# Define Operators Corresponding to Cobordisms Only Iff Each Connected Component of the Cobordism has Non-empty Outgoing Boundary. Drunken Risibility.

Define a category B whose objects are the oriented submanifolds of X, and whose vector space of morphisms from Y to Z is OYZ = ExtH(X)(H(Y), H(Z)) – the cohomology, as usual, has complex coefficients, and H(Y) and H(Z) are regarded as H(X)-modules by restriction. The composition of morphisms is given by the Yoneda composition of Ext groups. With this definition, however, it will not be true that OYZ is dual to OZY. (To see this it is enough to consider the case when Y = Z is a point of X, and X is a product of odd-dimensional spheres; then OYZ is a symmetric algebra, and is not self-dual as a vector space.)

We can do better by defining a cochain complex O’YZ of morphisms by

O’YZ = BΩ(X)(Ω(Y), Ω(Z)) —– (1)

where Ω(X) denotes the usual de Rham complex of a manifold X, and BA(B,C), for a differential graded algebra A and differential graded A- modules B and C, is the usual cobar resolution

Hom(B, C) → Hom(A ⊗ B, C) → Hom(A ⊗ A ⊗ B, C) → · · ·  —– (2)

in which the differential is given by

dƒ(a1 ⊗ · · · ⊗ ak ⊗ b) = 􏰝a1 ƒ(a2 ⊗ · · · ⊗ ak ⊗ b) + ∑(-1)i ƒ(a1 ⊗ · · · ⊗ aiai+1 ⊗ ak ⊗ b) + (-1)k ƒ(a1 ⊗ · · · ⊗ ak-1 ⊗ akb) —– (3)

whose cohomology is ExtA(B,C). This is different from OYZ = ExtH(X)(H(Y), H(Z)), but related to it by a spectral sequence whose E2-term is OYZ and which converges to H(O’YZ) = ExtΩ(X)(Ω(Y), Ω(Z)). But more important is that H(O’YZ) is the homology of the space PYZ of paths in X which begin in Y and end in Z. To be precise, Hp(O’YZ) ≅ Hp+dZ(PYZ), where dZ is the dimension of Z. On the cochain complexes the Yoneda composition is associative up to cochain homotopy, and defines a structure of an A category B’. The corresponding composition of homology groups

Hi(PYZ) × Hj(PZW) → Hi+j−dZ(PYW) —— (4)

is the composition of the Gysin map associated to the inclusion of the codimension dZ submanifold M of pairs of composable paths in the product PYZ × PZW with the concatenation map M → PYW.

Now let’s attempt to fit the closed string cochain algebra C to this A category. C is equivalent to the usual Hochschild complex of the differential graded algebra Ω(X), whose cohomology is the homology of the free loop space LX with its degrees shifted downwards by the dimension dX of X, so that the cohomology Hi(C) is potentially non-zero for −dX ≤ i < ∞. There is a map Hi(X) → H−i(C) which embeds the ordinary cohomology ring of X to the Pontrjagin ring of the based loop space L0X, based at any chosen point in X.

The structure is, however, not a cochain-level open and closed theory, as we have no trace maps inducing inner products on H(O’YZ). When one tries to define operators corresponding to cobordisms it turns out to be possible only when each connected component of the cobordism has non-empty outgoing boundary.

# Complicated Singularities – Why Should the Discriminant Locus Change Under Dualizing?

Consider the surface S ⊆ (C)2 defined by the equation z1 + z2 + 1 = 0. Define the map log : (C)2 → R2 by log(z1, z2) = (log|z1|, log|z2|). Then log(S) can be seen as follows. Consider the image of S under the absolute value map.

The line segment r1 + r2 = 1 with r1, r2 ≥ 0 is the image of {(−a, a−1)|0 < a < 1} ⊆ S; the ray r2 = r1 + 1 with r1 ≥ 0 is the image of {(−a, a−1)|a < 0} ⊆ S; and the ray r1 = r2 + 1 is the image of {(−a, a−1)|a > 1} ⊆ S. The map S → |S| is one-to-one on the boundary of |S| and two-to-one in the interior, with (z1, z2) and (z̄1, z̄2) mapping to the same point in |S|. Taking the logarithm of this picture, we obtain the amoeba of S, log(S) as depicted below.

Now consider S = S × {0} ⊆ Y = (C)2 × R = T2 × R3. We can now obtain a six-dimensional space X, with a map π : X → Y, an S1-bundle over Y\S degenerating over S, so that π−1(S) → S. We then have a T3-fibration on X, f : X → R3, by composing π with the map (log, id) : (C)2 × R → R3 = B. Clearly the discriminant locus of f is log(S) × {0}. If b is in the interior of log(S) × {0}, then f−1(b) is obtained topologically by contracting two circles {p1} × S1 and {p2} × S1 on T3 = T2 × S1 to points. These are the familiar conical singularities seen in the special Lagrangian situation.

If b ∈ ∂(log(S) × {0}), then f−1(b) has a slightly more complicated singularity, but only one. Let us examine how the “generic” singular fiber fits in here. In particular, for b in the interior of log(S) × {0}, locally this discriminant locus splits B into two regions, and these regions represent two different possible smoothings of f−1(b).

Assume now that f : X → B is a special Lagrangian fibration with topology and discriminant locus ∆ being an amoeba. Let b ∈ Int(∆), and set M = f−1(b). Set Mo = M\{x1, x2}, where x1, x2 are the two conical singularities of M. Suppose that the tangent cones to these two conical singularities, C1 and C2, are both cones of the form M0. Then the links of these cones, Σ1 and Σ2, are T2’s, and one expects that topologically these can be described as follows. Note that Mo ≅ (T2\{y1, y2}) × S1 where y1, y2 are two points in T2. We assume that the link Σi takes the form γi × S1, where γi is a simple loop around yi. If these assumptions hold, then to see how M can be smoothed, we consider the restriction maps in cohomology

H1(Mo, R) → H11, R) ⊕ H12, R)

The image of this map is two-dimensional. Indeed, if we write a basis ei1, ei2 of H1i, R) where ei1 is Poincaré dual to [γi] × pt and ei2 is Poincaré dual to pt × S1, it is not difficult to see the image of the restriction map is spanned by {(e11, e21)} and {(e12, −e22)}. Now this model of a topological fibration is not special Lagrangian, so in particular we don’t know exactly how the tangent cones to M at x1 and x2 are sitting inside C3, and thus can’t be compared directly with an asymptotically conical smoothing. So to make a plausibility argument, choose new bases fi1, fi2 of H1i, R) so that if M(a,0,0), M(0,a,0) and M(0,0,a) are the three possible smoothings of the two singular tangent cones at the singular points x1, x2 of M. Then Y(Mi(a,0,0)) = πafi1, Y(Mi(0,a,0)) = πafi2, and Y(Mi(0,0,a)) = −πa(fi1 + fi2).

Suppose that in this new basis, the image of the restriction map is spanned by the pairs (f11, rf22) and (rf12, f21) for r > 0, r ≠ 1. Then, there are two possible ways of smoothing M, either by gluing in M1(a,0,0) and M2(0,ra,0) at the singular points x1 and x2 respectively, or by gluing in M1(0,ra,0) and M2(a,0,0) at x1 and x2 respectively. This could correspond to deforming M to a fiber over a point on one side of the discriminant locus of f or the other side. This at least gives a plausibility argument for the existence of a special Lagrangian fibration of the topological type given by f. To date, no such fibrations have been constructed, however.

On giving a special Lagrangian fibration with codimension one discriminant and singular fibers with cone over T2 singularities, one is just forced to confront a codimension one discriminant locus in special Lagrangian fibrations. This leads inevitably to the conclusion that a “strong form” of the Strominger-Yau-Zaslow conjecture cannot hold. In particular, one is forced to conclude that if f : X → B and f’ : X’ → B are dual special Lagrangian fibrations, then their discriminant loci cannot coincide. Thus one cannot hope for a fiberwise definition of the dualizing process, and one needs to refine the concept of dualizing fibrations. Let us see why the discriminant locus must change under dualizing. The key lies in the behaviour of the positive and negative vertices, where in the positive case the critical locus of the local model of the fibration is a union of three holomorphic curves, while in the negative case the critical locus is a pair of pants. In a “generic” special Lagrangian fibration, we expect the critical locus to remain roughly the same, but its image in the base B will be fattened out. In the negative case, this image will be an amoeba. In the case of the positive vertex, the critical locus, at least locally, consists of a union of three holomorphic curves, so that we expect the discriminant locus to be the union of three different amoebas. The figure below shows the new discriminant locus for these two cases.

Now, under dualizing, positive and negative vertices are interchanged. Thus the discriminant locus must change. This is all quite speculative, of course, and underlying this is the assumption that the discriminant loci are just fattenings of the graphs. However, it is clear that a new notion of dualizing is necessary to cover this eventuality.

# Grothendieckian Construction of K-Theory with a Bundle that is Topologically Trivial and Class that is Torsion.

All relativistic quantum theories contain “antiparticles,” and allow the process of particle-antiparticle annihilation. This inspires a physical version of the Grothendieck construction of K-theory. Physics uses topological K-theory of manifolds, whose motivation is to organize vector bundles over a space into an algebraic invariant, that turns out to be useful. Algebraic K-theory started from Ki defined for i, with relations to classical constructions in algebra and number theory, followed by Quillen’s homotopy-theoretic definition ∀ i. The connections to algebra and number theory often persist for larger values of i, but in ways that are subtle and conjectural, such as special values of zeta- and L-functions.

One could also use the conserved charges of a configuration which can be measured at asymptotic infinity. By definition, these are left invariant by any physical process. Furthermore, they satisfy quantization conditions, of which the prototype is the Dirac condition on allowed electric and magnetic charges in Maxwell theory.

There is an elementary construction which, given a physical theory T, produces an abelian group of conserved charges K(T). Rather than considering the microscopic dynamics of the theory, all that is needed to be known is a set S of “particles” described by T, and a set of “bound state formation/decay processes” by which the particles combine or split to form other particles. These are called “binding processes.” Two sets of particles are “physically equivalent” if some sequence of binding processes convert the one to the other. We then define the group K(T) as the abelian group ZS of formal linear combinations of particles, quotiented by this equivalence relation.

Suppose T contains the particles S = {A,B,C}.

If these are completely stable, we could clearly define three integral conserved charges, their individual numbers, so K(T) ≅ Z3.

Introducing a binding process

A + B ↔ C —– (1)

with the bidirectional arrow to remind us that the process can go in either direction. Clearly K(T) ≅ Z2 in this case.

One might criticize this proposal on the grounds that we have assumed that configurations with a negative number of particles can exist. However, in all physical theories which satisfy the constraints of special relativity, charged particles in physical theories come with “antiparticles,” with the same mass but opposite charge. A particle and antiparticle can annihilate (combine) into a set of zero charge particles. While first discovered as a prediction of the Dirac equation, this follows from general axioms of quantum field theory, which also hold in string theory.

Thus, there are binding processes

B + B̄ ↔ Z1 + Z2 + · · · .

where B̄ is the antiparticle to a particle B, and Zi are zero charge particles, which must appear by energy conservation. To define the K-theory, we identify any such set of zero charge particles with the identity, so that

B + B̄ ↔ 0

Thus the antiparticles provide the negative elements of K(T).

Granting the existence of antiparticles, this construction of K-theory can be more simply rephrased as the Grothendieck construction. We can define K(T) as the group of pairs (E, F) ∈ (ZS, ZS), subject to the relations (E, F) ≅ (E+B, F +B) ≅ (E+L, F +R) ≅ (E+R, F +L), where (L, R) are the left and right hand side of a binding process (1).

Thinking of these as particles, each brane B must have an antibrane, which we denote by B̄. If B wraps a submanifold L, one expects that B̄ is a brane which wraps a submanifold L of opposite orientation. A potential problem is that it is not a priori obvious that the orientation of L actually matters physically, especially in degenerate cases such as L a point.

Now, let us take X as a Calabi-Yau threefold for definiteness. A physical A-brane, which are branes of the A-model topological string and thereby a TQFT shadow of the D-branes of the superstring, is specified by a pair (L, E) of a special Lagrangian submanifold L with a flat bundle E. The obvious question could be: When are (L1, E1) and (L2, E2) related by a binding process? A simple heuristic answer to this question is given by the Feynman path integral. Two configurations are connected, if they are connected by a continuous path through the configuration space; any such path (or a small deformation of it) will appear in the functional integral with some non-zero weight. Thus, the question is essentially topological. Ignoring the flat bundles for a moment, this tells us that the K-theory group for A-branes is H3(Y, Z), and the class of a brane is simply (rank E)·[L] ∈ H3(Y, Z). This is also clear if the moduli space of flat connections on L is connected.

But suppose it is not, say π1(L) is torsion. In this case, we need deeper physical arguments to decide whether the K-theory of these D-branes is H3(Y, Z), or some larger group. But a natural conjecture is that it will be K1(Y), which classifies bundles on odd-dimensional submanifolds. Two branes which differ only in the choice of flat connection are in fact connected in string theory, consistent with the K-group being H3(Y, Z). For Y a simply connected Calabi-Yau threefold, K1(Y) ≅ H3(Y, Z), so the general conjecture is borne out in this case

There is a natural bilinear form on H3(Y, Z) given by the oriented intersection number

I(L1, L2) = #([L1] ∩ [L2]) —– (2)

It has symmetry (−1)n. In particular, it is symplectic for n = 3. Furthermore, by Poincaré duality, it is unimodular, at least in our topological definition of K-theory.

D-branes, which are extended objects defined by mixed Dirichlet-Neumann boundary conditions in string theory, break half of the supersymmetries of the type II superstring and carry a complete set of electric and magnetic Ramond-Ramond charges. The product of the electric and magnetic charges is a single Dirac unit, and that the quantum of charge takes the value required by string duality. Saying that a D-brane has RR-charge means that it is a source for an “RR potential,” a generalized (p + 1)-form gauge potential in ten-dimensional space-time, which can be verified from its world-volume action that contains a minimal coupling term,

∫C(p + 1) —–(3)

where C(p + 1) denotes the gauge potential, and the integral is taken over the (p+1)-dimensional world-volume of the brane. For p = 0, C(1) is a one-form or “vector” potential (as in Maxwell theory), and thus the D0-brane is an electrically charged particle with respect to this 10d Maxwell theory. Upon further compactification, by which, the ten dimensions are R4 × X, and a Dp-brane which wraps a p-dimensional cycle L; in other words its world-volume is R × L where R is a time-like world-line in R4. Using the Poincaré dual class ωL ∈ H2n−p(X, R) to L in X, to rewrite (3) as an integral

R × X C(p + 1) ∧ ωL —– (4)

We can then do the integral over X to turn this into the integral of a one-form over a world-line in R4, which is the right form for the minimal electric coupling of a particle in four dimensions. Thus, such a wrapped brane carries a particular electric charge which can be detected at asymptotic infinity. Summarizing the RR-charge more formally,

LC = ∫XC ∧ ωL —– (5)

where C ∈ H∗(X, R). In other words, it is a class in Hp(X, R).

In particular, an A-brane (for n = 3) carries a conserved charge in H3(X, R). Of course, this is weaker than [L] ∈ H3(X, Z). To see this physically, we would need to see that some of these “electric” charges are actually “magnetic” charges, and study the Dirac-Schwinger-Zwanziger quantization condition between these charges. This amounts to showing that the angular momentum J of the electromagnetic field satisfies the quantization condition J = ħn/2 for n ∈ Z. Using an expression from electromagnetism, J⃗ = E⃗ × B⃗ , this is precisely the condition that (2) must take an integer value. Thus the physical and mathematical consistency conditions agree. Similar considerations apply for coisotropic A-branes. If X is a genuine Calabi-Yau 3-fold (i.e., with strict SU(3) holonomy), then a coisotropic A-brane which is not a special Lagrangian must be five-dimensional, and the corresponding submanifold L is rationally homologically trivial, since H5(X, Q) = 0. Thus, if the bundle E is topologically trivial, the homology class of L and thus its K-theory class is torsion.

If X is a torus, or a K3 surface, the situation is more complicated. In that case, even rationally the charge of a coisotropic A-brane need not lie in the middle-dimensional cohomology of X. Instead, it takes its value in a certain subspace of ⊕p Hp(X, Q), where the summation is over even or odd p depending on whether the complex dimension of X is even or odd. At the semiclassical level, the subspace is determined by the condition

(L − Λ)α = 0, α ∈ ⊕p Hp(X, Q)

where L and Λ are generators of the Lefschetz SL(2, C) action, i.e., L is the cup product with the cohomology class of the Kähler form, and Λ is its dual.

# The Closed String Cochain Complex C is the String Theory Substitute for the de Rham Complex of Space-Time. Note Quote.

In closed string theory the central object is the vector space C = CS1 of states of a single parameterized string. This has an integer grading by the “ghost number”, and an operator Q : C → C called the “BRST operator” which raises the ghost number by 1 and satisfies Q2 = 0. In other words, C is a cochain complex. If we think of the string as moving in a space-time M then C is roughly the space of differential forms defined along the orbits of the action of the reparametrization group Diff+(S1) on the free loop space LM (more precisely, square-integrable forms of semi-infinite degree). Similarly, the space C of a topologically-twisted N = 2 supersymmetric theory, is a cochain complex which models the space of semi-infinite differential forms on the loop space of a Kähler manifold – in this case, all square-integrable differential forms, not just those along the orbits of Diff+(S1). In both kinds of example, a cobordism Σ from p circles to q circles gives an operator UΣ,μ : C⊗p → C⊗q which depends on a conformal structure μ on Σ. This operator is a cochain map, but its crucial feature is that changing the conformal structure μ on Σ changes the operator UΣ,μ only by a cochain homotopy. The cohomology H(C) = ker(Q)/im(Q) – the “space of physical states” in conventional string theory – is therefore the state space of a topological field theory.

A good way to describe how the operator UΣ,μ varies with μ is as follows:

If MΣ is the moduli space of conformal structures on the cobordism Σ, modulo diffeomorphisms of Σ which are the identity on the boundary circles, then we have a cochain map

UΣ : C⊗p → Ω(MΣ, C⊗q)

where the right-hand side is the de Rham complex of forms on MΣ with values in C⊗q. The operator UΣ,μ is obtained from UΣ by restricting from MΣ to {μ}. The composition property when two cobordisms Σ1 and Σ2 are concatenated is that the diagram

commutes, where the lower horizontal arrow is induced by the map MΣ1 × MΣ2 → MΣ2 ◦ Σ1 which expresses concatenation of the conformal structures.

For each pair a, b of boundary conditions we shall still have a vector space – indeed a cochain complex – Oab, but it is no longer the space of morphisms from b to a in a category. Rather, what we have is an A-category. Briefly, this means that instead of a composition law Oab × Obc → Oac we have a family of ways of composing, parametrized by the contractible space of conformal structures on the surface of the figure:

In particular, any two choices of a composition law from the family are cochain homotopic. Composition is associative in the sense that we have a contractible family of triple compositions Oab × Obc × Ocd → Oad, which contains all the maps obtained by choosing a binary composition law from the given family and bracketing the triple in either of the two possible ways.

This is not the usual way of defining an A-structure. According to Stasheff’s original definition, an A-structure on a space X consists of a sequence of choices: first, a composition law m2 : X × X → X; then, a choice of a map

m3 : [0, 1] × X × X × X → X which is a homotopy between

(x, y, z) ↦ m2(m2(x, y), z) and (x, y, z) ↦ m2(x, m2(y, z)); then, a choice of a map

m4 : S4 × X4 → X,

where S4 is a convex plane polygon whose vertices are indexed by the five ways of bracketing a 4-fold product, and m4|((∂S4) × X4) is determined by m3; and so on. There is an analogous definition – applying to cochain complexes rather than spaces.

Apart from the composition law, the essential algebraic properties are the non-degenerate inner product, and the commutativity of the closed algebra C. Concerning the latter, when we pass to cochain theories the multiplication in C will of course be commutative up to cochain homotopy, but, the moduli space MΣ of closed string multiplications i.e., the moduli space of conformal structures on a pair of pants Σ, modulo diffeomorphisms of Σ which are the identity on the boundary circles, is not contractible: it has the homotopy type of the space of ways of embedding two copies of the standard disc D2 disjointly in the interior of D2 – this space of embeddings is of course a subspace of MΣ. In particular, it contains a natural circle of multiplications in which one of the embedded discs moves like a planet around the other, and there are two different natural homotopies between the multiplication and the reversed multiplication. This might be a clue to an important difference between stringy and classical space-times. The closed string cochain complex C is the string theory substitute for the de Rham complex of space-time, an algebra whose multiplication is associative and (graded)commutative on the nose. Over the rationals or the real or complex numbers, such cochain algebras model the category of topological spaces up to homotopy, in the sense that to each such algebra C, we can associate a space XC and a homomorphism of cochain algebras from C to the de Rham complex of XC which is a cochain homotopy equivalence. If we do not want to ignore torsion in the homology of spaces we can no longer encode the homotopy type in a strictly commutative cochain algebra. Instead, we must replace commutative algebras with so-called E-algebras, i.e., roughly, cochain complexes C over the integers equipped with a multiplication which is associative and commutative up to given arbitrarily high-order homotopies. An arbitrary space X has an E-algebra CX of cochains, and conversely one can associate a space XC to each E-algebra C. Thus we have a pair of adjoint functors, just as in rational homotopy theory. The cochain algebras of closed string theory have less higher commutativity than do E-algebras, and this may be an indication that we are dealing with non-commutative spaces that fits in well with the interpretation of the B-field of a string background as corresponding to a bundle of matrix algebras on space-time. At the same time, the non-degenerate inner product on C – corresponding to Poincaré duality – seems to show we are concerned with manifolds, rather than more singular spaces.

Let us consider the category K of cochain complexes of finitely generated free abelian groups and cochain homotopy classes of cochain maps. This is called the derived category of the category of finitely generated abelian groups. Passing to cohomology gives us a functor from K to the category of Z-graded finitely generated abelian groups. In fact the subcategory K0 of K consisting of complexes whose cohomology vanishes except in degree 0 is actually equivalent to the category of finitely generated abelian groups. But the category K inherits from the category of finitely generated free abelian groups a duality functor with properties as ideal as one could wish: each object is isomorphic to its double dual, and dualizing preserves exact sequences. (The dual C of a complex C is defined by (C)i = Hom(C−i, Z).) There is no such nice duality in the category of finitely generated abelian groups. Indeed, the subcategory K0 is not closed under duality, for the dual of the complex CA corresponding to a group A has in general two non-vanishing cohomology groups: Hom(A,Z) in degree 0, and in degree +1 the finite group Ext1(A,Z) Pontryagin-dual to the torsion subgroup of A. This follows from the exact sequence:

0 → Hom(A, Z) → Hom(FA, Z) → Hom(RA, Z) → Ext1(A, Z) → 0

derived from an exact sequence

0 → RA → FA → A → 0

The category K also has a tensor product with better properties than the tensor product of abelian groups, and, better still, there is a canonical cochain functor from (locally well-behaved) compact spaces to K which takes Cartesian products to tensor products.

# Morphism of Complexes Induces Corresponding Morphisms on Cohomology Objects – Thought of the Day 146.0

Let A = Mod(R) be an abelian category. A complex in A is a sequence of objects and morphisms in A

… → Mi-1 →di-1 Mi →di → Mi+1 → …

such that di ◦ di-1 = 0 ∀ i. We denote such a complex by M.

A morphism of complexes f : M → N is a sequence of morphisms fi : Mi → Ni in A, making the following diagram commute, where diM, diN denote the respective differentials:

We let C(A) denote the category whose objects are complexes in A and whose morphisms are morphisms of complexes.

Given a complex M of objects of A, the ith cohomology object is the quotient

Hi(M) = ker(di)/im(di−1)

This operation of taking cohomology at the ith place defines a functor

Hi(−) : C(A) → A,

since a morphism of complexes induces corresponding morphisms on cohomology objects.

Put another way, an object of C(A) is a Z-graded object

M = ⊕i Mi

of A, equipped with a differential, in other words an endomorphism d: M → M satisfying d2 = 0. The occurrence of differential graded objects in physics is well-known. In mathematics they are also extremely common. In topology one associates to a space X a complex of free abelian groups whose cohomology objects are the cohomology groups of X. In algebra it is often convenient to replace a module over a ring by resolutions of various kinds.

A topological space X may have many triangulations and these lead to different chain complexes. Associating to X a unique equivalence class of complexes, resolutions of a fixed module of a given type will not usually be unique and one would like to consider all these resolutions on an equal footing.

A morphism of complexes f: M → N is a quasi-isomorphism if the induced morphisms on cohomology

Hi(f): Hi(M) → Hi(N) are isomorphisms ∀ i.

Two complexes M and N are said to be quasi-isomorphic if they are related by a chain of quasi-isomorphisms. In fact, it is sufficient to consider chains of length one, so that two complexes M and N are quasi-isomorphic iff there are quasi-isomorphisms

M ← P → N

For example, the chain complex of a topological space is well-defined up to quasi-isomorphism because any two triangulations have a common resolution. Similarly, all possible resolutions of a given module are quasi-isomorphic. Indeed, if

0 → S →f M0 →d0 M1 →d1 M2 → …

is a resolution of a module S, then by definition the morphism of complexes

is a quasi-isomorphism.

The objects of the derived category D(A) of our abelian category A will just be complexes of objects of A, but morphisms will be such that quasi-isomorphic complexes become isomorphic in D(A). In fact we can formally invert the quasi-isomorphisms in C(A) as follows:

There is a category D(A) and a functor Q: C(A) → D(A)

with the following two properties:

(a) Q inverts quasi-isomorphisms: if s: a → b is a quasi-isomorphism, then Q(s): Q(a) → Q(b) is an isomorphism.

(b) Q is universal with this property: if Q′ : C(A) → D′ is another functor which inverts quasi-isomorphisms, then there is a functor F : D(A) → D′ and an isomorphism of functors Q′ ≅ F ◦ Q.

First, consider the category C(A) as an oriented graph Γ, with the objects lying at the vertices and the morphisms being directed edges. Let Γ∗ be the graph obtained from Γ by adding in one extra edge s−1: b → a for each quasi-isomorphism s: a → b. Thus a finite path in Γ∗ is a sequence of the form f1 · f2 ·· · ·· fr−1 · fr where each fi is either a morphism of C(A), or is of the form s−1 for some quasi-isomorphism s of C(A). There is a unique minimal equivalence relation ∼ on the set of finite paths in Γ∗ generated by the following relations:

(a) s · s−1 ∼ idb and s−1 · s ∼ ida for each quasi-isomorphism s: a → b in C(A).

(b) g · f ∼ g ◦ f for composable morphisms f: a → b and g: b → c of C(A).

Define D(A) to be the category whose objects are the vertices of Γ∗ (these are the same as the objects of C(A)) and whose morphisms are given by equivalence classes of finite paths in Γ∗. Define a functor Q: C(A) → D(A) by using the identity morphism on objects, and by sending a morphism f of C(A) to the length one path in Γ∗ defined by f. The resulting functor Q satisfies the conditions of the above lemma.

The second property ensures that the category D(A) of the Lemma is unique up to equivalence of categories. We define the derived category of A to be any of these equivalent categories. The functor Q: C(A) → D(A) is called the localisation functor. Observe that there is a fully faithful functor

J: A → C(A)

which sends an object M to the trivial complex with M in the zeroth position, and a morphism F: M → N to the morphism of complexes

Composing with Q we obtain a functor A → D(A) which we denote by J. This functor J is fully faithful, and so defines an embedding A → D(A). By definition the functor Hi(−): C(A) → A inverts quasi-isomorphisms and so descends to a functor

Hi(−): D(A) → A

establishing that composite functor H0(−) ◦ J is isomorphic to the identity functor on A.

# Categories of Pointwise Convergence Topology: Theory(ies) of Bundles.

Let H be a fixed, separable Hilbert space of dimension ≥ 1. Lets denote the associated projective space of H by P = P(H). It is compact iff H is finite-dimensional. Let PU = PU(H) = U(H)/U(1) be the projective unitary group of H equipped with the compact-open topology. A projective bundle over X is a locally trivial bundle of projective spaces, i.e., a fibre bundle P → X with fibre P(H) and structure group PU(H). An application of the Banach-Steinhaus theorem shows that we may identify projective bundles with principal PU(H)-bundles and the pointwise convergence topology on PU(H).

If G is a topological group, let GX denote the sheaf of germs of continuous functions G → X, i.e., the sheaf associated to the constant presheaf given by U → F(U) = G. Given a projective bundle P → X and a sufficiently fine good open cover {Ui}i∈I of X, the transition functions between trivializations P|Ui can be lifted to bundle isomorphisms gij on double intersections Uij = Ui ∩ Uj which are projectively coherent, i.e., over each of the triple intersections Uijk = Ui ∩ Uj ∩ Uk the composition gki gjk gij is given as multiplication by a U(1)-valued function fijk : Uijk → U(1). The collection {(Uij, fijk)} defines a U(1)-valued two-cocycle called a B-field on X,which represents a class BP in the sheaf cohomology group H2(X, U(1)X). On the other hand, the sheaf cohomology H1(X, PU(H)X) consists of isomorphism classes of principal PU(H)-bundles, and we can consider the isomorphism class [P] ∈ H1(X,PU(H)X).

There is an isomorphism

H1(X, PU(H)X) → H2(X, U(1)X) provided by the

boundary map [P] ↦ BP. There is also an isomorphism

H2(X, U(1)X) → H3(X, ZX) ≅ H3(X, Z)

The image δ(P) ∈ H3(X, Z) of BP is called the Dixmier-Douady invariant of P. When δ(P) = [H] is represented in H3(X, R) by a closed three-form H on X, called the H-flux of the given B-field BP, we will write P = PH. One has δ(P) = 0 iff the projective bundle P comes from a vector bundle E → X, i.e., P = P(E). By Serre’s theorem every torsion element of H3(X,Z) arises from a finite-dimensional bundle P. Explicitly, consider the commutative diagram of exact sequences of groups given by

where we identify the cyclic group Zn with the group of n-th roots of unity. Let P be a projective bundle with structure group PU(n), i.e., with fibres P(Cn). Then the commutative diagram of long exact sequences of sheaf cohomology groups associated to the above commutative diagram of groups implies that the element BP ∈ H2(X, U(1)X) comes from H2(X, (Zn)X), and therefore its order divides n.

One also has δ(P1 ⊗ P2) = δ(P1) + δ(P2) and δ(P) = −δ(P). This follows from the commutative diagram

and the fact that P ⊗ P = P(E) where E is the vector bundle of Hilbert-Schmidt endomorphisms of P . Putting everything together, it follows that the cohomology group H3(X, Z) is isomorphic to the group of stable equivalence classes of principal PU(H)-bundles P → X with the operation of tensor product.

We are now ready to define the twisted K-theory of the manifold X equipped with a projective bundle P → X, such that Px = P(H) ∀ x ∈ X. We will first give a definition in terms of Fredholm operators, and then provide some equivalent, but more geometric definitions. Let H be a Z2-graded Hilbert space. We define Fred0(H) to be the space of self-adjoint degree 1 Fredholm operators T on H such that T2 − 1 ∈ K(H), together with the subspace topology induced by the embedding Fred0(H) ֒→ B(H) × K(H) given by T → (T, T2 − 1) where the algebra of bounded linear operators B(H) is given the compact-open topology and the Banach algebra of compact operators K = K(H) is given the norm topology.

Let P = PH → X be a projective Hilbert bundle. Then we can construct an associated bundle Fred0(P) whose fibres are Fred0(H). We define the twisted K-theory group of the pair (X, P) to be the group of homotopy classes of maps

K0(X, H) = [X, Fred0(PH)]

The group K0(X, H) depends functorially on the pair (X, PH), and an isomorphism of projective bundles ρ : P → P′ induces a group isomorphism ρ∗ : K0(X, H) → K0(X, H′). Addition in K0(X, H) is defined by fibre-wise direct sum, so that the sum of two elements lies in K0(X, H2) with [H2] = δ(P ⊗ P(C2)) = δ(P) = [H]. Under the isomorphism H ⊗ C2 ≅ H, there is a projective bundle isomorphism P → P ⊗ P(C2) for any projective bundle P and so K0(X, H2) is canonically isomorphic to K0(X, H). When [H] is a non-torsion element of H3(X, Z), so that P = PH is an infinite-dimensional bundle of projective spaces, then the index map K0(X, H) → Z is zero, i.e., any section of Fred0(P) takes values in the index zero component of Fred0(H).

Let us now describe some other models for twisted K-theory which will be useful in our physical applications later on. A definition in algebraic K-theory may given as follows. A bundle of projective spaces P yields a bundle End(P) of algebras. However, if H is an infinite-dimensional Hilbert space, then one has natural isomorphisms H ≅ H ⊕ H and

End(H) ≅ Hom(H ⊕ H, H) ≅ End(H) ⊕ End(H)

as left End(H)-modules, and so the algebraic K-theory of the algebra End(H) is trivial. Instead, we will work with the Banach algebra K(H) of compact operators on H with the norm topology. Given that the unitary group U(H) with the compact-open topology acts continuously on K(H) by conjugation, to a given projective bundle PH we can associate a bundle of compact operators EH → X given by

EH = PH ×PU K

with δ(EH) = [H]. The Banach algebra AH := C0(X, EH) of continuous sections of EH vanishing at infinity is the continuous trace C∗-algebra CT(X, H). Then the twisted K-theory group K(X, H) of X is canonically isomorphic to the algebraic K-theory group K(AH).

We will also need a smooth version of this definition. Let AH be the smooth subalgebra of AH given by the algebra CT(X, H) = C(X, L1PH),

where L1PH = PH ×PUL1. Then the inclusion CT(X, H) → CT(X, H) induces an isomorphism KCT(X, H) → KCT(X, H) of algebraic K-theory groups. Upon choosing a bundle gerbe connection, one has an isomorphism KCT(X, H) ≅ K(X, H) with the twisted K-theory defined in terms of projective Hilbert bundles P = PH over X.

Finally, we propose a general definition based on K-theory with coefficients in a sheaf of rings. It parallels the bundle gerbe approach to twisted K-theory. Let B be a Banach algebra over C. Let E(B, X) be the category of continuous B-bundles over X, and let C(X, B) be the sheaf of continuous maps X → B. The ring structure in B equips C(X, B) with the structure of a sheaf of rings over X. We can therefore consider left (or right) C(X, B)-modules, and in particular the category LF C(X, B) of locally free C(X, B)-modules. Using the functor in the usual way, for X an equivalence of additive categories

E(B, X) ≅ LF (C(X, B))

Since these are both additive categories, we can apply the Grothendieck functor to each of them and obtain the abelian groups K(LF(C(X, B))) and K(E(B, X)). The equivalence of categories ensures that there is a natural isomorphism of groups

K(LF (C(X, B))) ≅ K(E(B, X))

This motivates the following general definition. If A is a sheaf of rings over X, then we define the K-theory of X with coefficients in A to be the abelian group

K(X, A) := K LF(A)

For example, consider the case B = C. Then C(X, C) is just the sheaf of continuous functions X → C, while E(C, X) is the category of complex vector bundles over X. Using the isomorphism of K-theory groups we then have

K(X, C(X,C)) := K(LF (C(X, C))) ≅ K (E(C, X)) = K0(X)

The definition of twisted K-theory uses another special instance of this general construction. For this, we define an Azumaya algebra over X of rank m to be a locally trivial algebra bundle over X with fibre isomorphic to the algebra of m × m complex matrices over C, Mm(C). An example is the algebra End(E) of endomorphisms of a complex vector bundle E → X. We can define an equivalence relation on the set A(X) of Azumaya algebras over X in the following way. Two Azumaya algebras A, A′ are called equivalent if there are vector bundles E, E′ over X such that the algebras A ⊗ End(E), A′ ⊗ End(E′) are isomorphic. Then every Azumaya algebra of the form End(E) is equivalent to the algebra of functions C(X) on X. The set of all equivalence classes is a group under the tensor product of algebras, called the Brauer group of X and denoted Br(X). By Serre’s theorem there is an isomorphism

δ : Br(X) → tor(H3(X, Z))

where tor(H3(X, Z)) is the torsion subgroup of H3(X, Z).

If A is an Azumaya algebra bundle, then the space of continuous sections C(X, A) of X is a ring and we can consider the algebraic K-theory group K(A) := K0(C(X,A)) of equivalence classes of projective C(X, A)-modules, which depends only on the equivalence class of A in the Brauer group. Under the equivalence, we can represent the Brauer group Br(X) as the set of isomorphism classes of sheaves of Azumaya algebras. Let A be a sheaf of Azumaya algebras, and LF(A) the category of locally free A-modules. Then as above there is an isomorphism

K(X, C(X, A)) ≅ K Proj (C(X, A))

where Proj (C(X, A)) is the category of finitely-generated projective C(X, A)-modules. The group on the right-hand side is the group K(A). For given [H] ∈ tor(H3(X, Z)) and A ∈ Br(X) such that δ(A) = [H], this group can be identified as the twisted K-theory group K0(X, H) of X with twisting A. This definition is equivalent to the description in terms of bundle gerbe modules, and from this construction it follows that K0(X, H) is a subgroup of the ordinary K-theory of X. If δ(A) = 0, then A is equivalent to C(X) and we have K(A) := K0(C(X)) = K0(X). The projective C(X, A)-modules over a rank m Azumaya algebra A are vector bundles E → X with fibre Cnm ≅ (Cm)⊕n, which is naturally an Mm(C)-module.

# Interleaves

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

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.

# Contact Geometry and Manifolds

Let M be a manifold of dimension 2n + 1. A contact structure on M is a distribution ξ ⊂ TM of dimension 2n, such that the defining 1-form α satisfies

α ∧ (dα)n ≠ 0 —– (1)

A 1-form α satisfying (1) is said to be a contact form on M. Let α be a contact form on M; then there exists a unique vector field Rα on M such that

α(Rα) = 1, ιRα dα = 0,

where ιRα dα denotes the contraction of dα along Rα. By definition Rα is called the Reeb vector field of the contact form α. A contact manifold is a pair (M, ξ) where M is a 2n + 1-dimensional manifold and ξ is a contact structure. Let (M, ξ) be a contact manifold and fix a defining (contact) form α. Then the 2-form κ = 1/2 dα defines a symplectic form on the contact structure ξ; therefore the pair (ξ, κ) is a symplectic vector bundle over M. A complex structure on ξ is the datum of J ∈ End(ξ) such that J2 = −Iξ.

Let α be a contact form on M, with ξ = ker α and let κ = 1/2 dα. A complex structure J on ξ is said to be κ-calibrated if gJ [x](·, ·) := κ[x](·, Jx ·) is a JxHermitian inner product on ξx for any x ∈ M.

The set of κ-calibrated complex structures on ξ will be denoted by Cα(M). If J is a complex structure on ξ = ker α, then we extend it to an endomorphism of TM by setting

J(Rα) = 0.

Note that such a J satisfies

J2 =−I + α ⊗ Rα

If J is κ-calibrated, then it induces a Riemannian metric g on M given by

g := gJ + α ⊗ α —– (2)

Furthermore the Nijenhuis tensor of J is defined by

NJ (X, Y) = [JX, JY] − J[X, JY] − J[Y, JX] + J2[X, Y] for any X, Y ∈ TM

A Sasakian structure on a 2n + 1-dimensional manifold M is a pair (α, J), where

• α is a contact form;

• J ∈ Cα(M) satisfies NJ = −dα ⊗ Rα

The triple (M, α, J) is said to be a Sasakian manifold. Let (M, ξ) be a contact manifold. A differential r-form γ on M is said to be basic if

ιRα γ = 0, LRα γ = 0,

where L denotes the Lie derivative and Rα is the Reeb vector field of an arbitrary contact form defining ξ. We will denote by ΛrB(M) the set of basic r-forms on (M, ξ). Note that

rB(M) ⊂ Λr+1B(M)

The cohomology HB(M) of this complex is called the basic cohomology of (M, ξ). If (M, α, J) is a Sasakian manifold, then

J(ΛrB(M)) = ΛrB(M), where, as usual, the action of J on r-forms is defined by

Jφ(X1,…, Xr) = φ(JX1,…, JXr)

Consequently ΛrB(M) ⊗ C splits as

ΛrB(M) ⊗ C = ⊕p+q=r Λp,qJ(ξ)

and, according with this gradation, it is possible to define the cohomology groups Hp,qB(M). The r-forms belonging to Λp,qJ(ξ) are said to be of type (p, q) with respect to J. Note that κ = 1/2 dα ∈ Λ1,1J(ξ) and it determines a non-vanishing cohomology class in H1,1B(M). The Sasakian structure (α, J) also induces a natural connection ∇ξ on ξ given by

ξX Y = (∇X Y)ξ if X ∈ ξ

= [Rα, Y] if X = Rα

where the subscript ξ denotes the projection onto ξ. One easily gets

ξX J = 0, ∇ξXgJ = 0, ∇ξX dα = 0, ∇ξX Y − ∇ξY X = [X,Y]ξ,

for any X, Y ∈ TM. Consequently we have Hol(∇ξ) ⊆ U(n).

The basic cohomology class

cB1(M) = 1/2π [ρT] ∈ H1,1B(M)

is called the first basic Chern class of (M, α, J) and, if it vanishes, then (M, α, J) is said to be null-Sasakian.

Furthermore a Sasakian manifold is called α-Einstein if there exist λ, ν ∈ C(M, R) such that

Ric = λg + να ⊗ α, where Ric is the Ricci Tensor.

A submanifold p: L ֒→ M of a 2n + 1-dimensional contact manifold (M, ξ) is said to be Legendrian if :

1) dimRL = n,

2) p(TL) ⊂ ξ

Observe that, if α is a defining form of the contact structure ξ, then condition 2) is equivalent to say that p(α) = 0. Hence Legendrian submanifolds are the analogue of Lagrangian submanifolds in contact geometry.

# Grothendieck’s Universes and Wiles Proof (Fermat’s Last Theorem). Thought of the Day 77.0

In formulating the general theory of cohomology Grothendieck developed the concept of a universe – a collection of sets large enough to be closed under any operation that arose. Grothendieck proved that the existence of a single universe is equivalent over ZFC to the existence of a strongly inaccessible cardinal. More precisely, 𝑈 is the set 𝑉𝛼 of all sets with rank below 𝛼 for some uncountable strongly inaccessible cardinal.

Colin McLarty summarised the general situation:

Large cardinals as such were neither interesting nor problematic to Grothendieck and this paper shares his view. For him they were merely legitimate means to something else. He wanted to organize explicit calculational arithmetic into a geometric conceptual order. He found ways to do this in cohomology and used them to produce calculations which had eluded a decade of top mathematicians pursuing the Weil conjectures. He thereby produced the basis of most current algebraic geometry and not only the parts bearing on arithmetic. His cohomology rests on universes but weaker foundations also suffice at the loss of some of the desired conceptual order.

The applications of cohomology theory implicitly rely on universes. Most number theorists regard the applications as requiring much less than their ‘on their face’ strength and in particular believe the large cardinal appeals are ‘easily eliminable’. There are in fact two issues. McLarty writes:

Wiles’s proof uses hard arithmetic some of which is on its face one or two orders above PA, and it uses functorial organizing tools some of which are on their face stronger than ZFC.

There are two current programs for verifying in detail the intuition that the formal requirements for Wiles proof of Fermat’s last theorem can be substantially reduced. On the one hand, McLarty’s current work aims to reduce the ‘on their face’ strength of the results in cohomology from large cardinal hypotheses to finite order Peano. On the other hand Macintyre aims to reduce the ‘on their face’ strength of results in hard arithmetic to Peano. These programs may be complementary or a full implementation of Macintyre’s might avoid the first.

McLarty reduces

1. ‘ all of SGA (Revêtements Étales et Groupe Fondamental)’ to Bounded Zermelo plus a Universe.
2. “‘the currently existing applications” to Bounded Zermelo itself, thus the con-sistency strength of simple type theory.’ The Grothendieck duality theorem and others like it become theorem schema.

The essential insight of the McLarty’s papers on cohomology is the role of replacement in giving strength to the universe hypothesis. A 𝑍𝐶-universe is defined to be a transitive set U modeling 𝑍𝐶 such that every subset of an element of 𝑈 is itself an element of 𝑈. He remarks that any 𝑉𝛼 for 𝛼 a limit ordinal is provable in 𝑍𝐹𝐶 to be a 𝑍𝐶-universe. McLarty then asserts the essential use of replacement in the original Grothendieck formulation is to prove: For an arbitrary ring 𝑅 every module over 𝑅 embeds in an injective 𝑅-module and thus injective resolutions exist for all 𝑅-modules. But he gives a proof in a system with the proof theoretic strength of finite order arithmetic that every sheaf of modules on any small site has an infinite resolution.

Angus Macintyre dismisses with little comment the worries about the use of ‘large-structure’ tools in Wiles proof. He begins his appendix,

At present, all roads to a proof of Fermat’s Last Theorem pass through some version of a Modularity Theorem (generically MT) about elliptic curves defined over Q . . . A casual look at the literature may suggest that in the formulation of MT (or in some of the arguments proving whatever version of MT is required) there is essential appeal to higher-order quantification, over one of the following.

He then lists such objects as C, modular forms, Galois representations …and summarises that a superficial formulation of MT would be 𝛱1m for some small 𝑚. But he continues,

I hope nevertheless that the present account will convince all except professional sceptics that MT is really 𝛱01.

There then follows a 13 page highly technical sketch of an argument for the proposition that MT can be expressed by a sentence in 𝛱01 along with a less-detailed strategy for proving MT in PA.

Macintyre’s complexity analysis is in traditional proof theoretic terms. But his remark that ‘genus’ is more a useful geometric classification of curves than the syntactic notion of degree suggests that other criteria may be relevant. McLarty’s approach is not really a meta-theorem, but a statement that there was only one essential use of replacement and it can be eliminated. In contrast, Macintyre argues that ‘apparent second order quantification’ can be replaced by first order quantification. But the argument requires deep understanding of the number theory for each replacement in a large number of situations. Again, there is no general theorem that this type of result is provable in PA.

# Rhizomatic Topology and Global Politics. A Flirtatious Relationship.

Deleuze and Guattari see concepts as rhizomes, biological entities endowed with unique properties. They see concepts as spatially representable, where the representation contains principles of connection and heterogeneity: any point of a rhizome must be connected to any other. Deleuze and Guattari list the possible benefits of spatial representation of concepts, including the ability to represent complex multiplicity, the potential to free a concept from foundationalism, and the ability to show both breadth and depth. In this view, geometric interpretations move away from the insidious understanding of the world in terms of dualisms, dichotomies, and lines, to understand conceptual relations in terms of space and shapes. The ontology of concepts is thus, in their view, appropriately geometric, a multiplicity defined not by its elements, nor by a center of unification and comprehension and instead measured by its dimensionality and its heterogeneity. The conceptual multiplicity, is already composed of heterogeneous terms in symbiosis, and is continually transforming itself such that it is possible to follow, and map, not only the relationships between ideas but how they change over time. In fact, the authors claim that there are further benefits to geometric interpretations of understanding concepts which are unavailable in other frames of reference. They outline the unique contribution of geometric models to the understanding of contingent structure:

Principle of cartography and decalcomania: a rhizome is not amenable to any structural or generative model. It is a stranger to any idea of genetic axis or deep structure. A genetic axis is like an objective pivotal unity upon which successive stages are organized; deep structure is more like a base sequence that can be broken down into immediate constituents, while the unity of the product passes into another, transformational and subjective, dimension. (Deleuze and Guattari)

The word that Deleuze and Guattari use for ‘multiplicities’ can also be translated to the topological term ‘manifold.’ If we thought about their multiplicities as manifolds, there are a virtually unlimited number of things one could come to know, in geometric terms, about (and with) our object of study, abstractly speaking. Among those unlimited things we could learn are properties of groups (homological, cohomological, and homeomorphic), complex directionality (maps, morphisms, isomorphisms, and orientability), dimensionality (codimensionality, structure, embeddedness), partiality (differentiation, commutativity, simultaneity), and shifting representation (factorization, ideal classes, reciprocity). Each of these functions allows for a different, creative, and potentially critical representation of global political concepts, events, groupings, and relationships. This is how concepts are to be looked at: as manifolds. With such a dimensional understanding of concept-formation, it is possible to deal with complex interactions of like entities, and interactions of unlike entities. Critical theorists have emphasized the importance of such complexity in representation a number of times, speaking about it in terms compatible with mathematical methods if not mathematically. For example, Foucault’s declaration that: practicing criticism is a matter of making facile gestures difficult both reflects and is reflected in many critical theorists projects of revealing the complexity in (apparently simple) concepts deployed both in global politics.  This leads to a shift in the concept of danger as well, where danger is not an objective condition but “an effect of interpretation”. Critical thinking about how-possible questions reveals a complexity to the concept of the state which is often overlooked in traditional analyses, sending a wave of added complexity through other concepts as well. This work seeking complexity serves one of the major underlying functions of critical theorizing: finding invisible injustices in (modernist, linear, structuralist) givens in the operation and analysis of global politics.

In a geometric sense, this complexity could be thought about as multidimensional mapping. In theoretical geometry, the process of mapping conceptual spaces is not primarily empirical, but for the purpose of representing and reading the relationships between information, including identification, similarity, differentiation, and distance. The reason for defining topological spaces in math, the essence of the definition, is that there is no absolute scale for describing the distance or relation between certain points, yet it makes sense to say that an (infinite) sequence of points approaches some other (but again, no way to describe how quickly or from what direction one might be approaching). This seemingly weak relationship, which is defined purely ‘locally’, i.e., in a small locale around each point, is often surprisingly powerful: using only the relationship of approaching parts, one can distinguish between, say, a balloon, a sheet of paper, a circle, and a dot.

To each delineated concept, one should distinguish and associate a topological space, in a (necessarily) non-explicit yet definite manner. Whenever one has a relationship between concepts (here we think of the primary relationship as being that of constitution, but not restrictively, we ‘specify’ a function (or inclusion, or relation) between the topological spaces associated to the concepts). In these terms, a conceptual space is in essence a multidimensional space in which the dimensions represent qualities or features of that which is being represented. Such an approach can be leveraged for thinking about conceptual components, dimensionality, and structure. In these terms, dimensions can be thought of as properties or qualities, each with their own (often-multidimensional) properties or qualities. A key goal of the modeling of conceptual space being representation means that a key (mathematical and theoretical) goal of concept space mapping is

associationism, where associations between different kinds of information elements carry the main burden of representation. (Conceptual_Spaces_as_a_Framework_for_Knowledge_Representation)

To this end,

objects in conceptual space are represented by points, in each domain, that characterize their dimensional values. A concept geometry for conceptual spaces

These dimensional values can be arranged in relation to each other, as Gardenfors explains that

distances represent degrees of similarity between objects represented in space and therefore conceptual spaces are “suitable for representing different kinds of similarity relation. Concept

These similarity relationships can be explored across ideas of a concept and across contexts, but also over time, since “with the aid of a topological structure, we can speak about continuity, e.g., a continuous change” a possibility which can be found only in treating concepts as topological structures and not in linguistic descriptions or set theoretic representations.