however, no example of an exotic sphere with positive sectional curvature has been found. In fact, until now, only one example of an exotic sphere with nonnegative sectional curvature was known, the so-called Gromoll-Meyer sphere [GM] in dimension 7. As one of our main results we prove:

Metric constraints on exotic spheres via Alexandrov geometry.

Metric constraints on exotic spheres via Alexandrov geometry.

Exotic spheres with positive Ricci curvature

We show that a certain class of manifolds admit metrics of positive Ricci curvature. This class includes many exotic spheres, including all homotopy spheres which represent elements of bP2n-.

Compact differentiable transformation groups on exotic spheres

0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 1 Statement of the main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 2 Higher dimensional subgroups of O(n + 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358 3 Higher dimensional transformation groups on homology spheres, geometric weight systems and orbit structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360 4 Higher dimensional transformation groups on homotopy spheres and the proof of Theorem B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 5 Degree of symmetry of exotic spheres revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 6 Large transformation groups on exotic spheres in bP.+ 1 . . . . . . . . . . . . . . . . . . . . . . . . . 384

Global gravitational instantons and their degrees of symmetry

It is shown that the n ten-dimensional very exotic spheres−or, equivalently, global gravitational instantons or solitons−admit only a finite number of involutions. They are realized as smooth SU(3) biaxial actions on Σ10 and are classified by the universal group manifold CP2. A formula, referred to as the degree of symmetry, which quantizes the maximal symmetry that a global gravitational instanton could possess is derived. Namely, it is established that Σ10 admits no more than 20 SU(3) involutions. These results, when combined with the ones found by the author and Lee [J. Math. Phys. 32, 2869 (1991)] yield a better appreciation of Witten’s conjecture [Commun. Math. Phys. 100, 197 (1985)].

On the global gravitational instanton and soliton that are homotopy spheres

A generalization of the Dirac string trick for higher-dimensional manifolds is presented. As a rather unexpected result, the procedure is shown to yield homotopy spheres. Because of their relevant interest to physical theories, particular emphasis is given to even-dimensional homotopy spheres. For instance, it is demonstrated that a nontrivial representation of 6–10-dimensional exotic spheres in dimensions 4k+2 can be performed by using the Pontrjagin–Thom construction. The physical content of the results is consistent with Witten’s hypothesis [Commun. Math. Phys., 100, 197–229 (1985)] that n–10-dimensional exotic spheres can be interpreted as gravitational instanton and/or soliton. It is also shown that three mutually diffeomorphic elements can be isolated from the 6–10-dimensional exotic spheres and their spectrum is provided.

A pinching theorem for homotopy spheres

The (length) excess of a triangle measures how much the triangle inequality fails to be an equality. This notion was first studied seriously in [AG]. We say that a (bounded) metric space X = (X, d) has excess :5 e if there are points p, q E X such that d(p, x)+d(x, q) ~ d(p, q)+e for all x EX. The excess, e(X) , of X is the infimum of all e ~ 0 where X has excess ~ e. Notice that the standard spheres have excess = O. In general, a closed Riemannian manifold M has excess = 0 if and only if there is a point p E M whoSe cutlocus C(P) is a single point q EM. Such a manifold is clearly a twisted sphere, i.e., it is the union of two discs glued together along their common boundary. Conversely ·according to a result of Weinstein [B; Appendix C.4], any twisted sphere carries a Riemannian metric for which there is a point whose cut-locus is another point. With this in mind, we are interested in closed Riemannian manifolds with small excess. However; without any further geometric restrictions, small excess has no topological significance (see Problem 8, though). In fact, any closed n-manifold M can be given a Riemannian metric withdiam M = 1C and arbitrarily small excess. Simply make sure that all topological complexities of M (i.e~, the complement of a topological n-disc) are contained in a sufficiently small metric ball and what remains is metrically close to the complement of a small ball in the unit sphere Sn (1). Our main result is Theorem A. Given k E R, D, v > 0, and an integer n ~ 2, there is an' t = e(n, k, D, v) such that any closed Riemannian n-mani/old M with sectional curvature KM ~ k, diamM ~ D, VolM ~ v, and e(M) ~ e is a homotopy sphere.

Negatively Curved Manifolds with Exotic Smooth Structures

Let $M$ denote a compact real hyperbolic manifold with dimension $m \geq 5$ and sectional curvature $K = - 1$, and let $\Sigma$ be an exotic sphere of dimension $m$. Given any small number $\delta > 0$, we show that there is a finite covering space $\widehat {M}$ of $M$ satisfying the following properties: the connected sum $\widehat {M}\# \Sigma$ is not diffeomorphic to $\widehat {M}$, but it is homeomorphic to $\widehat {M}$; $\widehat {M}\# \Sigma$ supports a Riemannian metric having all of its sectional curvature values in the interval $[ - 1 - \delta , - 1 + \delta ]$. Thus, there are compact Riemannian manifolds of strictly negative sectional curvature which are not diffeomorphic but whose fundamental groups are isomorphic. This answers Problem 12 of the list compiled by Yau [22]; i.e., it gives counterexamples to the Lawson-Yau conjecture. Note that Mostow’s Rigidity Theorem [17] implies that $\widehat {M}\# \Sigma$ does not support a Riemannian metric whose sectional curvature is identically -1 . (In fact, it is not diffeomorphic to any locally symmetric space.) Thus, the manifold $\widehat {M}\# \Sigma$ supports a Riemannian metric with sectional curvature arbitrarily close to -1 , but it does not support a Riemannian metric whose sectional curvature is identically -1 . More complicated examples of manifolds satisfying the properties of the previous sentence were first constructed by Gromov and Thurston [11].

Negatively curved manifolds with exotic smooth structures

Let M M denote a compact real hyperbolic manifold with dimension m ≥ 5 m \geq 5 and sectional curvature K = − 1 K = - 1 , and let Σ \Sigma be an exotic sphere of dimension m m . Given any small number δ > 0 \delta > 0 , we show that there is a finite covering space M ^ \widehat {M} of M M satisfying the following properties: the connected sum M ^ # Σ \widehat {M}\# \Sigma is not diffeomorphic to M ^ \widehat {M} , but it is homeomorphic to M ^ \widehat {M} ; M ^ # Σ \widehat {M}\# \Sigma supports a Riemannian metric having all of its sectional curvature values in the interval [ − 1 − δ , − 1 + δ ] [ - 1 - \delta , - 1 + \delta ] . Thus, there are compact Riemannian manifolds of strictly negative sectional curvature which are not diffeomorphic but whose fundamental groups are isomorphic. This answers Problem 12 of the list compiled by Yau [22]; i.e., it gives counterexamples to the Lawson-Yau conjecture. Note that Mostow’s Rigidity Theorem [17] implies that M ^ # Σ \widehat {M}\# \Sigma does not support a Riemannian metric whose sectional curvature is identically -1 . (In fact, it is not diffeomorphic to any locally symmetric space.) Thus, the manifold M ^ # Σ \widehat {M}\# \Sigma supports a Riemannian metric with sectional curvature arbitrarily close to -1 , but it does not support a Riemannian metric whose sectional curvature is identically -1 . More complicated examples of manifolds satisfying the properties of the previous sentence were first constructed by Gromov and Thurston [11].

On the construction of codimension two minimal immersions of exotic spheres into Euclidean spheres

It is, nowadays, a well-known fundamental fact in differential topology that there exist many smooth manifolds which are homeomorphic to but non-diffeomorphic to spheres. Such manifolds were christened exotic spheres by their founder, Milnor [223. The structures of those exotic spheres which bound some parallelizable manifolds were systematically analyzed by the techniques of surgery in [20]. For example, it is not difficult to show that such exotic spheres can be embedded as codimension two submanifolds of the unit spheres. In differential geometry and geometric variational theory, the study of closed minimal submanifolds of the euclidean n-sphere, S"(1), is directly related to that of the local structure of singularities of minimal subvarieties in the general Riemannian setting. Therefore, closed minimal submanifolds of S"(1) are not only interesting, nice, geometric objects by themselves, but are also of general theoretical importance in the study of geometric measure theory [16, 18, 24]. Among various problems on the closed minimal submanifolds of euclidean spheres, the following two problems naturally distinguish themselves as especially interesting, namely:

On the construction of constant mean curvature imbeddings of exotic and/or knotted spheres intoS n (1)

On the construction of constant mean curvature imbeddings of exotic and/or knotted spheres intoS n (1)

Smith equivalence of representations for odd order cyclic groups

THEOREM A. There exist odd order cyclic groups which have Smith equivalent but nonisomorphic representations. After a short background discussion, we shall give a list of conditions (1.1) which are important in the development of this paper and state more precise versions of our results. Theorem B provides nonisomorphic Smith equivalent representations if one has nonisomorphic representations of G which satisfy (1.1). Corollary C shows the existence of infinitely many groups which have nonisomorphic representations which satisfy (1.1). The orders of some cyclic groups of this type are described by Lemma 2.5 and Corollary 2.6. Taken together, Theorem B and Corollary C prove Theorem A. The concept of Smith equivalence is based upon a question posed by P. A. Smith in a 1960 survey article [32, p. 4061. In our terminology the question is whether Smith equivalent representations are linearly isomorphic. Atiyah-Bott [3] and Milnor [17] established an affirmative answer for cyclic groups of odd prime power order and for representations with semifree action. By definition a group acts semifreely if each point is left fixed by only the trivial group element or by each group element. An extension of this work by Sanchez [28] implied the answer also to be affirmative if G is cyclic of order pq where p and q are odd primes. Bredon [6] showed for 2-groups that Smith equivalent representations are isomorphic if their dimension is large in comparison to the order of the group. The first negative answers to Smith’s question were established by the second author for odd order abelian groups with at least four noncyclic Sylow subgroups. See [23], [24]. Subsequently a number of papers have established a negative answer for even order groups, but the case of odd order cyclic groups has remained open. For cyclic groups of even order see the papers [7], [25], [ 111, [3 13, For noncyclic abelian groups and nonabelian 2-groups, see [34] and [S]. There are many interesting results and ideas in these papers. Results and techniques used in answering Smith’s question varj markedly according to the parity of the order of G. In Remark 1.2(ii) we explain the difference of the geometric and number theoretic treatment of even and odd order groups as it arises from the Atiyah-Bott extension of the Lefschetz Fixed Point theorem. One interesting question, untouched here, is to relate the differentiable structure of a homotopy sphere Z and the isotropy representations of an action of G on Z with exactly two fixed points. Some general remarks on this appear in the problem section of the forthcoming proceedings of the 1983 conference on group actions at Boulder [393. It can be shown that the homotopy spheres with group actions that are constructed by our method are all standard spheres. The methods of Schultz (e.g. in [29] and other papers) are relevant to realizing these actions on exotic spheres.

Global gravitational anomalies

A general formula for global gauge and gravitational anomalies is derived. It is used to show that the anomaly free supergravity and superstring theories in ten dimensions are all free of global anomalies that might have ruined their consistency. However, it is shown that global anomalies lead to some restrictions on allowed compactifications of these theories. For example, in the case of O(32) superstring theory, it is shown that a global anomaly related to π7(O(32)) leads to a Dirac-like quantization condition for the field strength of the antisymmetric tensor field. Related to global anomalies is the question of the number of fermion zero modes in an instanton field. It is argued that the relevant gravitational instantons are exotic spheres. It is shown that the number of fermion zero modes in an instanton field is always even in ten dimensional supergravity.

Supergravity on seven-dimensional homotopy spheres

We discuss a spontaneous compactification scheme of eleven-dimensional supergravity on the exotic spheres Σ 7 and on the ordinary S 7 with the Hopf-fibering. These spheres are the nontrivial fibration of S 4 × SO(4)S 3 with the SO(4)-instantons. In particular two classical solutions are explicitly obtained for the Hopf-fibering of S 7 in the eleven-dimensional supergravity. The ratios of the curvatures in S AdS 4 × S 4′ × S 3 are (−1 : 1 : 1 4 ) and ( −9 5 : 1 : 5 4 ), and the corresponding super-symmetries are N = 8 and N = 1, respectively. They are gauge equivalent to the round and the squashed S 7.

Embedded Minimal Surfaces, Exotic Spheres, and Manifolds with Positive Ricci Curvature

Let N be a three dimensional Riemannian manifold. Let E be a closed embedded surface in N. Then it is a question of basic interest to see whether one can deform : in its isotopy class to some canonical embedded surface. From the point of view of geometry, a natural canonical surface will be the extremal surface of some functional defined on the space of embedded surfaces. The simplest functional is the area functional. The extremal surface of the area functional is called the minimal surface. Such minimal surfaces were used extensively by Meeks-Yau [MY21 in studying group actions on three dimensional manifolds. In [MY2], the theory of minimal surfaces was used to simplify and strengthen the classical Dehn's lemma, loop theorem and the sphere theorem. In the setting there, one minimizes area among all immersed surfaces and proves that the extremal object is embedded. In this paper, we minimize area among all embedded surfaces isotopic to a fixed embedded surface. In the category of these surfaces, we prove a general existence theorem (Theorem 1). A particular consequence of this theorem is that for irreducible manifolds an embedded incompressible surface is isotopic to an embedded incompressible surface with minimal area. We also prove that there exists an embedded sphere of least area enclosing a fake cell, provided the complementary volume is not a standard ball, and provided there exists no embedded one-sided RP2. By making use of the last result, and a cutting and pasting argument, we are able to settle a well-known problem in the theory of three dimensional manifolds. We prove that the covering space of any irreducible orientable three dimensional manifold is irreducible. It is possible to exploit our existence theorem to study

Exotic spheres as stationary sets of homotopy sphere involutions.

Exotic spheres as stationary sets of homotopy sphere involutions.

Smooth actions of the circle group on exotic spheres

Smooth actions of the circle group on exotic spheres

Some examples of non-regular almost contact structures on exotic spheres

Some examples of non-regular almost contact structures on exotic spheres

Some exotic spheres with positive Ricci curvature

Some exotic spheres with positive Ricci curvature

On a distance function between Differentiable structures

In this note, we investigate a relation between the connected sum of manifolds and the distance of manifolds ([2]). Since the smoothing of a piecewise linear equivalence is given by connected sum of exotic spheres ([1]), we have a certain estimate of the smoothing obstruction using the distance of manifolds (Proposition 3). In § 3, an application is given to show the impossibility of the 0.64-pinching of an exotic sphere.