Tensor-Centric Warfare V: Topology of Systems Confrontation

In this paper, as a new contribution to the tensor-centric warfare (TCW) series [1] [2] [3] [4], we extend the kinetic TCW-framework to include non-kinetic effects, by addressing a general systems confrontation [5], which is waged not only in the traditional physical Air-Land-Sea domains, but also simultaneously across multiple non-physical domains, including cyberspace and social networks. Upon this basis, this paper attempts to address a more general analytical scenario using rigorous topological methods to introduce a two-level topological representation of modern armed conflict; in doing so, it extends from the traditional red-blue model of conflict to a red-blue-green model, where green represents various neutral elements as active factions; indeed, green can effectively decide the outcomes from red-blue conflict. System confrontations at various stages of the scenario will be defined by the non-equilibrium phase transitions which are superficially characterized by sudden entropy growth. These will be shown to have the underlying topology changes of the systems-battlespace. The two-level topological analysis of the systems-battlespace is utilized to address the question of topology changes in the combined battlespace. Once an intuitive analysis of the combined battlespace topology is performed, a rigorous topological analysis follows using (co)homological invariants of the combined systems-battlespace manifold.


Introduction
The principal objective of the Modeling Complex Warfighting (MCW) Strategic The aim is to undertake analysis which closely represents the realities of modern warfare. When modeling complex modern combined battlefields it is therefore important to consider neutral forces-which we label "Green"-since they are much more now than in the past even the central feature of the strategic situation. Indeed, in modern asymmetric confrontations neutral or non-engaging groups have been known to side with one side or the other or even to engage actively in conflict, which rapidly changes the dynamics of the situation. Hence, social and psychological domains play an increasingly significant role in understanding the dynamics of modern armed conflict.
The dynamics of the combined effects of these and other factors mean that the statistical properties of such problem environments are non-stationary. As a result, the outcomes of battles are not predictable, since the battle is inherently non-ergodic; yet, it is possible to deal with such systems nonetheless by establishing conditions that are weaker than ergodicity, which have essentially topology-changing nature.
The tensor-centric warfare (TCW) series of papers (see [1]    that the battlefield entropy is conserved, 0 t S ∂ = (see [2]);  Entropic Red-Blue commutators, , for modeling warfare symmetry (see [2]), in which the entropy grows in the asymmetric case (>0) and stays conserved in the symmetric case (=0);  Hamilton-Langevin delta strikes, (on both sides) discrete striking spectra (slow-fire missiles) and continuous striking spectra (rapid-fire missiles), as well as bidirectional random strikes, Hamiltonian vectorfields, self-dissipation, opponent-caused dissipation and non-delta random forces (see [3] for the full explanation of all included temporally-confined kinetic strike/missile terms).
In the present paper, we extend the above kinetic Red-Blue framework to include non-kinetic effects, by addressing the general systems-confrontation [5], which is waged not only in the traditional physical Air-Land-Sea domains, but also in modern non-physical environments, such as cyberspace, electromagnetic spectrum, psychological and social network domains. This paper attempts to address this complex contemporary warfare situation using rigorous methods and techniques from modern algebraic topology; specifically, by extending the kinetic Red-Blue scenario ( Figure 1) into more general and more representative kinetic + non-kinetic Red-Blue-Green scenarios ( Figure 2 , and their corresponding flows, x , which consist of the integral lines of the vectorfields a R  and a B  obtained by their numerical integration starting from the chosen initial conditions, 0 These three quadratic forms are not necessarily positive-definite, which would be the necessary condition for the strict Riemannian geometry, but only non-degenerate, which is a weaker condition. Since we are working in the more general 2 Recall that John A. Wheeler was initially an assistant of Albert Einstein and later a supervisor of two future Nobel Laureates, Richard Feynman and Kip Thorne (even later, he became the "Godfather of Mathematica", as called by Stephen Wolfram). Einstein introduced local Riemannian geometry into physics, under his famous dictum: "Physics is simple only locally". Wheeler was not satisfied with this local view of physics-he felt that even when differential equations of all fields and motions are precisely defined-something important is still left missing-the global topology of the environmental configuration manifold in which these equations evolve (e.g., Einstein's gravitational equations are the same for elliptic surface of the Earth as for elliptic/hyperbolic surface of an apple-which is clearly not quite right, notwithstanding the magnificence of Einstein gravity theory).
So, using his famous slogan: "the boundary of a boundary is zero" (BBZ), Wheeler introduced global topological analysis into physical sciences, which goes hand-in-hand with local Riemannian geometry used by Einstein. Similarly, in our TCW systems-battle space, even when the local Red-Blue tensor equations are precisely defined, we are still missing the global topology of the systems-battle space with its dramatic changes. Redressing this is the objective of the present paper. 3 More correctly, according to the existence and uniqueness theorems for the sets of ODEs, the Red and Blue flows uniquely exist on the TM . However, for the present topological considerations, this subtle geometric difference can be neglected, or rather unified within the notion of pseudo-Riemannian geometry (see, e.g. [7] [8] and the references therein). 4 In this paper we present continuous, analytical approach to topological analysis of systems confrontation. Alternatively, a discrete, computational framework with networks of up to millions of nodes, based on persistent homology algorithms on directed simplices [9] has been developed as a Matlab toolbox supporting the cutting-edge topological research of brain cliques and cavities from computational neuroscience (the Blue Brain project [10] [11] functions applied to the bipartite-Red and tripartite-Blue adjacency matrices from the initial scenario 5 from [1] [12] (as depicted in Figure 1). This gives us the level of generality needed for a reasonable representation of the systems-battlespace, but can be easily generalized further to the system confrontation level by adding non-kinetic terms, mostly present within the combined Green tensor: The non-equilibrium phase transitions occurring at the battlefield at various stages of warfare, can be superficially characterized by sudden entropy growth.
However, these rapid changes of the systems-battlespace always have underlying structural topology changes (see [13] and the references therein). In this paper, we give a two-level topological analysis of the systems-battlespace. We start visually by giving a largely intuitive analysis of the systems-battlespace topology using Thom's cobordisms and Morse functions. Then we move into a rigorous topological analysis of the systems-battlespace by deriving its (co)homological invariants, which can be summarized by the famous dictum of John Wheeler: "The boundary of a boundary is zero (BBZ)". Specifically, we derive the Morse-Smale homology and the Morse-Witten cohomology of the systems-battlespace manifold. All the necessary geometrical and topological background is given in the self-content and comprehensive Appendix, which provides the Hodge-de Rham theory based on the Stokes theorem. Then we perform a rigorous analysis of the systems-battlespace topology with its dramatic spatial changes by deriving its (co)homological invariants 6 .

Components of the Systems-Battlespace Topology: Cobordisms and Morse Functions
In this section we develop the basic differential topology of the systems-battlespace, mainly following the work of the Fields medalist John Milnor [15] [16] [17].

Systems-Battlespace Cobordisms: Red-Blue versus Red-Blue-Green
To start with the systems-battlespace topology, we introduce an important concept from differential topology (and its gravitational-physics applications): 5 Recall that in our initial kinetic scenario (see [1] [12]) we have 30 n = aircraft on each side. In case of the Red force, they enter into the combat in the bipartite (15 + 15)-formation, while in the case of the Blue force, they enter into the combat in the tripartite (10 + 10 + 10)-formation. 6 The importance of the homological invariants here relates directly to the previously identified approach for dealing with uncertainty in non-ergodic problem environments by establishing weaker invariant conditions than ergodicity as the basis for reliable decision-making, within known limitations [14]. The context of this previous work was primarily autonomous systems development; however, the homological invariants derived here shows that the approach is applicable to modeling systems manifesting uncertainty more generally.   Figure 2).
Clearly, this operation can be extended to any number of parties, producing the so-called chain cobordism, between the first and the last manifold; e.g., in case of four parties/manifolds (Red, Blue, Green, Yellow), we have the following commutative chain cobordism: , etc.

Morse Functions of the Systems-Battlespace: Red-Blue versus Red-Blue-Green
Closely-related to the systems-battlespace cobordisms are the Morse functions of the systems-battlespace. Namely, on the Red-Blue systems-battlespace ( Figure   1), which is a pseudo-Riemannian n-manifold: More precisely, the c-level set of the Red-Blue Morse function RB : otherwise the topology of W does not change. The mechanism of topology change is attaching a λ -cell (the so-called "handlebody") to RB  11 We remark that the special handle body calculus has been developed to address the spatial topology changes (with applications in quantum gravity; see [19] and the references therein). However, this approach represents a further extension of Morse topology, too technical for the scope of the present paper-it might be addressed in our future research. 12 In general, the Morse number ( )  13 The nth Betti number represents the rank/dimension of the nth homology and nth cohomology groups, derived in the next section. 14 We can plot this landscape topology in a symmetric fashion, because any smooth (diffeomorphic) local perturbation will leave these principal characteristics invariant. Intelligent Control and Automation   Case 4 (depicted in Figure 6): both the Red and Blue quadratic forms are x , represented by its own pseudo-Riemannian quadratic form: which represents the Red-Blue-Green landscape. Its gradient vectorfield: according to the Morse lemma, defines the Red-Blue-Green level set 16 of equipotential contour lines (see Figure 7) passing through the critical points p of RBG f where the gradient vanishes: In addition, we need to consider only nondegenerate critical points of RBG f , that is, only those critical points p of the vanishing gradient, otherwise the topology of RBG W does not change. The mechanism of topology change is attaching a λ -cell/handlebody, completely determined by the index λ , at the critical points p of RBG f . Therefore, the index λ determines the topology changes of the Red-Blue-Green systems-battlespace RBG W (from Figure 2).
As in the case of cobordisms, this 3-party Morse function can be extended to address more players (e.g., various groups within Green, various coalition partners within Blue and Red, or even a third conflicting Yellow force) in the wargame.

Morse (Co)homology of the Systems-Battlespace
In this section, we move to the realm of (co)homology, which can be summarized by Wheeler's BBZ dictum: "the boundary of a boundary is zero".
We explore the systems-battlespace topology changes, using Morse (co)homology techniques. We will apply Morse (co)homology to the systems-battlespace-cobordism n-manifold W using two approaches, classical approach of Morse homology and modern approach of Morse cohomology (see the Appendix for the basic (co)homology definitions, all rooted in the BBZ dictum).

Morse-Smale Homology of the Systems-Battlespace
The Recall that the concept of homology as a rigorous mathematical method for defining and categorizing holes in a manifold was pioneered by Henri Poincaré in his seminal 1895-paper "Analysis situs" [21] (which introduced homology classes and relations; the possible configurations of orientable cycles are classified by the Betti numbers, which are refinement of the Euler characteristic of the manifold). Homology theory was developed as a way to analyze and classify manifolds according to their cycles. Informally, a cycle is a closed submanifold, a boundary is a cycle which is also the boundary of a submanifold and a homology class (which represents a hole) is an equivalence class of cycles modulo boundaries. A non-trivial equivalence class is thus represented by a cycle which is not the boundary of any submanifold. A hypothetical manifold whose boundary would be that particular cycle is "not there" which is why that cycle is indicative of the presence of a hole.
In local coordinates in a neighborhood of c , such that : , for every critical point c x (for the proof, see [17]). We develop the Morse homology of the systems-battlespace-cobordism n-manifold W in the following three steps: 1) On the systems-battlespace manifold W we define the negative gradient flow, ( From the work of Smale [22] [23], it follows that for a generic metric g W ∈ , the corresponding Hessian has only nondegenerate eigenvalues. 2) Using the negative gradient flow (3) , n x y is the number of points in the quotient manifold: The proof of the BBZ-condition: 2 0 ∂ = ∂ ∂ =  is based on gluing and cobordism arguments (see [20]  with lim and lim .

Morse-Witten Cohomology of the Systems-Battlespace
For technical details on structurally stable Morse-Smale dynamical systems, see [27] and the references therein. 20 The construction of the homology isomorphism:

Conclusions
Modern warfare, compared with its historical precedents, is marked by a shift from large-scale annihilation along defined fronts, and relatively little regard for 21 Note that in the previous subsection, the points on the manifold W were labeled by p and their Morse index by λ . However, in this subsection, the symbol λ is reserved for the eigenvalues of the Laplacian t ∆ , while p is reserved for the rank of the (co)homology groups. Intelligent Control and Automation neutral parties caught in the situation, to aims of causing system failure that undercuts an opposition's ability or willingness to fight, simultaneous conflict occurring across multiple domains without definable lines, and foundational international and national legal and social expectations about human, environmental and social consequences of armed conflict. Indeed, in contemporary conflict, social and humanitarian concerns can often both motivate confrontation and decide operational success. Arguably, this shift has been driven by a complex interwoven web of technological developments, social change, and legal, moral and ethical constraints, which first came to the fore in the modern sense during the soul-searching in post-Napoleonic Europe that simultaneously yielded the basis for both the modern professional military force and international humanitarian law. The combined effect of these factors is extreme nonlinearity, which makes approaches to modeling war and battle that represent simple attrition largely obsolete.
In this paper, we have extended the previously developed kinetic TCW-framework, to include non-kinetic effects, by addressing the general systems-confrontation, which means that our modeling of armed conflict includes interaction not only in the traditional physical Air-Land-Sea domains, but also in non-physical cyberspace, electromagnetic, psychological and social-network domains. In addition, we extend the TCW framework with the ability to represent "Green" neutral parties as richly as the main "Blue" and "Red" adversaries, and extend this to many factions, including coalition partners in Blue and Red and factions within Green, or even to situations with three or more main adversaries. In our formulation, Green may hold the ability to decide operational success from conflict between Blue and Red. This paper attempts to address this generic scenario representative of modern war and battle conditoins using rigorous methods and techniques from modern topology, specifically, by extending the kinetic Red-Blue scenario into this more general kinetic + non-kinetic Red-Blue-Green scenario. In particular, we have focussed here on the question of dramatic changes in the topology of the systems-battlespace, which appears as non-equilibrium phase transitions occurring at the battlefield at various stages of warfare, and is usually superficially characterized by sudden entropy growth. Such sudden changes have been long recognised as central features of war and battle; we thus have new modeling machinery with which to study their occurrence and effects.
We have performed a two-level topological analysis of the systems-battlespace. We have started gently with a largely intuitive analysis of the systems-battlespace topology using visual cobordisms and Morse functions. Then, we performed a rigorous topological analysis of the systems-battlespace by deriving its (co)homological invariants. Specifically, we derived the Morse-Smale homology and the Morse-Witten cohomology of the systems-battlespace manifold. All the necessary geometrical and topological background is given in the self-content and comprehensive Appendix, which provides the Hodge-de Rham theory based on the Stokes theorem.

Appendix: From Stokes-De Rham to Hodge Theory
Here we give a brief introduction to the Stokes-de Rham theory on arbitrary smooth manifolds, followed by its extension, the Hodge theory on Riemannian manifolds, all three standing at the crossroads of differential geometry, algebraic topology and modern physics, thus enriching all three disciplines (see [29]).

A.1. Stokes Theorem and Differential Forms
At the core of differential geometry (and its application to algebraic topology) lies the celebrated Stokes theorem. This fundamental result of modern mathematics (see, e.g. [30]) can be "softly" introduced in the following way.
Recall from multivariable calculus [31] that two differential forms (integrands in multiple integrals called the cochains in topology), A and dA 22 , defined in the Euclidean ( ) , x y -plane 2  (via two smooth functions 2 , P Q ∈  ) as:

1-form:
are related by the Green theorem in the closed region 2 C ∈  with the boundary C ∂ 23 : which can be rewritten as the Stokes theorem: -valid for any exterior differential p-form A in n  (as well as for all oriented 24 smooth n-manifolds). 22 The linear exterior derivative/differential operator d (also called the coboundary operator, or de Rham differential/homomorphism) represents a generalization of ordinary vector differential operators (grad, div and curl; see [32] [33]) that transforms p-forms w into (p + 1)-forms dw , with the fundamental closure property: the boundary of a boundary is zero (BBZ; see [34] [35]); formally, the exterior differential d is nilpotent: . For example, in 3  we have: 1) any scalar function ( ) , , f f x y z = is a 0-form; 2) the gradient f = d w of any smooth function f is a 1-form 3) the curl = a dw of any smooth 1-form w is a 2-form  23 The integration domain C is in topology called a chain, and C ∂ is a 1D boundary of a 2D chain C. In general, C ∂ is a (p − 1)-boundary of a p-chain C, governed by the BBZ property: Because of the common BBZ property, chains are dual to differential forms. 24 An orientation on an n-manifold is given by a nowhere vanishing exterior n-form. The integrands A and dA in the Stokes theorem (4) are the special 1D and 2D cases of general exterior differential p-forms, which are completely antisymmetric covariant tensors of rank p in n  (for p n ≤ ). Their "exterior calculus" can be introduced in the following "way of physics" where the most frequently used Euclidean n  space is 4  . Here in 4  , given the frame: where ∧ represents the exterior product 26 .

3-form
with components: or 25 In a smooth n-manifold M with local coordinates { } Furthermore, a nonlinear generalization of the Stokes theorem (5) to any oriented smooth manifold provides the general machinery for integration on smooth manifolds. It is based on the fundamental de Rham's duality between p-forms and p-chains, described in the dual language of (co)cycles and (co)boundaries, as follows.
Notation change: to improve the flow of the paper, we drop boldface letters from now on.
On a smooth n-manifold M, a cycle is a finite p-chain 27 Integration on a smooth manifold M should be thought of as a nondegenerate bilinear pairing , between p-forms and p-chains (spanning a finite domain on M). The duality of p-forms and p-chains on M is based on the de Rham period, the cycle, cocycle -pairing: (see [30] [32]). From the Poincaré lemma it follows that a closed p-form ω is exact iff , 0 C ω = .
Naturally, this fundamental topological duality is rooted in the Stokes theorem (5), as: 27 A p-chain C is a formal sum of the form: The chains and their boundaries are rigorously defined in simplicial and singular homology theories (see [36] [37]). 28 For this reason, the exterior differential d is also called the coboundary operator. 29 In general, a p-form β is called closed if its exterior derivative From this closure-condition one can see that the closed form, which is the kernel of the exterior derivative operator d, is a conserved quantity. Therefore, closed p-forms possess certain invariant properties, corresponding to the conservation laws in physics (see e.g., [38]). Also, a p-form β that is an exterior derivative of some (p − 1)-form α , that is, where C ∂ is the boundary of the p-chain C oriented on M coherently with C. While the boundary operator ∂ is a global operator, the coboundary operator d is local, and thus more suitable for applications. The dual BBZ-closure property:  is proved using the Stokes' theorem (6), in period notation as: or, in integral notation as:

A.2. De Rham's (Co)chain Complex and (Co)homology
In the Euclidean 3D space 3  we have the following short exact sequence 30 , called short de Rham cochain complex: Using the BBZ-closure property:   , we obtain the standard identities from vector calculus: curl grad 0 and div curl 0. = = As a duality in 3  , we also have another short exact sequence, called short chain complex: Its own BBZ-closure property:   implies the following three boundaries: 0  1  2  1  2  3  2  3 , , , where 0 0 C ∈  is a 0-boundary (or, a point), 1 1 C ∈  is a 1-boundary (or, a line), 2 2 C ∈  is a 2-boundary (or, a surface), and 3 3 C ∈  is a 3-boundary (or, a hypersurface). Similarly, the de Rham complex implies the following three coboundaries: where 0 0 C ∈ Ω is 0-form (or, a function), 1 1 C ∈ Ω is a 1-form, 2 2 C ∈ Ω is a 2-form, and 3 3 C ∈ Ω is a 3-form. 30 A short exact sequence of three vector spaces, or groups ( )

, ,
A B C , is governed by two linear maps, or homomorphisms, : f A B → (which is injective, or "one-to-one" map) and : g B C → (which is surjective, or "onto" map), and is written: 0 0 which can be rewritten, using the Hodge 2 L -inner product (9), as: Hodge codifferential δ .
The Hodge dual (or, formal adjoint) to the exterior derivative   As an example, to translate the notions from standard vector calculus in 3  , we first identify scalar functions with 0-forms, field intensity vectors with 1-forms, flux vectors with 2-forms and scalar densities with 3-forms. Then we have the following correspondence: Grad → d: on 0-forms; curl → d  : on 1-forms; Div → δ : on 1-forms; div grad → ∆ : on 0-forms; Curl curl-grad div → ∆ : on 1-forms.
We remark here that exact and coexact p-forms ( Hodge adjoints and self-adjoints. If α is a p-form and β is a (p + 1)-form then we have [32] [40]: