I have been working for a couple of weeks with the problem of defining the notion of p-adic manifold: this is one of the key challenges of TGD. The existing proposals by mathematicians are rather complicated and it seems that something is lacking. To my opinion, to identify this something it is essential to make the question "What p-adic numbers are supposed to describe?". This question has not bothered either matheticians or theoretical physicists proposing purely formal p-adic counterparts for the scattering amplitudes.
Without any answer to this question there are simply quite too many alternatives to consider and one ends up to the garden of branching paths. The text below is this introduction to the article and chapter about the topics.
This article was originally meant to be a summary of what I understand about the article "The p-Adic Icosahedron" in Notices of AMS. The original purpose was to summarize the basic ideas and discuss my own view about more technical aspects - in particular the generalization of Riemann sphere to p-adic context which is rather technical and leads to the notion of Bruhat Tits tree and Berkovich space. About Bruhat-Tits tree there is a nice web article titled p-Adic numbers and Bruhat-Tits tree describing also basics of p-adic numbers in a very concise form.
The notion of p-adic icosahedron leads to the challenge of constructing p-adic sphere, and more generally p-adic manifolds and this extended the intended scope of the article and led to consider the fundamental questions related to the construction of TGD.
Quite generally, there are two approaches to the construction of manifolds based on algebra resp. topology.
The attempt to construct p-adic manifolds by mimicking topological construction of real manifolds meets difficulties
The basic problem in the application of topological method to manifold construction is that p-adic disks are either disjoint or nested so that the standard construction of real manifolds using partially overlapping n-balls does not generalize to the p-adic context. The notions of Bruhat-Tits tree, building, and Berkovich disks and Berkovich space are represent attempts to overcome this problem. Berkovich disk is a generalization of the p-adic disk obtained by adding additional points so that the p-adic disk is a dense subset of it. Berkovich disk allows path connected topology which is not ultrametric. The generalization of this construction is used to construct p-adic manifolds using the modification of the topological construction in the real case. This construction provides also insights about p-adic integration.
The construction is highly technical and complex and pragmatic physicist could argue that it contains several un-natural features due to the forcing of the real picture to p-adic context. In particular, one must give up the p-adic topology whose ultra-metricity has a nice interpretation in the applications to both p-adic mass calculations and to consciousness theory.
I do not know whether the construction of Bruhat-Tits tree, which works for projective spaces but not for Qpn (!) is a special feature of projective spaces, whether Bruhat-Tits tree is enough so that no completion would be needed, and whether Bruhat-Tits tree can be deduced from Berkovich approach. What is remarkable that for M4× CP2 p-adic S2 and CP2 are projective spaces and allow Bruhat-Tits tree. This not true for the spheres associated with the light-cone boundary of D≠ 4-dimensional Minkowski spaces.
Two basic philosophies concerning the construction of p-adic manifolds
There exists two basic philosophies concerning the construction of p-adic manifolds: algebraic and topological approach. Also in TGD these approaches have been competing: algebraic approach relates real and p-adic space-time points by identifying common rationals. Finite pinary cutoff is however required to achieve continuity and has interpretation in terms of finite measurement resolution. Canonical identification maps p-adics to reals and vice versa in a continuous manner but is not consistent with field equations without pinary cutoff.
Number theoretical universality and the construction of p-adic manifolds
Construction of p-adic counterparts of manifolds is also one of the basic challenges of TGD. Here the basic vision is that one must take a wider perspective. One must unify real and various p-adic physics to single coherent whole and to relate them. At the level of mathematics this requires fusion of real and p-adic number fields along common rationals and the notion of algebraic continuation between number fields becomes a basic tool.
The number theoretic approach is essentially algebraic and based on the gluing of reals and various p-adic number fields to a larger structure along rationals and also along common algebraic numbers. A strong motivation for the algebraic approach comes from the fact that preferred extremals are characterized by a generalization of the complex structure to 4-D case both in Euclidian and Minkowskian signature. This generalization is independent of the action principle. This allows a straightforward identification of the p-adic counterparts of preferred extremals. The algebraic extensions of p-adic numbers play a key role and make it possible to realize the symmetries in the same manner as they are realized in the construction of p-adic icosahedron.
The lack of well-ordering of p-adic numbers implies strong constraints on the formulation of number theoretical universality.
How to achieve path connectedness?
The basic problem in the construction of p-adic manifolds is the total disconnectedness of the p-adic topology implied by ultrametricity. This leads also to problems with the notion of p-adic integration. Physically it seems clear that the notion of path connectedness should have some physical counterpart.
The notion of open set makes possib le path connectedness possible in the real context. In p-adic context Bruhat-Tits tree and Berkovich disk are introduced to achieve the same goal. One can of course ask whether Berkovich space could allow to achieve a more rigorous formulation for the p-adic counterparts of CP2, of partonic 2-surfaces, their light-like orbits, preferred extremals of Kähler action, and even the "world of classical worlds" (WCW). To me this construction does not look promising in TGD framework but I could be wrong.
TGD suggests two alternative approaches to the problem of path connectedness. They should be equivalent.
p-Adic manifold concept based on canonical identification
The TGD inspired solution to the construction of path connectd p-adic topology is based on the notion of canonical identification mapping reals to p-adics and vice versa in a continuous manner.
Could path connectedness have quantal description?
The physical content of path connectedness might also allow a formulation as a quantum physical rather than primarily topological notion, and could boil down to the non-triviality of correlation functions for second quantized induced spinor fields essential for the formulation of WCW spinor structure. Fermion fields and their n-point functions could become part of a number theoretically universal definition of manifold in accordance with the TGD inspired vision that WCW geometry - and perhaps even space-time geometry - allow a formulation in terms of fermions.
The natural question of physicist is whether quantum theory could provide a fresh number theoretically universal approach to the problem. The basic underlying vision in TGD framework is that second quantized fermion fields might allow to formulate the geometry of "world of classical worlds" (WCW) (for instance, Kähler action for preferred extremals and thus Kähler geometry of WCW would reduce to Dirac determinant. Maybe even the geometry of space-time surfaces could be expressed in terms of fermionic correlation functions.
This inspires the idea that second quantized fermionic fields replace the K-valued (K is algebraic extension of p-adic numbers) functions defined on p-adic disk in the construction of Berkovich. The ultrametric norm for the functions defined in p-adic disk would be replaced by the fermionic correlation functions and different Berkovich norms correspond to different measurement resolutions so that one obtains also a connection with hyper-finite factors of type II1. The existence of non-trivial fermionic correlation functions would be the counterpart for the path connectedness at space-time level. The 3-surfaces defining boundaries of a connected preferred extremal are also in a natural manner "path connected": the "path" is defined by the 4-surface. At the level of WCW and in zero energy ontology (ZEO) WCW spinor fields are analogous to correlation functions having collections of these disjoint 3-surfaces as arguments. There would be no need to complete p-adic topology to a path connected topology in this approach.
It must be emphasized that this apporach should be consistent with the first option and that it is much more speculative that the first option.
It is not easy to find readable literature from these topics. The Wikipedia article about Berkovich space is written with a jargon giving no idea about what is involved. There are video lectures about Berkovich spaces. The web article about Berkovich spaces by Temkin seems too technical for a non-specialist. The slides however give a concise bird's eye of view about the basic idea behind Berkovich spaces.
Topics of the article
The article was originally meant to discuss p-adic icosahedron. Although the focus was redirected to the notion of p-adic manifold - especially in TGD framework - I decided to keep the original starting point since it provides a concrete manner to end up with the deep problems of p-adic manifold theory and illustrates the group theoretical ideas.
For details see the new chapter What p-adic icosahedron could mean? And what about p-adic manifold? or the article with the same title.