Interleaves

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Many important spaces in topology and algebraic geometry have no odd-dimensional homology. For such spaces, functorial spatial homology truncation simplifies considerably. On the theory side, the simplification arises as follows: To define general spatial homology truncation, we used intermediate auxiliary structures, the n-truncation structures. For spaces that lack odd-dimensional homology, these structures can be replaced by a much simpler structure. Again every such space can be embedded in such a structure, which is the analogon of the general theory. On the application side, the crucial simplification is that the truncation functor t<n will not require that in truncating a given continuous map, the map preserve additional structure on the domain and codomain of the map. In general, t<n is defined on the category CWn⊃∂, meaning that a map must preserve chosen subgroups “Y ”. Such a condition is generally necessary on maps, for otherwise no truncation exists. So arbitrary continuous maps between spaces with trivial odd-dimensional homology can be functorially truncated. In particular the compression rigidity obstructions arising in the general theory will not arise for maps between such spaces.

Let ICW be the full subcategory of CW whose objects are simply connected CW-complexes K with finitely generated even-dimensional homology and vanishing odd-dimensional homology for any coefficient group. We call ICW the interleaf category.

For example, the space K = S22 e3 is simply connected and has vanishing integral homology in odd dimensions. However, H3(K;Z/2) = Z/2 ≠ 0.

Let X be a space whose odd-dimensional homology vanishes for any coefficient group. Then the even-dimensional integral homology of X is torsion-free.

Taking the coefficient group Q/Z, we have

Tor(H2k(X),Q/Z) = H2k+1(X) ⊗ Q/Z ⊕ Tor(H2k(X),Q/Z) = H2k+1(X;Q/Z) = 0.

Thus H2k(X) is torsion-free, since the group Tor(H2k(X),Q/Z) is isomorphic to the torsion subgroup of H2k(X).

Any simply connected closed 4-manifold is in ICW. Indeed, such a manifold is homotopy equivalent to a CW-complex of the form

Vi=1kSi2ƒe4

where the homotopy class of the attaching map ƒ : S3 → Vi=1k Si2 may be viewed as a symmetric k × k matrix with integer entries, as π3(Vi=1kSi2) ≅ M(k), with M(k) the additive group of such matrices.

Any simply connected closed 6-manifold with vanishing integral middle homology group is in ICW. If G is any coefficient group, then H1(M;G) ≅ H1(M) ⊗ G ⊕ Tor(H0M,G) = 0, since H0(M) = Z. By Poincaré duality,

0 = H3(M) ≅ H3(M) ≅ Hom(H3M,Z) ⊕ Ext(H2M,Z),

so that H2(M) is free. This implies that Tor(H2M,G) = 0 and hence H3(M;G) ≅ H3(M) ⊗ G ⊕ Tor(H2M,G) = 0. Finally, by G-coefficient Poincaré duality,

H5(M;G) ≅ H1(M;G) ≅ Hom(H1M,G) ⊕ Ext(H0M,G) = Ext(Z,G) = 0

Any smooth, compact toric variety X is in ICW: Danilov’s Theorem implies that H(X;Z) is torsion-free and the map A(X) → H(X;Z) given by composing the canonical map from Chow groups to homology, Ak(X) = An−k(X) → H2n−2k(X;Z), where n is the complex dimension of X, with Poincaré duality H2n−2k(X;Z) ≅ H2k(X;Z), is an isomorphism. Since the odd-dimensional cohomology of X is not in the image of this map, this asserts in particular that Hodd(X;Z) = 0. By Poincaré duality, Heven(X;Z) is free and Hodd(X;Z) = 0. These two statements allow us to deduce from the universal coefficient theorem that Hodd(X;G) = 0 for any coefficient group G. If we only wanted to establish Hodd(X;Z) = 0, then it would of course have been enough to know that the canonical, degree-doubling map A(X) → H(X;Z) is onto. One may then immediately reduce to the case of projective toric varieties because every complete fan Δ has a projective subdivision Δ, the corresponding proper birational morphism X(Δ) → X(Δ) induces a surjection H(X(Δ);Z) → H(X(Δ);Z) and the diagram

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commutes.

Let G be a complex, simply connected, semisimple Lie group and P ⊂ G a connected parabolic subgroup. Then the homogeneous space G/P is in ICW. It is simply connected, since the fibration P → G → G/P induces an exact sequence

1 = π1(G) → π1(G/P) → π0(P) → π0(G) = 0,

which shows that π1(G/P) → π0(P) is a bijection. Accordingly, ∃ elements sw(P) ∈ H2l(w)(G/P;Z) (“Schubert classes,” given geometrically by Schubert cells), indexed by w ranging over a certain subset of the Weyl group of G, that form a basis for H(G/P;Z). (For w in the Weyl group, l(w) denotes the length of w when written as a reduced word in certain specified generators of the Weyl group.) In particular Heven(G/P;Z) is free and Hodd(G/P;Z) = 0. Thus Hodd(G/P;G) = 0 for any coefficient group G.

The linear groups SL(n, C), n ≥ 2, and the subgroups S p(2n, C) ⊂ SL(2n, C) of transformations preserving the alternating bilinear form

x1yn+1 +···+ xny2n −xn+1y1 −···−x2nyn

on C2n × C2n are examples of complex, simply connected, semisimple Lie groups. A parabolic subgroup is a closed subgroup that contains a Borel group B. For G = SL(n,C), B is the group of all upper-triangular matrices in SL(n,C). In this case, G/B is the complete flag manifold

G/B = {0 ⊂ V1 ⊂···⊂ Vn−1 ⊂ Cn}

of flags of subspaces Vi with dimVi = i. For G = Sp(2n,C), the Borel subgroups B are the subgroups preserving a half-flag of isotropic subspaces and the quotient G/B is the variety of all such flags. Any parabolic subgroup P may be described as the subgroup that preserves some partial flag. Thus (partial) flag manifolds are in ICW. A special case is that of a maximal parabolic subgroup, preserving a single subspace V. The corresponding quotient SL(n, C)/P is a Grassmannian G(k, n) of k-dimensional subspaces of Cn. For G = Sp(2n,C), one obtains Lagrangian Grassmannians of isotropic k-dimensional subspaces, 1 ≤ k ≤ n. So Grassmannians are objects in ICW. The interleaf category is closed under forming fibrations.

Holonomies: Philosophies of Conjugacy. Part 1.

Figure-6-Holonomy-along-a-leafwise-path

Suppose that N is an irreducible 2n-dimensional Riemannian symmetric space. We may realise N as a coset space N = G/K with Gτ ⊂ K ⊂ (Gτ)0 for some involution τ of G. Now K is (a covering of) the holonomy group of N and similarly the coset fibration G → G/K covers the holonomy bundle P → N. In this setting, J(N) is associated to G:

J(N) ≅ G ×K J (R2n)

and if K/H is a K-orbit in J(R2n) then the corresponding subbundle is G ×K K/H = G/H and the projection is just the coset fibration. Thus, the subbundles of J(N) are just the orbits of G in J(N).

Let j ∈ J (N). Then G · j is an almost complex submanifold of J (N) on which J is integrable iff j lies in the zero-set of the Nijenhuis tensor NJ.

This focusses our attention on the zero-set of NJ which we denote by Z. In favourable circumstances, the structure of this set can be completely described. We begin by assuming that N is of compact type so that G is compact and semi-simple. We also assume that N is inner i.e. that τ is an inner involution of G or, equivalently, that rankG = rankK. The class of inner symmetric spaces include the even-dimensional spheres, the Hermitian symmetric spaces, the quaternionic Kähler symmetric spaces and indeed all symmetric G-spaces for G = SO(2n+1), Sp(n), E7, E8, F4 and G2. Moreover, all inner symmetric spaces are necessarily even-dimensional and so fit into our framework.

Let N = G/K be a simply-connected inner Riemannian symmetric space of compact type. Then Z consists of a finite number of connected components on each of which G acts transitively. Moreover, any G-flag manifold is realised as such an orbit for some N.

The proof for the above requires a detour into the geometry of flag manifolds and reveals an interesting interaction between the complex geometry of flag manifolds and the real geometry of inner symmetric spaces. For this, we begin by noting that a coset space of the form G/C(T) admits several invariant Kählerian complex structures in general. Using a complex realisation of G/C(T) as follows: having fixed a complex structure, the complexified group GC acts transitively on G/C(T) by biholomorphisms with parabolic subgroups as stabilisers. Conversely, if P ⊂ GC is a parabolic subgroup then the action of G on GC/P is transitive and G ∩ P is the centraliser of a torus in G. For the infinitesimal situation: let F = G/C(T) be a flag manifold and let o ∈ F. We have a splitting of the Lie algebra of G

gC = h ⊕ m

with m ≅ ToF and h the Lie algebra of the stabiliser of o in G. An invariant complex structure on F induces an ad h-invariant splitting of mC into (1, 0) and (0, 1) spaces mC = m+ ⊕ m− with [m+, m+] ⊂ m+ by integrability. One can show that m+ and m are nilpotent subalgebras of gC and in fact hC ⊕ m is a parabolic subalgebra of gC with nilradical m. If P is the corresponding parabolic subgroup of GC then P is the stabiliser of o and we obtain a biholomorphism between the complex coset space GC/P and the flag manifold F.

Conversely, let P ⊂ GC be a parabolic subgroup with Lie algebra p and let n be the conjugate of the nilradical of p (with respect to the real form g). Then H = G ∩ P is the centraliser of a torus and we have orthogonal decompositions (with respect to the Killing inner product)

p = hC ⊕ n, gC = hC ⊕ n ⊕ n

which define an invariant complex structure on G/H realising the biholomorphism with GC/P.

The relationship between a flag manifold F = GC/P and an inner symmetric space comes from an examination of the central descending series of n. This is a filtration 0 = nk+1 ⊂ nk ⊂…⊂ n1 = n of n defined by ni = [n, ni−1].

We orthogonalise this filtration using the Killing inner product by setting

gi = ni+1 ∩ ni

for i ≥ 1 and extend this to a decomposition of gC by setting g0 = hC = (g ∩ p)C and g−i = gfor i ≥ 1. Then

gC = ∑gi

is an orthogonal decomposition with

p = ∑i≤0 gi, n = ∑i>0 g

The crucial property of this decomposition is that

[gi, gj] ⊂ gi+j

which can be proved by demonstrating the existence of an element ξ ∈ h with the property that, for each i, adξ has eigenvalue √−1i on gi. This element ξ (necessarily unique since g is semi-simple) is the canonical element of p. Since ad ξ has eigenvalues in √−1Z, ad exp πξ is an involution of g which we exponentiate to obtain an inner involution τξ of G and thus an inner symmetric space G/K where K = (Gτξ)0. Clearly, K has Lie algebra given by

k = g ∩ ∑i g2i

Hyperbolic Brownian Sheet, Parabolic and Elliptic Financials. (Didactic 3)

Fig-3-Realizations-of-the-fractional-Brownian-sheet-on-the-plane-with-graph-dimensions

Financial and economic time series are often described to a first degree of approximation as random walks, following the precursory work of Bachelier and Samuelson. A random walk is the mathematical translation of the trajectory followed by a particle subjected to random velocity variations. The analogous physical system described by SPDE’s is a stochastic string. The length along the string is the time-to-maturity and the string configuration (its transverse deformation) gives the value of the forward rate f(t,x) at a given time for each time-to-maturity x. The set of admissible dynamics of the configuration of the string as a function of time depends on the structure of the SPDE. Let us for the time being restrict our attention to SPDE’s in which the highest derivative is second order. This second order derivative has a simple physical interpretation : the string is subjected to a tension, like a piano chord, that tends to bring it back to zero transverse deformation. This tension forces the “coupling” among different times-to-maturity so that the forward rate curve is at least continuous. In principle, the most general formulation would consider SPDE’s with terms of arbitrary derivative orders. However, it is easy to show that the tension term is the dominating restoring force, when present, for deformations of the string (forward rate curve) at long “wavelengths”, i.e. for slow variations along the time-to-maturity axis. Second order SPDE’s are thus generic in the sense of a systematic expansion.

In the framework of second order SPDE’s, we consider hyperbolic, parabolic and elliptic SPDE’s, to characterize the dynamics of the string along two directions : inertia or mass, and viscosity or subjection to drag forces. A string that has “inertia” or, equivalently, “mass” per unit length, along with the tension that keeps it continuous, is characterized by the class of hyperbolic SPDE’s. For these SPDE’s, the highest order derivative in time has the same order as the highest order derivative in distance along the string (time-to-maturity). As a consequence, hyperbolic SPDE’s present wave-like solutions, that can propagate as pulses with a “velocity”. In this class, we find the so-called “Brownian sheet” which is the direct generalization of Brownian motion to higher dimensions, that preserves continuity in time-to-maturity. The Brownian sheet is the surface spanned by the string configurations as time goes on. The Brownian sheet is however non-homogeneous in time-to-maturity.

If the string has no inertia, its dynamics are characterized by parabolic SPDE’s. These stochastic processes lead to smoother diffusion of shocks through time, along time-to-maturity. Finally, the third class of SPDE’s of second-order, namely elliptic partial differential equations. Elliptic SPDE’s give processes that are differentiable both in x and t. Therefore, in the strict limit of continuous trading, these stochastic processes correspond to locally riskless interest rates.

The general form of SPDE’s reads

A(t,x) ∂2f(t,x)/∂t2 + 2B(t,x) ∂2f(t,x)/∂t∂x + C(t,x) ∂2f(t,x)/∂x2 = F(t,x,f(t,x), ∂f(t,x)/∂t, ∂f(t,x)/∂x, S) —– (1)

where f (t, x) is the forward rate curve. S(t, x) is the “source” term that will be generally taken to be Gaussian white noise η(t, x) characterized by the covariance

Cov η(t, x), η(t′, x′) = δ(t − t′) δ(x − x′) —– (2)

where δ denotes the Dirac distribution. Equation (1) is the most general second-order SPDE in two variables. For arbitrary non-linear terms in F, the existence of solutions is not warranted and a case by case study must be performed. For the cases where F is linear, the solution f(t,x) exists and its uniqueness is warranted once “boundary” conditions are given, such as, for instance, the initial value of the function f(0,x) as well as any constraints on the particular form of equation (1).

Equation (1) is defined by its characteristics, which are curves in the (t, x) plane that come in two families of equation :

Adt = (B + √(B2 − AC))dx —– (3)

Adt = (B − √(B2 − AC))dx —– (4)

These characteristics are the geometrical loci of the propagation of the boundary conditions.

Three cases must be considered.

• When B2 > AC, the characteristics are real curves and the corresponding SPDE’s are called “hyperbolic”. For such hyperbolic SPDE’s, the natural coordinate system is formed from the two families of characteristics. Expressing (1) in terms of these two natural coordinates λ and μ, we get the “normal form” of hyperbolic SPDE’s :

2f/∂λ∂μ = P (λ,μ) ∂f/∂λ +Q (λ,μ) ∂f/∂μ + R (λ,μ)f + S(λ,μ) —– (5)

The special case P = Q = R = 0 with S(λ,μ) = η(λ,μ) corresponds to the so-called Brownian sheet, well studied in the mathematical literature as the 2D continuous generalization of the Brownian motion.

• When B2 = AC, there is only one family of characteristics, of equation

Adt = Bdx —– (6)

Expressing (1) in terms of the natural characteristic coordinate λ and keeping x, we get the “normal form” of parabolic SPDE’s :

2f/∂x2 = K (λ,μ)∂f/∂λ +L (λ,μ)∂f/∂x +M (λ,μ)f + S(λ,μ) —– (7)

The diffusion equation, well-known to be associated to the Black-Scholes option pricing model, is of this type. The main difference with the hyperbolic equations is that it is no more invariant with respect to time-reversal t → −t. Intuitively, this is due to the fact that the diffusion equation is not conservative, the information content (negentropy) continually decreases as time goes on.

• When B2 < AC, the characteristics are not real curves and the corresponding SPDE’s are called “elliptic”. The equations for the characteristics are complex conjugates of each other and we can get the “normal form” of elliptic SPDE’s by using the real and imaginary parts of these complex coordinates z = u ± iv :

2f/∂u2 + ∂2f/∂v2 = T ∂f/∂u + U ∂f/∂v + V f + S —– (8)

There is a deep connection between the solution of elliptic SPDE’s and analytic functions of complex variables.

Hyperbolic and parabolic SPDE’s provide processes reducing locally to standard Brownian motion at fixed time-to-maturity, while elliptic SPDE’s give locally riskless time evolutions. Basically, this stems from the fact that the “normal forms” of second-order hyperbolic and parabolic SPDE’s involve a first-order derivative in time, thus ensuring that the stochastic processes are locally Brownian in time. In contrast, the “normal form” of second-order elliptic SPDE’s involve a second- order derivative with respect to time, which is the cause for the differentiability of the process with respect to time. Any higher order SPDE will be Brownian-like in time if it remains of order one in its time derivatives (and higher-order in the derivatives with respect to x).