K-theory (of Grothendieck) as a theoretical framework of E-infinity and extension of the Riemann-Roch index theorem

19th January, 2011.
E-infinity Communication No. 73
K-theory (of Grothendieck) as a theoretical framework of E-infinity and extension of the Riemann-Roch index theorem

As is well known, K-theory as well as the work of M. Atiyah are extensions of the celebrated Riemann-Roch index theorem. K-theory is largely due to the mathematician (almost cult figure) Alexander Grothendieck (born 1928, Field Medal 1966) who presently lives as a recluse and who declined receiving the Crafoord Prize or in fact any prizes since 1988 because he was disillusioned with the scientific community and society at large. He even resigned his prestigious position at the very prestigious IHES in France. Those interested in the very deep mathematical foundations of E-infinity theory should consider the role of the Riemann-Roch index theorem in the work of Mohamed El Naschie. During his Cambridge time, El Naschie published in 2000 a paper entitled “On the unification of Heterotic strings, M-theory and E-infinity theory”, CS&F, Vol. 11 (2000), pp. 2397-2408. In this paper El Naschie dealt with various mathematical aspects including the Riemann-Roch theorem which he evaluated on page 2406 for n = 4 and q = 1 + ϕ where = is the golden mean and found that the index is
Index = (2 n ̶ 1)(q ̶ 1)
= (7)(ϕ)
= 4.3262.
Numerical evaluation taking 1200 times by Castro and Granik using gamma function to find the average Hausdorff dimension of E-infinity spacetime gave 4.32 which is very close to the above exact theoretical value. Now the exact Hausdorff dimension of E-infinity spacetime core is well established and is given by 4.236067977 which is quite close to both the Riemann-Roch index as well as the gamma function approximation using these functions as weights in analogy to the golden mean weight of the E-infinity derivation. The reason for this proximity between 4.23606799 of E-infinity and 4.3262 of the Riemann-Roch index and the 4.32 of the gamma function numerical approximation is explained as follows by Prof. El Naschie: As observed by André Weil there is a deep connection between the topological characteristic of a variety and its number theoretic aspect and this blend between numbers and topology is a main line in E-infinity theory. The same is true for K-theory as we will discuss shortly. In a second paper published a year later in 2001 El Naschie returned to Riemann-Roch, this time in connection to moduli spaces. The paper is entitled “Remarks to moduli spaces, virtual dimensions and Heterotic strings”, CS&F, 12 (2001), pp. 1607-1610. He considers on page 1607 moduli spaces of bundles and derives the dimensions of super strings in integer form. Subsequently he derives the exact transfinite values including the Euler characteristic of E-infinity, namely 26 + k = 26.18033989. We repeatedly stressed that El Naschie showed that Penrose tiling is the prototype example for both Connes’ noncommutative geometry as well as E-infinity theory.
Let us look a little closer at the K-theory connection of Penrose tiling. E-infinity researchers are familiar with the construction of a Cantor set. However only those working in deterministic chaos will be familiar with Steve Smale’s horseshoe. From this construction of a horseshoe one will realize the relation between numerical sequences and a Cantor set. The Smale construction as a mapping between a Cantor set and infinite sequences of zeros and ones. There is an important result permitting to parameterize a Penrose tiling with a set K of infinite sequence of zero and one satisfying certain conditions which will turn out to describe the Connes dimensional function or El Naschie’s bijection formula. This K is a compact and of course totally disjoint space homeomorphic to a Cantor set and we also have a relation of equivalence on R. The space x of Penrose tiling is the quotient space x = K/R. In E-infinity theory, it is easily shown that the dimension of K is given by the dimension of the real line plus the dimension of the resulting Cantor set while the dimension of R is equal to that of the real minus the dimension of the Cantor set. Assuming a random Cantor set with a Hausdorff dimension equal ϕ as per Mauldin-Williams theorem we find that
dim x = dim K/dim R
= (1 + ϕ)/(1 ̶ ϕ)
= 4 + the golden mean to power 3
= 4.236067977.
This is the well known Hausdorff dimension of E-infinity and corresponds to a 4 Menger-Urysohn topological dimension. For further more technical discussion of the relation of Penrose tiling, noncommutative geometry and K-theory, the reader is referred to the nice mathematical papers of D. Bigatti for instance. El Naschie was frequently asked ‘what is the simplest most direct way to start making contact with E-infinity theory?’ Here is his answer. The deep problems are the continuum, the denumerable infinity and the not denumerable infinity which is related to the cardinality of the continuum. In E-infinity we have our cake and eat it so to speak. We have countably infinite numbers of Cantor sets which we sum over. In each Cantor set we have uncountably infinite Cantor points. Thus we are counting not over points but over equivalence classes. In his latest work to salvage quantum field theory ‘t Hooft adopted this view point and introduced a dilaton field akin to that of El Naschie’s compactified Klein modular space. This space has an inbuilt dilaton field and is homeomorphic to Penrose fractal tiling, i.e. to the x = K/R space. For a new theory connecting string states with the theory of instantons using x = K/R, the reader is referred to A. Elokaby’s paper “On the deep connection between instantons and string states encoded in Klein’s modular space”, CS&F, 42 (2009), pp. 303-305.
E-infinity Group.