Canonical Actions on Bundles – Philosophizing Identity Over Gauge Transformations.

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In physical applications, fiber bundles often come with a preferred group of transformations (usually the symmetry group of the system). The modem attitude of physicists is to regard this group as a fundamental structure which should be implemented from the very beginning enriching bundles with a further structure and defining a new category.

A similar feature appears on manifolds as well: for example, on ℜ2 one can restrict to Cartesian coordinates when we regard it just as a vector space endowed with a differentiable structure, but one can allow also translations if the “bigger” affine structure is considered. Moreover, coordinates can be chosen in much bigger sets: for instance one can fix the symplectic form w = dx ∧ dy on ℜ2 so that ℜ2 is covered by an atlas of canonical coordinates (which include all Cartesian ones). But ℜ2 also happens to be identifiable with the cotangent bundle T*ℜ so that we can restrict the previous symplectic atlas to allow only natural fibered coordinates. Finally, ℜ2 can be considered as a bare manifold so that general curvilinear coordinates should be allowed accordingly; only if the full (i.e., unrestricted) manifold structure is considered one can use a full maximal atlas. Other choices define instead maximal atlases in suitably restricted sub-classes of allowed charts. As any manifold structure is associated with a maximal atlas, geometric bundles are associated to “maximal trivializations”. However, it may happen that one can restrict (or enlarge) the allowed local trivializations, so that the same geometrical bundle can be trivialized just using the appropriate smaller class of local trivializations. In geometrical terms this corresponds, of course, to impose a further structure on the bare bundle. Of course, this newly structured bundle is defined by the same basic ingredients, i.e. the same base manifold M, the same total space B, the same projection π and the same standard fiber F, but it is characterized by a new maximal trivialization where, however, maximal refers now to a smaller set of local trivializations.

Examples are: vector bundles are characterized by linear local trivializations, affine bundles are characterized by affine local trivializations, principal bundles are characterized by left translations on the fiber group. Further examples come from Physics: gauge transformations are used as transition functions for the configuration bundles of any gauge theory. For these reasons we give the following definition of a fiber bundle with structure group.

A fiber bundle with structure group G is given by a sextuple B = (E, M, π; F ;>.., G) such that:

  • (E, M, π; F) is a fiber bundle. The structure group G is a Lie group (possibly a discrete one) and λ : G —–> Diff(F) defines a left action of G on the standard fiber F .
  • There is a family of preferred trivializations {(Uα, t(α)}α∈I of B such that the following holds: let the transition functions be gˆ(αβ) : Uαβ —–> Diff(F) and let eG be the neutral element of G. ∃ a family of maps g(αβ) : Uαβ —–> G such

    that, for each x ∈ Uαβγ = Uα ∩ Uβ ∩ Uγ

    g(αα)(x) = eG

    g(αβ)(x) = [g(βα)(x)]-1

    g(αβ)(x) . g(βγ)(x) . g(γα)(x) = eG

    and

    (αβ)(x) = λ(g(αβ)(x)) ∈ Diff(F)

The maps g(αβ) : Uαβ —–> G, which depend on the trivialization, are said to form a cocycle with values in G. They are called the transition functions with values in G (or also shortly the transition functions). The preferred trivializations will be said to be compatible with the structure. Whenever dealing with fiber bundles with structure group the choice of a compatible trivialization will be implicitly assumed.

Fiber bundles with structure group provide the suitable framework to deal with bundles with a preferred group of transformations. To see this, let us begin by introducing the notion of structure bundle of a fiber bundle with structure group B = (B, M, π; F; x, G).

Let B = (B, M, π; F; x, G) be a bundle with a structure group; let us fix a trivialization {(Uα, t(α)}α∈I and denote by g(αβ) : Uαβ —–> G its transition functions. By using the canonical left action L : G —–> Diff(G) of G onto itself, let us define gˆ(αβ) : Uαβ —–> Diff(G) given by gˆ(αβ)(x) = L (g(αβ)(x)); they obviously satisfy the cocycle properties. Now by constructing a (unique modulo isomorphisms) principal bundle PB = P(B) having G as structure group and g(αβ) as transition functions acting on G by left translation Lg : G —> G.

The principal bundle P(B) = (P, M, p; G) constructed above is called the structure bundle of B = (B, M, π; F; λ, G).

Notice that there is no similar canonical way of associating a structure bundle to a geometric bundle B = (B, M, π; F), since in that case the structure group G is at least partially undetermined.

Each automorphism of P(B) naturally acts over B.

Let, in fact, {σ(α)}α∈I be a trivialization of PB together with its transition functions g(αβ) : Uαβ —–> G defined by σ(β) = σ(α) . g(αβ). Then any principal morphism Φ = (Φ, φ) over PB is locally represented by local maps ψ(α) : Uα —> G such that

Φ : [x, h]α ↦ [φ(α)(x), ψ(α)(x).h](α)

Since Φ is a global automorphism of PB for the above local expression, the following property holds true in Uαβ.

φ(α)(x) = φ(β)(x) ≡ x’

ψ(α)(x) = g(αβ)(x’) . ψ(β)(x) . g(βα)(x)

By using the family of maps {(φ(α), ψ(α))} one can thence define a family of global automorphisms of B. In fact, using the trivialization {(Uα, t(α)}α∈I, one can define local automorphisms of B given by

Φ(α)B : (x, y) ↦ (φ(α)(x), [λ(ψ(α)(x))](y))

These local maps glue together to give a global automorphism ΦB of the bundle B, due to the fact that g(αβ) are also transition functions of B with respect to its trivialization {(Uα, t(α)}α∈I.

In this way B is endowed with a preferred group of transformations, namely the group Aut(PB) of automorphisms of the structure bundle PB, represented on B by means of the canonical action. These transformations are called (generalized) gauge transformations. Vertical gauge transformations, i.e. gauge transformations projecting over the identity, are also called pure gauge transformations.

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Principal Bundles Preserve Structures…

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A bundle P = (P, M ,π; G) is a principal bundle if the standard fiber is a Lie group G and ∃ (at least) one trivialization the transition functions of which act on G by left translations Lg : G → G : h ↦ f  g . h (where . denotes here the group multiplication).

The principal bundles are slightly different from affine bundles and vector bundles. In fact, while in affine bundles the fibers π-1(x) have a canonical structure of affine spaces and in vector bundles the fibers π-1(x) have a canonical structure of vector spaces, in principal bundles the fibers have no canonical Lie group structure. This is due to the fact that, while in affine bundles transition functions act by means of affine transformations and in vector bundles transition functions act by means of linear transformations, in principal bundles transition functions act by means of left translations which are not group automorphisms. Thus the fibers of a principal bundle do not carry a canonical group structure, but rather many non-canonical (trivialization-depending) group structures. In the fibers of a vector bundle there exists a preferred element (the “zero”) the definition of which does not depend on the local trivialization. On the contrary, in the fibers of a principal bundle there is no preferred point which is fixed by transition functions to be selected as an identity. Thus, while in affine bundles affine morphisms are those which preserve the affine structure of the fibers and in vector bundles linear morphisms are the ones which preserve the linear structure of the fibers, in a principal bundle P = (P, M, π; G) principal morphisms preserve instead a structure, the right action of G on P.

Let P = (P, M, π; G) be a principal bundle and {(Uα, t(α)}α∈I a trivialization. We can locally consider the maps

R(α)g : π-1(Uα) → π-1(Uα) : [x, h](α) ↦ [x, h . g](α) —– (1)

∃ a (global) right action Rg of G on P which is free, vertical and transitive on fibers; the local expression in the given trivialization of this action is given by R(α)g .

Using the local trivialization, we set p = [x, h](α) = [x, g(βα)(x) . h]β following diagram commutes:

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which clearly shows that the local expressions agree on the overlaps Uαβ, to define a right action. This is obviously a vertical action; it is free because of the following:

Rgp = p => [x, h . g](α) = [x, h](α) => h · g = h => g = e —– (2)

Finally, if p = [x, h1](α) and q = [x, h2](α) are two points in the same fiber of p, one can choose g = h2-1 . h1 (where · denotes the group multiplication) so that p = Rgq. This shows that the right action is also transitive on the fibers.

On the contrary, that a global left action cannot be defined by using the local maps

L(α)g : π-1(Uα) → π-1(Uα) : [x, h](α) ↦ [x, g . h](α) —– (3)

since these local maps do not satisfy a compatibility condition analogous to the condition of the commuting diagram.

let P = (P, M, π; G) and P’ = (P’, M’, π’ ; G’ ) be two principal bundles and θ : G → G’ be a homomorphism of Lie groups. A bundle morphism Φ = (Φ, φ) : P → P’ is a principal morphism with respect to θ if the following diagram is commutative:

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When G = G’ and θ = idG we just say that Φ is a principal morphism.

A trivial principal bundle (M x G, M, π; G) naturally admits the global unity section I ∈ Γ(M x G), defined with respect to a global trivialization, I : x ↦ (x, e), e being the unit element of G. Also, principal bundles allow global sections iff they are trivial. In fact, on principal bundles there is a canonical correspondence between local sections and local trivializations, due to the presence of the global right action.

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.

Functoriality in Low Dimensions. Note Quote.

Let CW be the category of CW-complexes and cellular maps, let CW0 be the full subcategory of path connected CW-complexes and let CW1 be the full subcategory of simply connected CW-complexes. Let HoCW denote the category of CW-complexes and homotopy classes of cellular maps. Let HoCWn denote the category of CW-complexes and rel n-skeleton homotopy classes of cellular maps. Dimension n = 1: It is straightforward to define a covariant truncation functor

t<n = t<1 : CW0 → HoCW together with a natural transformation

emb1 : t<1 → t<∞,

where t<∞ : CW0 → HoCW is the natural “inclusion-followed-by-quotient” functor given by t<∞(K) = K for objects K and t<∞(f) = [f] for morphisms f, such that for all objects K, emb1∗ : H0(t<1K) → H0(t<∞K) is an isomorphism and Hr(t<1K) = 0 for r ≥ 1. The details are as follows: For a path connected CW-complex K, set t<1(K) = k0, where k0 is a 0-cell of K. Let emb1(K) : t<1(K) = k0 → t<∞(K) = K be the inclusion of k0 in K. Then emb1∗ is an isomorphism on H0 as K is path connected. Clearly Hr(t<1K) = 0 for r ≥ 1. Let f : K → L be a cellular map between objects of CW0. The morphism t<1(f) : t<1(K) = k0 → l0 = t<1(L) is the homotopy class of the unique map from a point to a point. In particular, t<1(idK) = [idk0] and for a cellular map g : L → P we have t<1(gf) = t<1(g) ◦ t<1(f), so that t<1 is indeed a functor. To show that emb1 is a natural transformation, we need to see that

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that is

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commutes in HoCW. This is where we need the functor t<1 to have values only in HoCW, not in CW, because the square need certainly not commute in CW. (The points k0 and l0 do not know anything about f, so l0 need not be the image of k0 under f.) Since L is path connected, there is a path ω : I → L from l0 = ω(0) to f (k0) = ω(1). Then H : {k0} × I → L, H(k0, t) = ω(t), defines a homotopy from

k0 → l0 → L to k0 → K →f L.

Dimension n = 2: We will define a covariant truncation functor t<n = t<2 : CW1 → HoCW

together with a natural transformation
emb2 : t<2 → t<∞,

where t<∞ : CW1 → HoCW is as above (only restricted to simply connected spaces), such that for all objects K, emb2∗ : Hr(t<2K) → Hr(t<∞K) is an isomorphism for r = 0, 1, and Hr(t<2K) = 0 for r ≥ 2. For a simply connected CW-complex K, set t<2(K) = k0, where k0 is a 0-cell of K. Let emb2(K) : t<2(K) = k0 → t<∞(K) = K be the inclusion as in the case n = 1. It follows that emb2∗ is an isomorphism both on H0 as K is path connected and on H1 as H1(k0) = 0 = H1(K), while trivially Hr(t<2K) = 0 for r ≥ 2. On a cellular map f, t<2(f) is defined as in the case n = 1. As in the case n = 1, this yields a functor and emb2 is a natural transformation.

Homotopically Truncated Spaces.

The Eckmann–Hilton dual of the Postnikov decomposition of a space is the homology decomposition (or Moore space decomposition) of a space.

A Postnikov decomposition for a simply connected CW-complex X is a commutative diagram

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such that pn∗ : πr(X) → πr(Pn(X)) is an isomorphism for r ≤ n and πr(Pn(X)) = 0 for r > n. Let Fn be the homotopy fiber of qn. Then the exact sequence

πr+1(PnX) →qn∗ πr+1(Pn−1X) → πr(Fn) → πr(PnX) →qn∗ πr(Pn−1X)

shows that Fn is an Eilenberg–MacLane space K(πnX, n). Constructing Pn+1(X) inductively from Pn(X) requires knowing the nth k-invariant, which is a map of the form kn : Pn(X) → Yn. The space Pn+1(X) is then the homotopy fiber of kn. Thus there is a homotopy fibration sequence

K(πn+1X, n+1) → Pn+1(X) → Pn(X) → Yn

This means that K(πn+1X, n+1) is homotopy equivalent to the loop space ΩYn. Consequently,

πr(Yn) ≅ πr−1(ΩYn) ≅ πr−1(K(πn+1X, n+1) = πn+1X, r = n+2,

= 0, otherwise.

and we see that Yn is a K(πn+1X, n+2). Thus the nth k-invariant is a map kn : Pn(X) → K(πn+1X, n+2)

Note that it induces the zero map on all homotopy groups, but is not necessarily homotopic to the constant map. The original space X is weakly homotopy equivalent to the inverse limit of the Pn(X).

Applying the paradigm of Eckmann–Hilton duality, we arrive at the homology decomposition principle from the Postnikov decomposition principle by changing:

    • the direction of all arrows
    • π to H
    • loops Ω to suspensions S
    • fibrations to cofibrations and fibers to cofibers
    • Eilenberg–MacLane spaces K(G, n) to Moore spaces M(G, n)
    • inverse limits to direct limits

A homology decomposition (or Moore space decomposition) for a simply connected CW-complex X is a commutative diagram

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such that jn∗ : Hr(X≤n) → Hr(X) is an isomorphism for r ≤ n and Hr(X≤n) = 0 for

r > n. Let Cn be the homotopy cofiber of in. Then the exact sequence

Hr(X≤n−1) →in∗ Hr(X≤n) → Hr(Cn) →in∗ Hr−1(X≤n−1) → Hr−1(X≤n)

shows that Cn is a Moore space M(HnX, n). Constructing X≤n+1 inductively from X≤n requires knowing the nth k-invariant, which is a map of the form kn : Yn → X≤n.

The space X≤n+1 is then the homotopy cofiber of kn. Thus there is a homotopy cofibration sequence

Ynkn X≤nin+1 X≤n+1 → M(Hn+1X, n+1)

This means that M(Hn+1X, n+1) is homotopy equivalent to the suspension SYn. Consequently,

H˜r(Yn) ≅ Hr+1(SYn) ≅ Hr+1(M(Hn+1X, n+1)) = Hn+1X, r = n,

= 0, otherwise

and we see that Yn is an M(Hn+1X, n). Thus the nth k-invariant is a map kn : M(Hn+1X, n) → X≤n

It induces the zero map on all reduced homology groups, which is a nontrivial statement to make in degree n:

kn∗ : Hn(M(Hn+1X, n)) ∼= Hn+1(X) → Hn(X) ∼= Hn(X≤n)

The original space X is homotopy equivalent to the direct limit of the X≤n. The Eckmann–Hilton duality paradigm, while being a very valuable organizational principle, does have its natural limitations. Postnikov approximations possess rather good functorial properties: Let pn(X) : X → Pn(X) be a stage-n Postnikov approximation for X, that is, pn(X) : πr(X) → πr(Pn(X)) is an isomorphism for r ≤ n and πr(Pn(X)) = 0 for r > n. If Z is a space with πr(Z) = 0 for r > n, then any map g : X → Z factors up to homotopy uniquely through Pn(X). In particular, if f : X → Y is any map and pn(Y) : Y → Pn(Y) is a stage-n Postnikov approximation for Y, then, taking Z = Pn(Y) and g = pn(Y) ◦ f, there exists, uniquely up to homotopy, a map pn(f) : Pn(X) → Pn(Y) such that

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homotopy commutes. Let X = S22 e3 be a Moore space M(Z/2,2) and let Y = X ∨ S3. If X≤2 and Y≤2 denote stage-2 Moore approximations for X and Y, respectively, then X≤2 = X and Y≤2 = X. We claim that whatever maps i : X≤2 → X and j : Y≤2 → Y such that i : Hr(X≤2) → Hr(X) and j : Hr(Y≤2) → Hr(Y) are isomorphisms for r ≤ 2 one takes, there is always a map f : X → Y that cannot be compressed into the stage-2 Moore approximations, i.e. there is no map f≤2 : X≤2 → Y≤2 such that

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commutes up to homotopy. We shall employ the universal coefficient exact sequence for homotopy groups with coefficients. If G is an abelian group and M(G, n) a Moore space, then there is a short exact sequence

0 → Ext(G, πn+1Y) →ι [M(G, n), Y] →η Hom(G, πnY) → 0,

where Y is any space and [−,−] denotes pointed homotopy classes of maps. The map η is given by taking the induced homomorphism on πn and using the Hurewicz isomorphism. This universal coefficient sequence is natural in both variables. Hence, the following diagram commutes:

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Here we will briefly write E2(−) = Ext(Z/2,−) so that E2(G) = G/2G, and EY (−) = Ext(−, π3Y). By the Hurewicz theorem, π2(X) ∼= H2(X) ∼= Z/2, π2(Y) ∼= H2(Y) ∼= Z/2, and π2(i) : π2(X≤2) → π2(X), as well as π2(j) : π2(Y≤2) → π2(Y), are isomorphisms, hence the identity. If a homomorphism φ : A → B of abelian groups is onto, then E2(φ) : E2(A) = A/2A → B/2B = E2(B) remains onto. By the Hurewicz theorem, Hur : π3(Y) → H3(Y) = Z is onto. Consequently, the induced map E2(Hur) : E23Y) → E2(H3Y) = E2(Z) = Z/2 is onto. Let ξ ∈ E2(H3Y) be the generator. Choose a preimage x ∈ E23Y), E2(Hur)(x) = ξ and set [f] = ι(x) ∈ [X,Y]. Suppose there existed a homotopy class [f≤2] ∈ [X≤2, Y≤2] such that

j[f≤2] = i[f].

Then

η≤2[f≤2] = π2(j)η≤2[f≤2] = ηj[f≤2] = ηi[f] = π2(i)η[f] = π2(i)ηι(x) = 0.

Thus there is an element ε ∈ E23Y≤2) such that ι≤2(ε) = [f≤2]. From ιE2π3(j)(ε) = jι≤2(ε) = j[f≤2] = i[f] = iι(x) = ιEY π2(i)(x)

we conclude that E2π3(j)(ε) = x since ι is injective. By naturality of the Hurewicz map, the square

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commutes and induces a commutative diagram upon application of E2(−):

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It follows that

ξ = E2(Hur)(x) = E2(Hur)E2π3(j)(ε) = E2H3(j)E2(Hur)(ε) = 0,

a contradiction. Therefore, no compression [f≤2] of [f] exists.

Given a cellular map, it is not always possible to adjust the extra structure on the source and on the target of the map so that the map preserves the structures. Thus the category theoretic setup automatically, and in a natural way, singles out those continuous maps that can be compressed into homologically truncated spaces.

Disjointed Regularity in Open Classes of Elementary Topology

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Let x, y, … denote first-order structures in St𝜏, x ≈ y will denote isomorphism.

x ∼n,𝜏 y means that there is a sequence 0 ≠ I0 ⊆ …. ⊆ In of sets of 𝜏-partial isomorphism of finite domain so that, for i < j ≤ n, f ∈ Ii and a ∈ x (respectively, b ∈ y), there is g ∈ Ij such that g ⊇ f and a ∈ Dom(g) (respectively, b ∈ Im(g)). The later is called the extension property.

x ∼𝜏 y means the above holds for an infinite chain 0 ≠ I0 ⊆ …. ⊆ In ⊆ …

Fraïssé’s characterization of elementary equivalence says that for finite relational vocabularies: x ≡ y iff x ∼n,𝜏 y. To have it available for vocabularies containing function symbols add the complexity of terms in atomic formulas to the quantifier rank. It is well known that for countable x, y : x ∼𝜏 y implies x ≈ y.

Given a vocabulary 𝜏 let 𝜏 be a disjoint renaming of 𝜏. If x, y ∈ St𝜏 have the same power, let y be an isomorphic copy of y sharing the universe with x and renamed to be of type 𝜏. In this context, (x, y) will denote the 𝜏 ∪ 𝜏-structure that results of expanding x with the relations of y.

Lemma: There is a vocabulary 𝜏+ ⊇ 𝜏 ∪ 𝜏 such that for each finite vocabulary 𝜏0 ⊆ 𝜏 there is a sequence of elementary classes 𝛥1 ⊇ 𝛥2 ⊇ 𝛥3 ⊇ …. in St𝜏+ such that if 𝜋 = 𝜋𝜏+,𝜏∪𝜏 then (1) 𝜋(𝛥𝑛) = {(x,y) : |x| = |y| ≥ 𝜔, x ≡n,𝜏0 y}, (2) 𝜋(⋂n 𝛥n) = {(x, y) : |x| = |y| ≥ 𝜔, x ∼𝜏0 y}. Moreover, ⋂n𝛥n is the reduct of an elementary class.

Proof. Let 𝛥 be the class of structures (x, y, <, a, I) where < is a discrete linear order with minimum but no maximum and I codes for each c ≤ a a family Ic = {I(c, i, −, −)}i∈x of partial 𝜏0-𝜏0–isomorphisms from x into y, such that for c < c’ ≤ a : Ic ⊆ Ic and the extension property holds. Describe this by a first-order sentence 𝜃𝛥 of type 𝜏+ ⊇ 𝜏0 ∪ 𝜏0 and set 𝛥𝑛 = ModL(𝜃𝛥 ∧ ∃≥n x(x ≤ a)}. Then condition (1) in the Lemma is granted by Fraïssé’s characterization and the fact that x being (2) is granted because (x, y, <, a, I) ∈ ⋂n𝛥n iff < contains an infinite increasing 𝜔-chain below a, a ∑11 condition.

A topology on St𝜏 is invariant if its open (closed) classes are closed under isomorphic structures. Of course, it is superfluous if we identify isomorphic structures.

Theorem: Let Γ be a regular compact topology finer than the elementary topology on each class St𝜏 such that the countable structures are dense in St𝜏 and reducts and renamings are continuous for these topologies. Then Γ𝜏 is the elementary topology ∀ 𝜏.

Proof: We show that any pair of disjoint closed classes C1, C2 of Γ𝜏 may be separated by an elementary class. Assume this is not the case since Ci are compact in the topology Γ𝜏 then they are compact for the elementary topology and, by regularity of the latter, ∃ xi ∈ Ci such that x1 ≡ x2 in L𝜔𝜔(𝜏). The xi must be infinite, otherwise they would be isomorphic contradicting the disjointedness of the Ci. By normality of Γ𝜏, there are towers Ui ⊆ Ci ⊆ Ui ⊆ Ci, i = 1,2, separating the Ci with Ui, Ui open and Ci, Ci closed in Γ𝜏 and disjoint. Let I be a first-order sentence of type 𝜏 ⊇ 𝜏 such that (z, ..) |= I ⇔ z is infinite, and let π be the corresponding reduct operation. For fixed n ∈ ω and the finite 𝜏0  ⊆ 𝜏, let t be a first-order sentence describing the common ≡n,𝜏0 – equivalence class of x1, x2. As,

(xi,..) ∈ Mod𝜏(I) ∩ π-1 Mod(t) ∩ π-1Ui, i = 1, 2,..

and this class is open in Γ𝜏‘ by continuity of π, then by the density hypothesis there are countable xi ∈ Ui , i = 1, 2, such that x1n,𝜏 x2. Thus for some expansion of (x1, x2),

(x, x,..) ∈ 𝛥n,𝜏0 ∩ 𝜋1−1(𝐶1) ∩ (𝜌𝜋2)−1(C2) —– (1)

where 𝛥𝑛,𝜏0 is the class of Lemma, 𝜋1, 𝜋2 are reducts, and 𝜌 is a renaming:

𝜋1(x1, x2, …) = x1 𝜋1 : St𝜏+ → St𝜏∪𝜏 → St𝜏

𝜋2(x1, x2, …) = x2 𝜋2 : St𝜏+ → St𝜏∪𝜏 → St𝜏

𝜌(x2) = x2 𝜌 : St𝜏 → St𝜏

Since the classes (1) are closed by continuity of the above functors then ⋂n𝛥n,𝜏0 ∩ 𝜋1−1(C1) ∩ (𝜌𝜋2)−1(C2) is non-emtpy by compactness of Γ𝜏+. But ⋂n𝛥n,𝜏0 = 𝜋(V) with V elementary of type 𝜏++ ⊇ 𝜏+. Then

V ∩ π-1π1-1(U1) ∩ π-1(ρπ2)-1 (U2) ≠ 0

is open for ΓL++ and the density condition it must contain a countable structure (x1, x*2, ..). Thus (x1, x*2, ..) ∈ ∩n 𝛥𝑛,𝜏0, with xi ∈ Ui ⊆ Ci. It follows that x1 ~𝜏0 x2 and thus x1 |𝜏0 ≈ x2 |𝜏0. Let δ𝜏0 be a first-order sentence of type 𝜏 ∪ 𝜏* ∪{h} such that (x, y*, h) |= δ𝜏0 ⇔ h : x |𝜏0 ≈ y|𝜏0. By compactness,

(∩𝜏0fin𝜏 Mod𝜏∪𝜏*∪{f}𝜏0)) ∩ π1-1(C1) ∩ (ρπ2)-1(C2) ≠ 0

and we have h : x1 ≈ x2, xi ∈ Ci, contradicting the disjointedness of Ci. Finally, if C is a closed class of Γ𝜏 and x ∉ C, clΓ𝜏{x} is disjoint from C by regularity of Γ𝜏. Then clΓ𝜏{x} and C may be separated by open classes of elementary topology, which implies C is closed in this topology.

How are Topological Equivalences of Structures Homeomorphic?

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Given a first-order vocabulary 𝜏, 𝐿𝜔𝜔(𝜏) is the set of first-order sentences of type 𝜏. The elementary topology on the class 𝑆𝑡𝜏 of first-order structures type 𝜏 is obtained by taking the family of elementary classes

𝑀𝑜𝑑(𝜑) = {𝑀:𝑀 |= 𝜑}, 𝜑 ∈ 𝐿𝜔𝜔(𝜏)

as an open basis. Due to the presence of classical negation, this family is also a closed basis and thus the closed classes of 𝑆𝑡𝜏 are the first-order axiomatizable classes 𝑀𝑜𝑑(𝑇), 𝑇 ⊆ 𝐿𝜔𝜔(𝜏). Possible foundational problems due to the fact that the topology is a class of classes may be settled observing that it is indexed by a set, namely the set of theories of type 𝜏.

The main facts of model theory are reflected by the topological properties of these spaces. Thus, the downward Löwenheim-Skolem theorem for sentences amounts to topological density of the subclass of countable structures. Łoś theorem on ultraproducts grants that U-limits exist for any ultrafilter 𝑈, condition well known to be equivalent to topological compactness, and to model theoretic compactness in this case.

These spaces are not Hausdorff or T1, but having a clopen basis they are regular; that is, closed classes and exterior points may be separated by disjoint open classes. All properties or regular compact spaces are then available: normality, complete regularity, uniformizability, the Baire property, etc.

Many model theoretic properties are related to the continuity of natural operations between classes of structures, where operations are seen to be continuous and play an important role in abstract model theory.

A topological space is regular if closed sets and exterior points may be separated by open sets. It is normal if disjoint closed sets may be separated by disjoint open sets. Thus, normality does not imply regularity here. However, a regular compact space is normal. Actually, a regular Lindelöf space is already normal

Consider the following equivalence relation in a space 𝑋: 𝑥 ≡ 𝑦 ⇔ 𝑐𝑙{𝑥} = 𝑐𝑙{𝑦}

where 𝑐𝑙 denotes topological adherence. Clearly, 𝑥 ≡ 𝑦 iff 𝑥 and 𝑦 belong to the same closed (open) subsets (of a given basis). Let 𝑋/≡ be the quotient space and 𝜂 : 𝑋 → 𝑋/≡ the natural projection. Then 𝑋/≡ is T0 by construction but not necessarily Hausdorff. The following claims thus follow:

a) 𝜂 : 𝑋 → 𝑋/≡ induces an isomorphism between the respective lattices of Borel subsets of 𝑋 and 𝑋/≡. In particular, it is open and closed, preserves disjointedness, preserves and reflects compactness and normality.

b) The assignment 𝑋 → 𝑋/≡ is functorial, because ≡ is preserved by continuous functions and thus any continuous map 𝑓 : 𝑋 → 𝑌 induces a continuous assignment 𝑓/≡ : 𝑋/≡ → 𝑌/≡ which commutes with composition.

c) 𝑋 → 𝑋/≡ preserves products; that is, (𝛱𝑖𝑋𝑖)/≡ is canonically homeomorphic to 𝛱𝑖(𝑋𝑖/≡) with the product topology (monomorphisms are not preserved).

d) If 𝑋 is regular, the equivalence class of 𝑥 is 𝑐𝑙{𝑥} (this may fail in the non-regular case).

e) If 𝑋 is regular, 𝑋/≡ is Hausdorff : if 𝑥 ≢ 𝑦 then 𝑥 ∉ 𝑐𝑙{𝑦} by (d); thus there are disjoint open sets 𝑈, 𝑉 in 𝑋 such that 𝑥 ∈ 𝑈, 𝑐𝑙{𝑦} ⊆ 𝑉, and their images under 𝜂 provide an open separation of 𝜂𝑥 and 𝜂𝑦 in 𝑋/≡ by (a).

f) If 𝐾1 and 𝐾2 are disjoint compact subsets of a regular topological space 𝑋 that cannot be separated by open sets there exist 𝑥𝑖 ∈ 𝐾𝑖, 𝑖 = 1, 2, such that 𝑥1 ≡ 𝑥2. Indeed, 𝜂𝐾1 and 𝜂𝐾2 are compact in 𝑋/≡ by continuity and thus closed because 𝑋/≡ is Hausdorff by (e). They can not be disjoint; otherwise, they would be separated by open sets whose inverse images would separate 𝐾1 and 𝐾2. Pick 𝜂𝑥 = 𝜂𝑦 ∈ 𝜂𝐾1 ∩ 𝜂𝐾2 with 𝑥 ∈ 𝐾1, 𝑦 ∈ 𝐾2.

Clearly then, for the elementary topology on 𝑆𝑡𝜏, the relation ≡ coincides with elementary equivalence of structures and 𝑆𝑡𝜏/≡ is homeomorphic to the Stone space of complete theories.