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The Non-Existent Complex 6-Sphere
Abstract
The possible existence of a complex structure on the 6-sphere has been a famous unsolved problem for over 60 years. In that time many ``solutions'' have been put forward, in both directions. Mistakes have always been found. In this paper I present a short proof of the non-existence, based on ideas developed, but not fully exploited, over 50 years ago.
... These reality conditions are relevant in the twistor approach to integrable systems (see §6), integral geometry, twistor inspired computations of scattering amplitudes (see §7), as well as recent applications [16] of the Index Theorem [23] which do not rely on positivity of the metric. The discussion in this subsection has assumed real analyticity of M R . ...
... 16) where D µ = ∂ µ + [A µ , ·], and Ψ = Ψ(w, z,w,z, λ) is the fundamental matrix solution. Computing the commutator of the Lax pair (L, M) yields[L, M] = Fzw − λ(F ww − F zz ) + λ 2 F wz = 0,and the vanishing of the coefficients of various powers of λ gives (5.14). ...
We review aspects of twistor theory, its aims and achievements spanning the last five decades. In the twistor approach, space-time is secondary with events being derived objects that correspond to compact holomorphic curves in a complex threefold-the twistor space. After giving an elementary construction of this space, we demonstrate how solutions to linear and nonlinear equations of mathematical physics-anti-self-duality equations on Yang-Mills or conformal curvature-can be encoded into twistor cohomology. These twistor correspondences yield explicit examples of Yang-Mills and gravitational instantons, which we review. They also underlie the twistor approach to integrability: the solitonic systems arise as symmetry reductions of anti-self-dual (ASD) Yang-Mills equations, and Einstein-Weyl dispersionless systems are reductions of ASD conformal equations. We then review the holomorphic string theories in twistor and ambitwistor spaces, and explain how these theories give rise to remarkable new formulae for the computation of quantum scattering amplitudes. Finally, we discuss the Newtonian limit of twistor theory and its possible role in Penrose's proposal for a role of gravity in quantum collapse of a wave function.
... Despite different proofs of this claim, based on distinct arguments, presented by Etesi, the mathematical community does not seem to have found a common opinion concerning his results -we refer to the discussions on mathoverflow [1, 2]. On the other hand, Sir Michael Atiyah posted recently a preprint on the arXiv, see [5], in which he claims to have proved the non existence of a complex structure on S 6 . In this case as well, the community of experts does not seem to find unity, see again the mathoverflow discussions [3,4]. ...
This short note serves as a historical introduction to the Hopf problem: "Does there exist a complex structure on ?" This unsolved mathematical question was the subject of the Conference "MAM 1 (Non-)Existence of Complex Structures on ", which took place at Philipps-Universit\"at Marburg, Germany, between March 27th and March 30th, 2017.
... Despite different proofs of this claim, based on distinct arguments, presented by Etesi, the mathematical community does not seem to have found a common opinion concerning his results -we refer to the discussions on mathoverflow [1, 2]. On the other hand, Sir Michael Atiyah posted recently a preprint on the arXiv, see [5], in which he claims to have proved the non existence of a complex structure on S 6 . In this case as well, the community of experts does not seem to find unity, see again the mathoverflow discussions [3,4]. ...
This short note serves as a historical introduction to the Hopf problem: "Does there exist a complex structure on ?" This unsolved mathematical question was the subject of the Conference "MAM 1 (Non-)Existence of Complex Structures on ", which took place at Philipps-Universit\"at Marburg, Germany, between March 27th and March 30th, 2017.
... The total Chern scalar curvature of ω g is (2.2) X s C · ω n g = n Ric(ω g ) ∧ ω n−1 g , 1 Recently, it was announced in [4] by Sir Michael Atiyah that there is no complex structure on S 6 . ...
Let (X, g) be a compact Riemannian manifold with quasi-positive Riemannian scalar curvature. If there exists a complex structure J compatible with g, then the Kodaira dimension of (X, J) is equal to and the canonical bundle is not pseudo-effective. We also introduce the complex Yamabe number for compact complex manifold X, and show that if is greater than 0, then is equal to ; moreover, if X is also spin, then the Hirzebruch A-hat genus is zero.
... In other recent work of the first author, [2], related to the famous question on the existence of complex structures on the sphere S 6 , odd and even modules for the quaternion group of order eight are considered, where the odd modules are faithful quaternionic representations, with value −1 on the center, while the even ones descend to abelian modules and have value +1 on the center. We expect that these odd and even types will also play a role in the geometric models of matter, where they may be related to topological insulators. ...
We show that the "geometric models of matter" approach proposed by the first author can be used to construct models of anyon quasiparticles with fractional quantum numbers, using 4-dimensional edge-cone orbifold geometries with orbifold singularities along embedded 2-dimensional surfaces. The anyon states arise through the braid representation of surface braids wrapped around the orbifold singularities, coming from multisections of the orbifold normal bundle of the embedded surface. We show that the resulting braid representations can give rise to a universal quantum computer.
We show the non-existence of complex structure on spheres of any dimension other than two, in particular, on . Therefore we solves the open problem on the non-existence of complex .
The paper is devoted to an algebraic analogue of a geometric approach to the classical notion of complex dilatation suggested in the paper arXiv:1701.06259 [math.CV] by the author. At the same time it provides an invariant version of this geometric approach. From the algebraic point of view it is only natural to work with a general field extension K/k of degree 2 instead of the fields of real and complex number (under the assumption that the characteristic is not equal to 2). Given a k-linear map between two K-vector spaces of dimension 1 over K, there are two natural measures of deviation of this map from being K-linear: its conformal dilatation, defined in terms of quadratic forms over k, and its Beltrami form, directly generalizing the classical complex dilatation. It turns out that these two measures are related in the same way as in the classical case. Working with a general field extension does not lead to any new difficulties compared to the classical case, but only clarifies the algebraic aspects of the theory.
On a compact complex manifold X it is an interesting problem to compare the continuous and holomorphic vector bundles. The case of line-bundles is classical and is well understood in the framework of sheaf theory. On the other hand for bundles E with dimE>dimX we are in the stable topological range and one can use K-theory. Much is known in this direction, for example the topological and holomorphic K-groups of all complex projective spaces are isomorphic. This paper deals with what is perhaps the simplest case not covered by the methods indicated above. We shall consider 2-dimensional complex vector bundles over the 3-dimensional complex projective space P3. Our aim is to prove (1.1) Theorem. Every continuous 2-dimensional vector bundle over P3 admits a holomorphic structure. The corresponding result for P2 was proved by Schwarzenberger [13], but this falls into the class of stable problems. In particular 2-dimensional vector bundles over P2 are determined by their Chern classes c 1 , c 2 . This is no longer true on P3 and therein lies the main difficulty and also the interest of this paper. In fact Horrocks in [-10] has already constructed holomorphic (actually algebraic) bundles with arbitrary cl, c 2 subject only to the topologically necessary condition that c~c 2 be even [8; p. 166]. It is not hard to see that, topologically, there are at most two bundles on P3 with given cl, c 2 . The two possibilities arise because the homotopy group n 5 (U(2)) ~ n 5 (S 3) which classifies 2-dimensional bundles over S 6, and acts on the bundles over P3, has order 2. It turns out that there are two sharply different cases depending on the parity of c~. In w 2 we study the case of even c 1 . By tensoring with line-bundles one reduces to the case of cl =0 in which case the structure group is SU(2)~-Sp(1). We view our 2-dimensional complex vector bundle as a quaternion line-bundle and this simplifies the classification because quaternion line-bundles over P3 are already
THE ^-theory of complex vector bundles (2, 5) has many variants and refinements. Thus there are: (1) ^-theory of real vector bundles, denoted by KO, (2) ^"-theory of self-conjugate bundles, denoted by KC (1) or KSC (7)
In 2003, S.-s. Chern began a study of almost-complex structures on the
6-sphere, with the idea of exploiting the special properties of its well-known
almost-complex structure invariant under the exceptional group . While he
did not solve the (currently still open) problem of determining whether there
exists an integrable almost-complex structure on the 6-sphere, he did prove a
significant identity that resolves the question for an interesting class of
almost-complex structures on the 6-sphere.
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- M F Atiyah
- R Bott
- A Shapiro
M.F. Atiyah, R. Bott and A. Shapiro. Clifford Modules. Topology 3 (Suppl. 1)
(1964), 3-38.