MANIFOLDS

John Rognes

January 12, 1999

1. TOP, PL and DIFF manifolds

1.1. TOP manifolds. Let ℝ_{+}^{n} ⊂ ℝ^{n} be the closed upper half-space of points (x_{1}, …, x_{n}) with x_{n} ≥ 0.

A TOP (topological) n-manifold is a metrizable space M that is locally homeomorphic to ℝ_{+}^{n}.

It is covered by open subsets U_{α} ⊆ M that admit embeddings ϕ_{α}: U_{α} → ℝ_{+}^{n} onto open subsets ϕ_{α}(U_{α}) ⊆ ℝ_{+}^{n}. We call the pair (U_{α}, ϕ_{α}) a chart. A collection of charts such that the U_{α}’s cover M is called an atlas. Two atlases are compatible if their union is also an atlas. Each atlas is contained in a unique maximal compatible atlas.

The boundary ∂M of M is the (possibly empty) (n − 1)-manifold of points corresponding under a map ϕ_{α} to points in the hyperplane ℝ^{n−1} ⊂ ℝ_{+}^{n} where x_{n} = 0.

The interior of M is the complement of the boundary in M. It is the open submanifold of M of dimension n of points corresponding under a ϕ_{α} to points in the open half-space where x_{n} > 0.

We say that a manifold M is closed if it is compact and has no boundary, i.e., ∂M = ∅.

1.2. PL manifolds. Let Δ^{n} be the standard affine n-simplex. It is PL homeomorphic to the PL n-ball B^{n} = [−1, 1]^{n} ⊂ ℝ^{n}. Its boundary is PL homeomorphic to the PL (n − 1)-sphere ∂B^{n}.

A PL (piecewise linear) n-manifold is a polyhedron M that is locally PL homeomorphic to B^{n}.

It has an atlas of charts (U_{α}, ϕ_{α}) with ϕ_{α}: U_{α} → ℝ^{n} a PL embedding onto an open subset ϕ_{α}(U_{α}) ⊆ ℝ^{n}. Then each coordinate change map

ϕ_{α}ϕ_{β}^{−1} : ϕ_{β}(U_{α} ∩ U_{β}) ⟶ ϕ_{α}(U_{α} ∩ U_{β})

is a PL homeomorphism. Conversely, an atlas on a topological manifold such that each coordinate change is a PL-homeomorphism between open subsets of ℝ^{n} determines a polyhedral structure on M, making it a PL manifold. We call such an atlas a PL atlas. Each PL atlas is contained in a unique maximal compatible PL atlas. A PL structure on a manifold M is a choice of a maximal PL atlas on M.

A PL manifold can also be characterized as a combinatorial manifold, i.e., as a polyhedron such that the ‘link’ of each simplex is PL homeomorphic to a PL sphere or a PL ball.

Typeset by AMS-TEX

A PL structure on a topological manifold M is determined by a choice of homeomorphism h: N ≅ M from a PL manifold N. If the (U_{α}, ϕ_{α}) are the charts of a PL atlas on N, then the (h(U_{α}), ϕ_{α}h^{−1}) are charts for a PL atlas on M.

A PL structure on a TOP manifold is also called a triangulation.

1.3. DIFF manifolds. A DIFF (smooth) n-manifold is a space M equipped with an atlas of charts (U_{α}, ϕ_{α}) such that each coordinate change map

ϕ_{α}ϕ_{β}^{−1} : ϕ_{β}(U_{α} ∩ U_{β}) ⟶ ϕ_{α}(U_{α} ∩ U_{β})

is a (C^{∞}) diffeomorphism. Such an atlas is called a smooth atlas, and it is contained in a unique maximal compatible smooth atlas. A smooth structure on a manifold M is a choice of a maximal smooth atlas on M.

A DIFF structure on a topological manifold M is determined by a choice of homeomorphism h: N ≅ M from a DIFF manifold N. If the (U_{α}, ϕ_{α}) are the charts of a smooth atlas on N, then the (h(U_{α}), ϕ_{α}h^{−1}) are charts for a smooth atlas on M.

A DIFF structure on a TOP or PL manifold is also called a smoothing.

(Manifolds with corners ?)

1.4. Examples. Open subsets of ℝ^{n}, the disc D^{n}, the sphere S^{n}, the torus T^{n}, real projective space ℝP^{n} and complex projective space ℂP^{n} give common examples of manifolds. These all have standard TOP, PL and DIFF structures. The sphere, torus and projective spaces are closed.

1.5. Forgetful functors. Let TOP, PL and DIFF be the categories of topological, PL and smooth manifolds and homeomorphisms, PL homeomorphisms and diffeomorphisms, respectively. Let CAT denote either of these three categories.

By theorems of S.S. Cairns (1935) and J.H.C. Whitehead (1940), each DIFF manifold M admits a triangulation such that the inclusion of each simplex σ ⊂ M is a smooth map, and any two such triangulations are PL homeomorphic. And any DIFF or PL manifold is a TOP manifold by neglect of structure. Hence there are essentially forgetful functors:

DIF F ⟶ P L ⟶ T OP

Are these functors equivalences ?

2. Triangulation, smoothing- and surgery problems

2.1. Triangulations. Given a TOP manifold M, does there exist a PL manifold N and a homeomorphism h: N ≅ M ? This is the triangulation problem: does each manifold admit a triangulation ?

And if so, is this PL structure unique ? Given two PL manifolds N and N′ and homeomorphisms h: N ≅ M and h′: N′ ≅ M, will h^{−1}h′ be a PL homeomorphism ?

Answering these questions positively would prove the:

Hauptvermutung. Each TOP manifold M admits a unique PL structure.

This is in fact true for n-manifolds when n = 1, 2, 3, (easy, ?, E.E. Moise) but is false for n ≥ 4 (R.C. Kirby and L.C. Siebenmann, 1969).

Theorem (Kirby-Siebenmann). Let M be a topological manifold of dimension ≥ 6, or dimension ≥ 5 if ∂M = ∅.

(a) There is a naturally defined cohomology class ks(M) ∈ H^{4}(M; ℤ/2) such that M admits a PL structure if and only if ks(M) = 0.

(b) If ks(M) = 0 then the ‘isotopy classes’ of PL structures on M are in bijection with the elements of H^{3}(M; ℤ/2).

We shall return to this result later.

2.2. Smoothings. Given a PL manifold M, does there exist a smooth manifold N and a PL homeomorphism h: N ≅ M ? This is the smoothing problem: does each PL manifold admit a smoothing ? And if so, are any two smoothings diffeomorphic ?

Again both answers are yes for n ≤ 3, but not in general. There is a continuum (Taubes and Gompf?) of smooth structures on ℝ^{4} that are pairwise not diffeomorphic. The first examples of such are due to S. Donaldson (1982), using Gauge theory and results on simply-connected topological 4-manifolds due to M. Friedman (1982).

Restricting to closed manifolds, there is a smooth structure on S^{7} that is not diffeomorphic to the usual smooth structure (J.W. Milnor, 1956). In fact there are several, and such smooth manifolds Σ^{n} that are homeomorphic but not diffeomorphic to the standard S^{n} are called exotic spheres. Conversely, there are examples of closed PL 8-manifolds that do not admit a smooth structure (M.A. Kervaire, 1960).

We are left with the problem: Given a PL manifold M, how many inequivalent smooth structures does it admit, if any ?

Smoothing theory provides an answer to this question, in terms of homotopy theory. In particular there is a space denoted P L/O, such that for n ≥ 5 the ‘isotopy classes’ of smooth structures on the n-sphere S^{n} are in bijection with the elements of the nth homotopy group π_{n}(P L/O). For n = 5, …, 11 these groups are (M.A. Kervaire and J.W. Milnor, 1963)

0, 0, ℤ/28, ℤ/2, (ℤ/2)^{3}, ℤ/6, ℤ/992 .

The 28 isotopy classes of smoothings of S^{7} (probably) fall into 15 diffeomorphisms classes, providing 14 exotic spheres in addition to the standard 7-sphere. These were constructed by Milnor, quite simply as unit sphere bundles of various ℝ^{4}-vector bundles over S^{4}.

The Abelian group of ‘isotopy classes’ of smooth structures on S^{n} is denoted Θ_{n} or θ^{n}. This equals the group of ‘homotopy spheres’, or the group of diffeomorphisms of S^{n−1} modulo the subgroup of those that extend over D^{n}, which is denoted Γ_{n} or Γ^{n}.

We will explain these points in more detail below.

2.3. Poincaré complexes. The (co-)homology of a closed oriented n-manifold M satisfies Poincaré duality: There is a fundamental class [M] ∈ H_{n}(M) such that cap product with [M] defines an isomorphism

P D: H^{p}(M) ⟶^{≅} H_{n−p}(M)

for all p. Here P D(a) = a ∩ [M].

Consider a space X. Does there exist a closed oriented n-manifold M and a homotopy equivalence h: M ≃ X ? Then X will have the homotopy type of a finite simplicial complex. Furthermore there are isomorphisms h_{∗}: H_{∗}(M) ≅ H_{∗}(X) and h^{∗}: H^{∗}(X) ≅ H^{∗}(M), so there must exist a class [X] ∈ H_{n}(X) such that cap product with [X] defines an isomorphism

(−) ∩ [X]: H^{p}(X) ⟶^{≅} H_{n−p}(X)

for all p. A Poincaré complex of dimension n is a finite simplicial complex X equipped with such a cohomology class [X] ∈ H_{n}(X). (Some additional conditions may be imposed when X is not simply-connected.)

Now we may ask again: given a Poincaré complex (X, [X] ∈ H_{n}(X)), does there exist a closed oriented n-manifold M and a homotopy equivalence h: M ≃ X ? We might also require that h_{∗}([M]) = [X]. If the answer is yes, we might try do decide how many inequivalent manifolds there exist of this kind.

The surgery theory of W. Browder, S.P. Novikov, D. Sullivan and C.T.C. Wall provides a two-step resolution to the existence problem, and if positive, a description of the ‘structure set’ of ‘concordance classes’ of manifolds homotopy equivalent to the given Poincaré complex.

We shall discuss surgery theory later.

3. Automorphism spaces

3.1. TOP automorphisms. The existence question of when two topological manifolds M and M′ are homeomorphic can be refined to the question of how many homeomorphisms there are. If there are any, we can replace M′ by M using one of them, and are led to consider the homeomorphisms of M with itself, also known as the TOP automorphisms of M.

Let T OP (M) be the space of homeomorphisms h: M ≅ M, given the compactopen topology as a subspace of the space of continuous maps M → M. By a theorem of A.V. Černavskii (1968), this space is locally contractible.

To simplify such questions about the local structure of this automorphism space, and its PL and DIFF analogues, we will replace T OP (M) by a simplicial group.

Note that a singular q-simplex σ: Δ^{q} → T OP (M) determines a homeomorphism

σ′: M × Δ^{q} ⟶^{≅} M × Δ^{q}

over Δ^{q}, i.e., one that respects projection to the second coordinate. Here σ′(x, ξ) = (σ(ξ)(x), ξ) for x ∈ M, ξ ∈ Δ^{q}. Conversely each such homeomorphism of M × Δ^{q} over Δ^{q} corresponds to a singular q-simplex σ.

Definition. Let T OP (M)_{•} be the simplicial group with q-simplices the discrete group of homeomorphisms M × Δ^{q} → M × Δ^{q} over Δ^{q}. The ith face map d_{i} is determined by restriction to the face M × δ_{i}(Δ^{q}), and similarly for the jth degeneracy map s_{j}.

The natural map |T OP (M)_{•}| → T OP (M) is a weak homotopy equivalence. Later on we will only work with the simplicial group model, and then use T OP (M) to denote it, rather than the mapping space.

3.2. PL automorphisms. Let M be a PL manifold. It is not so clear how to give the set P L(M) of PL homeomorphisms of M the structure of a topological group. Instead we note that a PL map Δ^{q} → P L(M) ought to correspond to a PL automorphism of M × Δ^{p} over Δ^{p}.

Definition. Let P L(M)_{•} be the simplicial group with q-simplices the discrete group of PL homeomorphisms M × Δ^{q} → M × Δ^{q} over Δ^{q}.

3.3. DIFF automorphisms. Let M be a smooth manifold. The set DIF F (M) of diffeomorphisms M ≅ M can be given a topology, called the Whitney C^{∞}- topology, such that the simplicial set of singular simplices σ: Δ^{q} → DIF F (M) contains as a deformation retract the simplicial set of smooth automorphisms of M ×Δ^{q} over Δ^{q}. (See Appendix A of Burghelea-Lashof-Rothenberg.) Thus we can replace the topological group DIF F (M) with a simplicial construction:

Definition. Let DIF F (M)_{•} be the simplicial group with q-simplices the discrete group of diffeomorphisms M × Δ^{q} → M × Δ^{q} over Δ^{q}.

Then the natural map |DIF F (M)_{•}| → DIF F (M) is a weak homotopy equivalence, as in the TOP case.

3.4. Homotopy automorphisms. Let X be any space. We give the space of continuous maps X → X the compact-open topology. Let G(X) be the subspace of homotopy equivalences X ≃ X. This is a set of path components of the whole mapping space. It is a topological monoid, i.e., has a continuous, associative and unital composition law, but is typically not a group. (Homotopy equivalences have homotopy inverses rather than true inverses.)

A singular q-simplex Δ^{q} → G(X) corresponds bijectively to a homotopy automorphism of X × Δ^{q} over Δ^{q}.

Definition. Let G(X)_{•} be the simplicial monoid with q-simplices the discrete monoid of homotopy equivalences X × Δ^{q} → X × Δ^{q} over Δ^{q}.

Again there is a weak homotopy equivalence |G(X)_{•}| → G(X).

3.5. Comparison maps. From now on we shall only use the simplicial models for the automorphism spaces T OP (M), P L(M), DIF F (M) and G(X), so let us omit the subscript _{•} from their notations.

There are now essentially forgetful maps of simplicial groups or monoids

DIF F (M) ⟶ P L(M) ⟶ T OP (M) ⟶ G(M)

defined if M is a smooth manifold. If M is only PL or topological, one must omit the first one or two spaces in this diagram.

(Reference for how to triangulate diffeomorphisms ?)

4. Isotopy and concordance

We need to consider some notions of equivalence among automorphisms, and among smoothings or triangulations.

4.1. Isotopy of automorphisms. Consider CAT automorphisms M ≅ M, i.e., homeomorphisms, PL homeomorphisms or diffeomorphisms according to the category. A CAT isotopy is a 1-simplex in the space of such automorphisms, or equivalently, a CAT automorphism

Φ: M × I ⟶^{≅} M × I

respecting projection to I. Writing Φ(x, t) = (ϕ_{t}(x), t), the family t ↦ ϕ_{t} describes a path of CAT automorphisms M ≅ M. Then Φ is called an isotopy from ϕ_{0} to ϕ_{1}, and we say that ϕ_{0} and ϕ_{1} are isotopic. This defines an equivalence relation on automorphisms M ≅ M.

If M has boundary, we typically insist that Φ fixes ∂M × I pointwise, i.e., that Φ(x, t) = (x, t) for all x ∈ ∂M, t ∈ I. This defines an equivalence relation on the automorphisms fixing the boundary.

Now consider the TOP case, with M. We know that the space of homeomorphisms M ≅ M is locally contractible. Hence each homeomorphism ϕ that is sufficiently close to the identity map 1_{M} can be connected with a path t ↦ ϕ_{t} such that ϕ_{0} = 1_{M} and ϕ_{1} = ϕ. There is a corresponding isotopy Φ: (x, t) ↦ (ϕ_{t}(x), t). Thus ϕ is isotopic to the identity. On the other hand, homeomorphisms further removed from the identity need not be isotopic to the identity. For example, a reflection of a sphere is a homeomorphism, but has degree −1, hence is neither isotopic nor homotopic to the identity.

4.2. Concordance of automorphisms. Let M be a CAT manifold. A CAT concordance of M is a CAT automorphism

Ψ: M × I ≅ M × I

fixing the subspace M × {0} ∪ ∂M × I. Then Ψ takes the part M × {1} of the boundary of M × I to itself. Thus Φ(x, 1) = (ϕ_{1}(x), 1) for some automorphism ψ_{1} : M → M fixing ∂M.

We do not insist that Ψ respects projection to I. If it does, it is nothing more than an isotopy from the identity to ψ_{1}. Thus concordances generalize isotopies from the identity, and are also called pseudoisotopies.

Thus isotopy (to the identity) implies concordance for automorphisms. It is an important question when the converse holds.

4.3. Isotopy of structures. Consider a manifold M, with two CAT manifold structures Σ and Σ′. For instance, one may consider two CAT manifolds N and N′ and homeomorphisms N ≅ M and N′ ≅ M.

Suppose there is a TOP isotopy (= path of homeomorphisms)

Φ: M × I ≅ M × I

from the identity to ϕ: M ≅ M, such that ϕ is a CAT isomorphism from M with structure Σ to M with structure Σ′. Then we say that the CAT structures Σ and Σ′ are isotopic.

Clearly isotopy implies isomorphism. However, also in this case there are manifolds with non-isotopic but isomorphic CAT structures. For example, the 28 isotopy classes of smooth structures on S^{7} form a cyclic group ℤ/28 with respect to connected sum #. Reflection in a hyperplane defines an automorphism ρ of S^{7}, that takes the ith smooth structure to the (28 − i)th. For if Σ is a smooth 7-manifold homeomorphic to S^{7} that generates this group, and Σ′ is such that ρ: Σ ≅ Σ′ is a diffeomorphism, then Σ#Σ′ ≅ S^{7} is a diffeomorphism, so ρ takes the generator to its additive inverse. Thus there are at most 15 isomorphism classes of smooth structures on S^{7}.

If M has boundary, we might insist that Σ and Σ′ restrict to the same structure on ∂M, and ask that Φ fixes ∂M × I.

4.4. Concordance of structures. Consider a manifold M with two CAT structures Σ and Σ′. A CAT concordance from Σ to Σ′ is a CAT structure Γ on the product M × I, such that Γ restricts to the CAT structure Σ × {0} on M × {0} and to Σ′ × {1} on M × {1}.

Again this defines an equivalence relation on the CAT structures on M. Isotopy implies concordance, because given an isotopy Φ from the identity to an isomorphism ϕ from Σ to Σ′, we can define a CAT structure Γ on M × I by making Φ a CAT isomorphism from the product CAT structure Σ × I to Γ. Then Γ is a concordance from Σ to Σ′.

Conversely, we have the ‘concordance implies isotopy’ theorems of J. Munkres (1966) for DIFF structures on PL manifolds, and of Kirby and Siebenmann (1970?) for smoothings or triangulations of topological manifolds:

Theorem (Kirby-Siebenmann). Let M be a TOP manifold of dimension ≥ 6, or dimension ≥ 5 if ∂M = ∅. Two DIFF (resp. PL) structures Σ and Σ′ on M are concordant if and only if they are isotopic.

5. Spaces of embeddings, immersions

Let Emb(M, N) be the space of CAT embeddings M → N, defined as the simplicial set with q-simplices the set of CAT embeddings

f: M × Δ^{q} → N × Δ^{q}

that respect projection to Δ^{q}.

Let Imm(M, N) be the space of CAT immersions M → N, defined as the simplicial set with q-simplices the set of CAT immersions

f: M × Δ^{q} → N × Δ^{q}

that respect projection to Δ^{q}.

There is a map Emb(M, N) → Imm(M, N). Also CAT (M) and CAT (N) act continuously on the right and left of Emb(N), respectively.

Let E → B and E′ → B′ be two vector bundles. Let Inj(E, E′) be the space of injective bundle maps, with points the bundle maps

\xymatrix{E \ar[r]^{F} \ar[d] & E′ \ar[d] \\ B \ar[r]^{f} & B′}

that are fiberwise injective, i.e., such that F_{x}: E_{x} → E′_{f(x)} is injective for each x ∈ B.

The following result is known as Smale-Hirsch immersion theory.

Theorem (Smale, Hirsch). Let M and N be smooth manifolds, with tangent bundles T M → M and T N → N. The map

Imm(M, N) ⟶^{D} Inj(T M, T N)

that takes an immersion f: M → N to its derivative Df: T M → T N, is a homotopy equivalence.

(Must M be compact ? Stability ?)

A similar result holds for PL or TOP manifolds, replacing tangent vector bundles by ‘microbundles’.

Understanding the spaces of embeddings is a harder problem, of current interest. Note that Emb(S^{1}, ℝ^{3}) is the space of (smooth) knots in ℝ^{3}.

Example. Let i: S^{2} ⊂ ℝ^{3} be the inclusion. Both i and −i: p ↦ −p are embeddings S^{2} → ℝ^{3}, hence also immersions. The induced maps of tangent bundles Di: T S^{2} → T ℝ^{3} and D(−i): T S^{2} → T ℝ^{3} are homotopic as injective bundle maps. An explicit homotopy t ↦ h_{t} for t ∈ [0, 1] is given by

h_{t}(p, v) = ((1 − 2t)p, r_{tπ}(v))

where r_{tπ}: T_{p}S^{2} → T_{p}S^{2} ⊂ ℝ^{3} rotates the oriented plane T_{p}S^{2} through an angle tπ. Here p ∈ S^{2} and v ∈ T_{p}S^{2}. Then h_{0} = Di and h_{1} = D(−i), as required.

Hence by Smale-Hirsch theory i and −i ∈ Imm(S^{2}, ℝ^{3}) are in the same path component. This implies that there exists a regular homotopy, i.e., a path of immersions, from i: S^{2} ⊂ ℝ^{3} to −i: S^{2} → ℝ^{3}. This means that the 2-sphere can be turned inside-out in ℝ^{3}, by passing through immersions only. (For embeddings this is, of course, impossible.)

A display in Palais de la Découverte in Paris shows how this can be done.

6. The h-cobordism theorem

6.1. Cobordism. Let M and M′ be closed CAT manifolds. A CAT cobordism (W; M, M′) from M to M′ is a compact CAT manifold W with boundary, with an isomorphism ∂W ≅ M ∐ M′ of CAT manifolds.

If M and M′ are oriented, a CAT oriented cobordism from M to M′ is a CAT cobordism W that is oriented, such that the induced orientation on ∂W agrees with that of M and is opposite to that of M′. Writing −M′ for M′ with the opposite orientation, we should have ∂W ≅ M ∐(−M′) as oriented manifolds.

If M and M′ are compact CAT manifolds with isomorphic boundaries ∂M ≅ ∂M′, then a CAT cobordism from M to M′ is a compact CAT manifold W with boundary, such that

∂W ≅ M ∪ (∂M × I) ∪ M′

identified along ∂M ≅ ∂M × {0}, and along ∂M × {1} ≅ ∂M′.

Cobordism is an equivalence relation. The cylinder W = M × I provides a cobordism from M to itself, called the trivial cobordism. Given a cobordism W from M to M′ and a cobordism W′ from M′ to M′′, the union W ∪_{M′} W′ is a cobordism from M to M′′. (In the DIFF category, the union is formed by aligning two collars of M′ in a sensible way.) A compact manifold W can be thought of as a cobordism from ∂M to the empty manifold ∅.

In each dimension n the set of cobordism classes of CAT n-manifolds forms an Abelian group N_{n}^{CAT}, with the sum operation induced by disjoint sum of manifolds. So writing [M] for the cobordism class of a CAT manifold M, we define [M]+[N] = [M ∐ N]. The graded Abelian group

N_{∗}^{CAT} = ⨁_{n≥0} N_{n}^{CAT}

is furthermore a graded ring, with product [M]·[N] = [M ×N] induced by Cartesian product of manifolds. This is a ℤ/2-algebra, since 2[M] = [M]+[M] = [∂(M ×I)] = 0 for all closed M.

Theorem (Thom, 1954). N_{∗}^{DIF F} is a polynomial algebra over ℤ/2 with one generator in each dimension n ≠ 2^{i} − 1.

Likewise for oriented cobordism. The set of oriented cobordism classes of oriented CAT n-manifolds is denoted Ω_{n}^{CAT}, and

Ω_{∗}^{CAT} = ⨁_{n≥0} Ω_{n}^{CAT}

is a graded ring. Ω_{0}^{CAT} = ℤ, while Ω_{n}^{CAT} = 0 for n = 1, 2, 3. However Ω_{4}^{DIF F} ≅ ℤ, generated by the 4-manifold ℂP^{2}.

Theorem (Thom, 1954). Ω_{∗}^{DIF F} ⊗ ℚ is a polynomial algebra over ℚ with generators [ℂP^{2n}] for each n ≥ 1.

6.2. h-cobordisms. A cobordism W from M to M′ is called an h-cobordism if the inclusions M → W and M′ → W are homotopy equivalences.

The trivial cobordism M × I gives one example of an h-cobordism from M to itself. Are there other examples ?

DIFF h-cobordism theorem (Smale, 1960). Let W be a DIFF h-cobordism from M to M′. Suppose n = dim(M) ≥ 6, or n ≥ 5 if M is closed. Suppose further that M is 1-connected, i.e., π_{1}(M) = 0. Then W is diffeomorphic to M × I. So M ≅ M′ as smooth manifolds.

A similar result holds in the PL category (Stallings, Zeeman ?). Smale’s original proof uses handle theory, as presented in Kosinski’s book. Milnor gave a proof using Morse theory. See books of Cohen or Rourke and Sanderson for proofs in the PL category.

Clearly M is 1-connected if and only if W is 1-connected if and only if M′ is 1connected, when W is an h-cobordism. If W is any cobordism between closed manifolds M and M′, such that all three spaces are 1-connected, and H_{∗}(W, M) = 0, then by Lefschetz duality H_{∗}(W, M′) = 0. So by 1-connectedness and the Whitehead theorem W is an h-cobordism. Thus 1-connectedness, H_{∗}(W, M) = 0 and n ≥ 5 constitute alternative hypotheses for the theorem.

The h-cobordism theorem tells us that, in dimensions ≥ 5, any h-cobordism from M to M′ is isomorphic to the trivial one.

6.3. The Poincaré conjecture.

Generalized Poincaré Conjecture. Let Σ be a closed n-manifold homotopy equivalent to S^{n}. Then Σ is homeomorphic to S^{n}.

This is clearly true for n = 1, 2. Poincaré’s original conjecture concerns dimension n = 3, where the hypothesis that Σ ≃ S^{3} can be replaced by asking that Σ be 1-connected. That case remains open, and is the most famous open problem in topology.

For n ≥ 4 triangulation theory tells us that the conclusion can be replaced by ‘Σ is PL homeomorphic to S^{n}’. But the presence of exotic spheres tells us that we cannot replace the conclusion by ‘Σ is diffeomorphic to S^{n}’, except for n = 5, 6.

S. Smale proved the smooth Poincaré conjecture in dimensions n ≥ 6 in 1960. Shortly thereafter J. Stallings obtained the PL conjecture in dimensions n ≥ 5. By rather different methods M. Friedman proved the Poincaré conjecture in dimension 4 in 1982. Thus only the original case, n = 3, remains.

A closed n-manifold of the homotopy type of a sphere is called a homotopy sphere. Thus homotopy spheres are spheres, except possibly in dimension 3. The proof in dimensions n ≥ 6 is an easy application of the h-cobordism theorem:

Proof. Let Σ be a closed smooth (n + 1)-manifold, homotopy equivalent to S^{n+1}, with n ≥ 5. Choose two disjoint embedded (n + 1)-discs D ⊂ Σ and D′ ⊂ Σ. Let

W = Σ \ (int(D) ∪ int(D′))

be the closure of the complement of D and D′. Let M = ∂D and M′ = ∂D′. Then W is a cobordism from M ≅ S^{n} to M′ ≅ S^{n}. In fact it is an h-cobordism. For by excision H_{∗}(D′, M′) ≅ H_{∗}(Σ, W ∪ D) and by hypothesis H_{∗}(S^{n+1}, ∗) ≅ H_{∗}(Σ, D). In the long exact sequence for the triple (Σ, W ∪ D, D) the natural map

H_{∗}(S^{n+1}, ∗) ≅ H_{∗}(Σ, D) ⟶ H_{∗}(Σ, W ∪ D) ≅ H_{∗}(D′, M′)

can be checked to be an isomorphism. Hence the third terms H_{∗}(W ∪ D, D) ≅ H_{∗}(W, M) are zero by exactness and excision. Now apply the h-cobordism theorem to W, and deduce W ≅ M × I ≅ S^{n} × I. Hence there are homeomorphisms

Σ = D ∪_{M} W ∪_{M′} D′ ≅ D^{n+1} ∪_{S^{n}} (S^{n} × I) ∪_{S^{n}} D^{n+1} ≅ S^{n+1} .

This completes the proof. □

6.4. Concordance of h-structures. A CAT h-structure on a space X is a CAT manifold M and a homotopy equivalence f: M → X. (These are the objects surgery theory searches for and classifies.) One can think of an h-cobordism as a concordance between two h-structures on X. For given homotopy equivalences f: M → X and f′: M′ → X from two CAT manifolds two X, a concordance between them is an h-structure F: W → X × I on X × I, that restricts to f and f′ at the two ends. This means that ∂W is the disjoint union of M over X × {0} and M′ over X × {1}. Since f, f′ and F are homotopy equivalences, it follows that M → W and M′ → W are homotopy equivalences.

6.5. The group Θ_{n}. Let Θ_{n} be the group of h-cobordism classes of smooth homotopy spheres. For n ≠ 3 they are all topological spheres, by the proven cases of the Poincaré conjecture, hence may be viewed as smooth structures on S^{n}. Clearly a concordance between two smooth structures on S^{n} yields an h-cobordism between the two smooth manifolds. Conversely, for n ≥ 5 the h-cobordism theorem tells us that h-cobordism implies concordance. Thus h-cobordism is equivalent to concordance, which in turn is equivalent to isotopy. Thus we may think of Θ_{n} an either h-cobordism classes, concordance classes or isotopy classes of smooth homotopy spheres. The notions are equivalent (for n ≥ 5.)

7. Whitehead torsion, the s-cobordism theorem

See J. Milnor’s survey article (Bull. AMS., 1966) for everything about Whitehead torsion.

7.1. Torsion for chain complexes. Let R be any ring, and C_{∗} a finitely generated complex of free R-modules:

0 → C_{m} ⟶^{d_{m}} C_{m−1} ⟶ … ⟶ C_{1} ⟶^{d_{1}} C_{0} → 0

Suppose that C_{∗} is exact, and that each module C_{n} is equipped with an ordered R-basis, denoted c_{n}. Let B_{n} = im(d_{n+1}) ⊆ C_{n}, so that

0 → B_{n} → C_{n} ⟶^{d_{n}} B_{n−1} → 0

is short exact. Then B_{0} = C_{0} and by induction each B_{n} is stably free, i.e., B_{n}⊕R^{s} ≅ R^{t} for some r, t. By adding copies of complexes like 0 → R ⟶^{1} R → 0 to C_{∗} we may assume each B_{n} is in fact free.

Choose an ordered R-module basis b_{n} for each B_{n}. Then we can form an ordered basis denoted b_{n}b_{n−1} for C_{n} by lifting the elements in b_{n−1} over d_{n}, and including the elements in b_{n} into C_{n}. Let b_{n}b_{n−1}/c_{n} denote the change of basis matrix comparing the two bases b_{n}b_{n−1} and c_{n} for C_{n}. Different choices of liftings of elements in b_{n−1} will alter this matrix by triangular matrices, hence elementary matrices. But the equivalence class

[b_{n}b_{n−1}/c_{n}] ∈ K_{1}(R) = GL(R)/E(R)

is well defined. Changing the ordering of any basis may alter this class by a sign. We think of K_{1}(R) as an additive group.

Definition. The torsion τ(C_{∗}) of the exact chain complex C_{∗}, with chosen bases c_{n} for C_{n}, is the alternating sum

τ(C_{∗}) = ∑_{n}(−1)^{n}[b_{n}b_{n−1}/c_{n}] ∈ K_{1}(R) .

The sum is independent of the choice of bases b_{n} for the B_{n}.

When m = 1, so C_{∗} is a complex 0 → C_{1} ⟶^{d_{1}} C_{0} → 0 with d_{1} an isomorphism, τ(C_{∗}) is the class in K_{1}(R) of the matrix of d_{1} with respect to the given bases c_{0} and c_{1}. The definition above extends this construction to make sense for longer chain complexes.

In particular consider the case when π is a group and R = ℤ[π] is its integral group ring. Let C_{∗} be a finitely generated complex of free ℤ[π]-modules. Suppose first that C_{∗} is exact, and that each module C_{n} is equipped with an ordered basis c_{n} as a ℤ[π]-module. Then its torsion is a class in K_{1}(ℤ[π]). The basis c_{n} can be altered by multiplying any one element by an element of ±π ⊂ ℤ[π], and still remain a basis. The resulting torsion will then change by an element in the image of the composition:

{±π} ⊂ GL_{1}(ℤ[π]) → GL(ℤ[π]) → K_{1}(ℤ[π])

Definition. The Whitehead group W h(π) of π is the quotient

W h(π) = K_{1}(ℤ[π])/{±π}

of K_{1}(ℤ[π]) by the image of the homomorphism above.

Hence an exact finitely generated free complex of ℤ[π]-modules C_{∗}, equipped with an unordered ℤ[π]-basis c_{n} in each degree, where each element is given up to multiples by π and sign, has a well-defined Whitehead torsion element τ(C_{∗}) ∈ W h(π). As an important special case, when π = 1 is trivial, K_{1}(ℤ) ≅ {±1} and W h(1) = 0.

7.2. Torsion for relative CW-complexes. Let X be a space, and let C(X) be the category of cofibrant spaces under X, i.e., of spaces Y equipped with a cofibration i: X → Y. The morphisms from (Y, i) to (Y′, i′) are maps f: Y → Y′ such that fi = i′. We think of Y as a space containing X, thus the letter ‘C’.

Let C_{f}(X) be the subcategory of finite relative CW-complexes (Y, X) and cellular maps under X. This is the subcategory of spaces Y that can be built from X by adjoining a finite number of cells, always attaching new cells to X and cells of lower dimension.

Let C_{f}^{h}(X) be the full subcategory of finite relative CW-complexes (Y, X) such that the inclusion X → Y is a homotopy equivalence. Then X is a deformation retract of Y by the homotopy extension property.

Suppose X is path connected, and let π = π_{1}(X) be its fundamental group. The Whitehead torsion τ(Y, X) ∈ W h(π) of a pair (Y, X) ∈ C_{f}^{h}(X) can be defined as follows:

Let Ỹ → Y be the universal cover of Y, with fibers isomorphic to π_{1}(Y), and let X̃ → X be its restriction over the subspace X ⊆ Y. Since X ≃ Y we have π_{1}(X) ≅ π_{1}(Y) ≅ π, so X̃ → X is also the universal cover of X. Then (Ỹ , X̃) is a relative CW-complex, and Ỹ is built from X̃ by attaching one n-cell for each relative n-cell of (Y, X) and each element of π. Hence each relative n-cell of (Y, X) is covered by a number of relative n-cells in (Ỹ , X̃), namely one for each element in π.

Let C_{∗} = C_{∗}(Ỹ , X̃) be the cellular chain complex. So each C_{n} has a ℤ-basis with one element for each relative n-cell of (Ỹ , X̃). Now π acts by deck transformations, freely permuting these n-cells. Choose one representative cell in each π-orbit. This amounts to choosing one lifted n-cell in (Ỹ , X̃) for each n-cell in (Y, X). Their classes provide a basis for C_{n} as a free ℤ[π]-module, which is finite since (Y, X) is a finite relative CW-complex. (Different choices of lifted cells may alter the basis for C_{n} only by multiples of π. Different orderings will affect the sign.) And X̃ → Ỹ is a homotopy equivalence, since X → Y is one. Thus the cellular complex C_{∗} has homology H_{∗}(Ỹ , X̃) = 0, and is exact. Thus we can make sense of:

τ(Y, X) = τ(C_{∗}(Ỹ , X̃)) ∈ W h(π)

The following lemmas are useful.

Lemma 7.2.1. Let Y ∈ C_{f}^{h}(X). Suppose that each component of Y \ X is simplyconnected. Then τ(Y, X) = 0 in W h(π).

Lemma 7.2.2. Let Y ∈ C_{f}^{h}(X) and Z ∈ C_{f}^{h}(Y). Then Z ∈ C_{f}^{h}(X) and τ(Z, X) = τ(Y, X) + τ(Z, Y) in W h(π).

7.3. Simple homotopy theory. Let X be a CW-complex with an embedded n-cell D_{1}^{n} ⊆ X. Form a CW-pair (Y, X) by attaching another n-cell D_{2}^{n} along ∂D_{1}^{n}, and filling in with an (n + 1)-cell along D_{1}^{n} ∪_{∂} D_{2}^{n} ≅ S^{n}:

Y = X ∪ D_{2}^{n} ∪ D^{n+1}

Then Y deformation retracts to X, with the n-cell D_{2}^{n} traversing over D^{n+1}. We say there is an elementary collapse of Y on X. The collapse is across D^{n+1} onto D_{1}^{n} from the complementary face D_{2}^{n}. Conversely we say there is an elementary expansion from X to Y.

The relation that there is an elementary collapse of Y on X generates an equivalence relation on CW-complexes, called simple homotopy equivalence, denoted X ≃_{s} Y. So two CW-complexes X and X′ are simple homotopy equivalent if there is a finite chain of elementary collapses and elementary expansions from X to X′. This is in general a more restrictive equivalence relation than homotopy equivalence.

By Lemma 7.2.1, τ(Y, X) = 0 when there is an elementary collapse of Y on X. Hence by Lemma 7.2.2, Whitehead torsion satisfies τ(Y, X) = τ(Z, X) if Y and Z are simply homotopy equivalent spaces containing X. Thus Whitehead torsion is a simple homotopy invariant.

Let s′C_{f}^{h}(X) be the subcategory of C_{f}^{h}(X) with the same objects, but morphisms only the f: Y → Y′ under X that are elementary collapses, and their composites. The geometric realization |s′C_{f}^{h}(X)| has path components equal to the simple homotopy classes of finite CW-pairs (Y, X) where Y deformation retracts to X.

Given two objects Y, Y′ ∈ C_{f}^{h}(X) we may form their sum Y ∪_{X} Y′ ∈ C_{f}^{h}(X) by gluing along X.

Proposition. Let π = π_{1}(X). Whitehead torsion defines an additive bijection:

τ: π_{0}(|s′C_{f}^{h}(X)|) ⟶^{≅} W h(π)

So for simply-connected spaces, simple homotopy equivalence equals homotopy equivalence. In general this is not so.

7.4. Torsion for h-cobordisms. It is clear that Whitehead torsion is an invariant under cellular isomorphisms, or PL-isomorphisms for polyhedra viewed as CWcomplexes. J.H.C. Whitehead (1950) showed that the torsion is also invariant under subdivision, hence is a PL-homeomorphism invariant. More is true:

Theorem (T.A. Chapman). Whitehead torsion is a topological invariant.

(Make this more precise.)

Now let (W; M, M′) be a CAT h-cobordism from M, and let π = π_{1}(M). If CAT=PL or DIFF we have a PL-triangulation of the pair (W, M), making it a CWpair in C_{f}^{h}(M). Hence its torsion τ(W, M) ∈ W h(π) is well defined. Chapman’s theorem implies that the same applies when CAT=TOP.

Definition. An s-cobordism (W; M, M′) is an h-cobordism such that τ(W, M) = 0 in W h(π).

Given τ ∈ W h(π) one can construct an h-cobordism (W; M, M′) with τ(W, M) = τ. The Whitehead torsion map from h-cobordisms is thus surjective, in each dimension n ≥ 5.

Clearly there is no distinction between h- and s-cobordisms when W h(π) = 0, as when π = 1, i.e., when M is 1-connected.

7.5. The s-cobordism theorem. The following was proven independently by Barden, Mazur and Stallings in 1963. See Kervaire (1965) for a proof.

DIFF s-cobordism theorem. Let (W; M, M′) be a DIFF h-cobordism from M to M′ and suppose n = dim(M) ≥ 6, or n ≥ 5 if M is closed. Then W is diffeomorphic to M ×I if and only if W is an s-cobordism from M, i.e., the torsion τ(W, M) vanishes. Then M ≅ M′ as smooth manifolds.

The diffeomorphism classes of h-cobordisms from M are thus in bijection with the Whitehead group W h(π) for π = π_{1}(M), classified by their torsion.

Corollary. Each h-cobordism (W; M, M′) admits a codimension 0 embedding in a cylinder M × I, taking M to M × {0}.

Proof. Let τ = τ(W, M) ∈ W h(π) and choose an h-cobordism (W′; M′, M′′) with torsion τ(W′′, M′) = −τ. Then W ∪_{M′} W′ is an h-cobordism from M with torsion τ − τ = 0, i.e., an s-cobordism, so W ∪_{M′} W′ ≅ M × I by the s-cobordism theorem. The embedding takes W into W ∪_{M′} W′ ≅ M × I by this diffeomorphism. □

The same considerations apply for PL and TOP manifolds. It is remarkable that the obstruction group W h(π) is the same for all three categories. (See Kirby-Siebenmann for references.)

8. Concordance spaces

8.1. Concordances and pseudoisotopies.

Definition. Let M be a CAT manifold, possibly with boundary ∂M. The space of CAT concordances of M is the space

C^{CAT} (M) = CAT (M × I rel M × {0} ∪ ∂M × I)

of CAT automorphisms of M × I fixing (a neighborhood of) M × {0} and ∂M × I.

Such an automorphism Ψ must map M ×{1} by an isomorphism to itself, because Ψ takes the boundary of M × I to itself, and is the identity on the remainder of this boundary. Hence Ψ ∈ C^{CAT} (M) determines an automorphism ψ_{1} ∈ CAT (M) by Ψ(x, 1) = (ψ_{1}(x), 1), i.e., by restriction to the top end. There is a map

r_{1} : C^{CAT} (M) ⟶ CAT (M)

with r_{1}(Ψ) = ψ_{1}. Its fiber r_{1}^{−1}(1_{M}) at the identity of M is the topological group of automorphisms of M × I fixing the entire boundary, i.e., CAT (M × I). By a theorem of Cerf (reference ?) the diagram

CAT (M × I) ⟶ C^{CAT} (M) ⟶^{r_{1}} CAT (M)

is a principal fiber bundle over the path components in the image of r_{1}.

A point Ψ in C^{CAT} (M) is also a pseudoisotopy from the identity of M to r_{1}(Ψ) = ψ_{1}. Hence C^{CAT} (M) is also called the CAT pseudoisotopy space of M, and is often denoted P^{CAT} (M). The image of r_{1} consists of the automorphisms of M that are pseudoisotopic to the identity. The path component of the identity in CAT (M) consists of the automorphisms of M that are isotopic to the identity.

8.2. The Alexander trick. The support of an automorphism Ψ: N ≅ N is the closure in N of the set {x ∈ N | Ψ(x) ≠ x} where Ψ differs from the identity. Given an isomorphism Φ: N ≅ M, and an automorphism Ψ: N ≅ N, the composite ΦΨΦ^{−1} : M ≅ M is an automorphism of M. We say that ΦΨΦ^{−1} is obtained from Ψ by conjugation along Φ. Then Φ maps the support of Ψ to the support of ΦΨΦ^{−1}.

\xymatrix{N \ar[r]^{Ψ} \ar[d]^{Φ} & N \ar[d]^{Φ} \\ M \ar[r]_{ΦΨΦ^{−1}} & M}

Theorem (Alexander trick). The spaces C^{T OP} (D^{n}) and C^{P L}(D^{n}) are contractible.

Proof. There is an explicit contraction of C^{CAT} (D^{n}) to the identity, which gradually shrinks the support of each automorphism Ψ of D^{n} × I towards the center point (0, 1) of the top face, by conjugation with an isomorphism

Φ_{s}: D^{n} × I ⟶^{≅} sD^{n} × [1 − s, 1]

for s ∈ (0, 1]. Here Φ_{s}(x, t) = (sx, 1 − s(1 − t)) for (x, t) ∈ D^{n} × I. Given Ψ let

Ψ_{s}(x, t) = \{ \begin{matrix}(Φ_{s} ∘ Ψ ∘ Φ_{s}^{−1})(x, t) & for (x, t) ∈ sD^{n} × [1 − s, 1], \\ (x, t) & otherwise. \end{matrix}

Then Ψ_{1} = Ψ, while for each (x, t), Φ_{s}(x, t) converges to (x, t) as s → 0^{+}. In the TOP and PL categories, we can define Ψ_{0} = 1 (the identity map), and obtain a continuous contraction (Ψ, s) ↦ Ψ_{s} of C^{CAT} (D^{n}) to a point. □

Remark. In the DIFF category, the function

(x, t, s) ↦ Ψ_{s}(x, t)

is typically not smooth at s = 0, hence does not define a continuous path in C^{DIF F} (D^{n}). Thus the above argument fails.

In fact, π_{i}(C^{DIF F} (D^{n})) ⊗ ℚ ≅ ℚ for i ≡ 3 mod 4 when n ≫ i, so C^{DIF F} (D^{n}) is not contractible for n large. This is a consequence of the relationship between concordance spaces and Waldhausen’s A-theory, a rational equivalence between A-theory and algebraic K-theory, and Borel’s rational calculation of algebraic Ktheory. Thus the DIFF Alexander trick fails. To quote Hatcher: ‘DIFF’ is for ‘different’.

(Explain Hatcher’s homotopy equivalence DIF F (D^{n}) ≃ O(n) × C(D^{n−1}).)

8.3. Collars. Let M be a CAT manifold with boundary ∂M. The behavior of M near ∂M is always standard, i.e., like that of ∂M × I near ∂M × {0}.

Definition. A CAT collar of ∂M in M is a CAT embedding

c: ∂M × I ⟶ M

such that c(x, 0) = x and c(∂M × [0, 1)) is an open neighborhood of ∂M in M.

There is a space of collars Coll^{CAT} (∂M ⊂ M), which is contractible. We say that ∂M always has a collar, defined up to contractible choice.

To see this in the DIFF category, note that a collar c determines an inward pointing vector field X = c_{∗}(∂/∂t) in a neighborhood of ∂M in M. Then Coll^{CAT} (∂M ⊂ M) is homotopy equivalent to a space of ‘germs of collars’, which is homotopy equivalent to a space of ‘germs of inward pointing vector fields’ near ∂M, which is a convex space.

(Also in the PL and TOP cases ?)

8.4. Boundary conditions. There are several alternative definitions of the space of concordances that are useful. One is

C^{CAT} (M)′ = CAT (M × I rel M × {0}) .

Here we do not require the automorphisms to fix ∂M ×I. When M has no boundary, this is irrelevant. But the two definitions are homotopy equivalent also when M has nonempty boundary.

To see this, first assume the boundary of M comes equipped with a collar c: ∂M× I → M. Let U = c(∂M × [0, 1)) be the open collar neighborhood of the boundary, with closure Ū. Then there is a PL-homeomorphism of CAT manifold pairs

Φ(c): (M × I, M × {0}) ⟶^{≅} (M × I, M × {0} ∪ ∂M × I)

that is the identity on (M \ U) × I. It stretches

Ū × {0} ≅ ∂M × I × {0}

PL-homeomorphically to

Ū × {0} ∪ ∂M × I ≅ ∂M × (I × {0} ∪ {0} × I)

by introducing a corner along ∂M × (0, 0). At the same time it contracts

∂M × I ∪ Ū × {1} ≅ ∂M × ({0} × I ∪ I × {1})

PL-homeomorphically to

Ū × {1} ≅ ∂M × I × {1}

by smoothing the corner along ∂M ×(0, 1). This can all be done by using a standard PL-automorphism of I × I relative to {1} × I.

Conjugation by Φ(c) defines a homeomorphism

CAT (M × I rel M × {0}) ⟶^{≅} CAT (M × I rel M × {0} ∪ ∂M × I)

between the two models for C^{CAT} (M). In the DIFF case, Φ(c) is not smooth at the corners, but this does not matter if we require the concordances to fix a neighborhood of the indicated subspaces.

Now there are homotopy equivalences

C^{CAT} (M)′ ≃ C^{CAT} (M)′ × Coll^{CAT} (∂M ⊂ M) ≅

C^{CAT} (M) × Coll^{CAT} (∂M ⊂ M) ≃ C^{CAT} (M)

as desired. The middle homeomorphism takes a concordance Ψ and collar c to the conjugate concordance by means of the PL-homeomorphism Φ(c) defined by c.

8.5. Stabilization. The second definition of the concordance space lets us define stabilization maps:

C^{CAT} (M)′ ⟶ C^{CAT} (M × N)′

for CAT manifolds M and N, taking Ψ: M × I ≅ M × I to Ψ × 1_{N}. If Ψ fixes (a neighborhood of) M × {0} then Ψ × 1_{N} fixes (a neighborhood of) M × N × {0}. However, if Ψ also fixes ∂M × I and N has a boundary, it does not follow that Ψ × 1_{N} fixes ∂(M × N) × I.

The case N = I is of particular interest, and we write

Σ′: C^{CAT} (M)′ ⟶ C^{CAT} (M × I)′

for the stabilization map in this case. We can also iterate, and pass to the colimit.

Definition. Let

C^{CAT} (M)′ = hocolim_{Σ′} C^{CAT} (M × I^{j})′

be the stable CAT concordance space.

Via the homotopy equivalences C^{CAT} (M)′ ≃ C^{CAT} (M) one may also construct explicit stabilization maps

Σ: C^{CAT} (M) ⟶ C^{CAT} (M × I)

and form the homotopy colimit

C^{CAT} (M) = hocolim_{Σ} C^{CAT} (M × I^{j}) .

Then C^{CAT} (M)′ ≃ C^{CAT} (M) are two models for the same homotopy type. This is also called the stable CAT pseudoisotopy space, denoted P^{CAT} (M).

Given a pseudoisotopy Ψ: M × I → M × I, the stabilization Σ(Ψ): M × I × I → M × I × I multiplies Ψ by 1_{I} away from a collar of the boundary of M × I, but bends copies of M × I from being vertical to horizontal over the collar.

(Draw a picture !)

It is a surprising fact that C^{CAT} is a homotopy functor, i.e., only depends on the homotopy type of M (Hatcher, Waldhausen). Furthermore C^{T OP} (M) ≃ C^{P L}(M) for PL manifolds M by triangulation theory, while also stably C^{DIF F} (M) is different from the other two. By the Alexander trick, C^{T OP} (∗) ≃ C^{P L}(∗) ≃ ∗, but by our previous remark, C^{DIF F} (∗) has nontrivial rational homotopy.

8.6. Stability theorems. The interest in the stabilization maps stems from the result that the concordance spaces converge to their stable analogs in a controlled fashion. This is the stability theorem for CAT concordances.

Definition. An integer k is in the CAT concordance stable range for M if the stabilization maps

Σ: C^{CAT} (M × I^{j}) → C^{CAT} (M × I^{j+1})

are k-connected for all j ≥ 0. Then also

C^{CAT} (M) ⟶ C^{CAT} (M)

is k-connected.

Theorem (Igusa). If M is smooth and n = dim(M) ≥ max{2k + 7, 3k + 4}, then k is in the DIFF, PL and TOP concordance stable range for M. Hence

C^{CAT} (M) ⟶ C^{CAT} (M)

is at least k-connected when 2k + 7 ≤ n and 3k + 4 ≤ n.

(Check the TOP assertion.)

Hatcher (Ann. of Math., 1975) first claimed to have proved the PL stability theorem, from which Burghelea and Lashof (Trans. A.M.S., 1974) deduced a DIFF stability theorem. However, Hatcher’s argument contained a mistake regarding PL mapping cylinders. Later, K. Igusa (K-Theory, 1988, 355 pages) provided a careful proof of the DIFF stability theorem. From this one may deduce a PL stability theorem for smooth manifolds, by reversing the argument of Burghelea and Lashof, and according to Goodwillie (unpublished?), the stability range is the same.

Note that the known concordance stable range for a DIFF manifold M only depends on its dimension. There is a function

ϕ(n) = min{(n − 7)/2, (n − 4)/3}

such that π_{k}(C^{CAT} (M)) ≅ π_{k}(C^{CAT} (M)) for k < ϕ(n) with n = dim(M). The optimal concordance stable range may turn out to be much better; possibly ϕ(n) ∼ n.

8.7. Pseudoisotopy modulo isotopy. There is also a third model for the CAT concordance space that is useful.

A block isotopy in a CAT manifold M is a CAT automorphism Ψ of M × I that takes M × {0} to itself and M × {1} to itself. These form a topological group CAT (M × (I, 0, 1)), which later will be considered as the 1-simplices in the ‘block automorphism space’ { \atop CAT} (M). An isotopy in M is such a CAT automorphism of M × I that commutes with projection to I. This amounts to a path I → CAT (M), or a 1-simplex in the automorphism space CAT (M).

If M has boundary, we may or may not demand that these automorphisms leave ∂M×I pointwise fixed. The two cases yield homotopy equivalent results by ‘bending collars’, as above. For concreteness, let us not make any demands on the behavior along ∂M × I. This simplifies stabilization, but complicates the involution.

We get an inclusion of topological groups CAT (M)^{I} ⟶ CAT (M × (I, 0, 1)) and let

C^{CAT} (M)′′ = CAT (M × (I, 0, 1))/CAT (M)^{I}

be the (right) orbit space. There is a natural homotopy equivalence C^{CAT} (M)′ ≃ C^{CAT} (M)′′ which we now describe.

The CAT automorphisms of M act on the right on CAT (M × (I, 0, 1)) by precomposition in the M-coordinate. This agrees with the right group multiplication by the constant isotopy in CAT (M)^{I}. Pseudoisotopies are block isotopies that start from the identity, while block isotopies can be uniquely factored as a constant isotopy followed by a pseudoisotopy. Hence we can form the composite

C^{CAT} (M)′ ⟶ CAT (M × (I, 0, 1)) ⟶ CAT (M × (I, 0, 1))/CAT (M)

which is an isomorphism. The inclusion CAT (M) → CAT (M)^{I} is a homotopy equivalence, so the natural map of orbit spaces

CAT (M × (I, 0, 1))/CAT (M) ⟶ C^{CAT} (M)′′

is also a homotopy equivalence. Thus the three definitions are compatible. There are stabilization maps

Σ′′: C^{CAT} (M)′′ ⟶ C^{CAT} (M × I)′′

multiplying an automorphism by 1_{I}, and again C^{CAT} (M)′′ ≃ C^{CAT} (M).

This model for the CAT concordance space does not favor the top or bottom end of the cylinder to the other, and is suitable for describing the involution on C^{CAT} (M) that turns concordances ‘up-side down’. The definition also shows that concordances somehow interpolate between block automorphisms and genuine automorphisms.

8.8. The involution on concordances. Let M be a CAT manifold. There is a map

C^{CAT} (M)′′ ⟶^{−} C^{CAT} (M)′′

denoted Ψ ↦ Ψ̄. It conjugates a block isotopy Ψ: M × (I, 0, 1) ≅ M × (I, 0, 1) by the reflection R(x, t) = (x, 1 − t). Then ¯ Ψ̄ = Ψ, so we can say that the map is an involution, has order two, or defines a C_{2}-action on concordances.

The reflection in the I-coordinate is compatible with stabilization, hence there is also an involution

C^{CAT} (M)′′ ⟶^{−} C^{CAT} (M)′′

on stable CAT concordances.

When ∂M = ∅ we can translate the involution on C^{CAT} (M)′′ to the model C^{CAT} (M)′ = CAT (M × I rel M × 0) via the equivalences explained above. This yields the following description:

Let Ψ ∈ C^{CAT} (M)′ be a CAT concordance, and let ψ_{1} = r_{1}(Ψ) ∈ CAT (M) be its restriction to the top edge. Turning Ψ up-side down yields an automorphism RΨR of M × I that fixes M × {1}, but restricts to ψ_{1} along M × {0}. Let Ψ̄ be RΨR precomposed by ψ_{1}^{−1} in the M-coordinate. So

Ψ̄(x, t) = (RΨR)(ψ_{1}^{−1}(x), t) .

Then Ψ̄ ∈ C^{CAT} (M)′ as desired. Also r_{1}(Ψ̄) = r_{1}(Ψ)^{−1}, and ¯ Ψ̄ = Ψ so this defines an involution on C^{CAT} (M)′.

If ∂M ≠ ∅ we need to ‘bend collars’ again to make r_{1} well-defined, and work with C^{CAT} (M). In particular this is necessary for stabilization.

By stabilization, one obtains an involution on C^{CAT} (M). Up to homotopy the resulting involution can be seen (Vogell ?) to only depend on the tangential homotopy type of M, i.e., the homotopy type of the tangent bundle T M → M.

8.9. Partitions.

We introduce some not(at)ions from Waldhausen’s 1982 paper: ‘Algebraic Ktheory of spaces, a manifold approach.’

Let M be a compact CAT manifold, with boundary ∂M. By the corollary in §7.5, each h-cobordism (W; M, M′) admits a codimension 0 embedding in M × I that identifies M with M × {0}.

Definition. The CAT h-cobordism space H^{CAT} (M) is the space of compact codimension 0 CAT submanifolds W ⊂ M × I such that

(1) There is a neighborhood of ∂M × I whose intersection with W equals its intersection with M × [0, t] for some t ∈ (0, 1).

(2) W is an h-cobordism from M ≅ M × {0} to a CAT submanifold M′ ⊂ M × (0, 1), relative to the boundary ∂M.

To make sense of the topology on H^{CAT} (M), we may instead consider the simplicial set H^{CAT} (M)_{•} with q-simplices the locally trivial families of h-cobordisms as above, parametrized on Δ^{q}. Thus a q-simplex is a compact codimension 0 CAT submanifold V ⊂ M × I × Δ^{q} such that the projection

p: V ⊂ M × I × Δ^{q} ⟶ Δ^{q}

is a locally trivial bundle of CAT manifolds, and each fiber V_{ξ} = p^{−1}(ξ) over ξ ∈ Δ^{q} satisfies (1) and (2) above. Here t and M′ may vary with ξ.

We shall see that the h-cobordism space H^{CAT} (M) is a delooping of the concordance space C^{CAT} (M).

The space of collars Coll^{CAT} (M ⊂ M ×[0, 1)) is the space of embeddings c: M × I → M × [0, 1), that are the identity on M × {0}, and are standard near ∂M × I in a suitable sense. As noted before, this is a contractible space.

Given a collar c, the image c(M ×I) ⊂ M ×I is an h-cobordism W in H^{CAT} (M), in fact an s-cobordism. Hence there is a forgetful map

Coll^{CAT} (M) ⟶^{π} H^{CAT} (M)

taking a collar to the corresponding closed collar neighborhood, forgetting the choice of parametrization implicit in the collar. Its image is the connected component of H^{CAT} (M) consisting of h-cobordisms isomorphic to a cylinder, i.e., the component consisting of s-cobordisms.

The group C^{CAT} (M) of concordances acts freely on Coll^{CAT} (M) by precomposition. Given a concordance Ψ and collar c, the composite

c ∘ Ψ: M × I ⟶^{Ψ} M × I ⟶^{c} M × [0, 1)

is again a collar. In fact π is a principal fiber bundle with structure group C^{CAT} (M), for two collars with the same underlying collar neighbourhood differ by precomposition with a unique concordance. Hence there is a fiber sequence

C^{CAT} (M) ⟶ Coll^{CAT} (M) ⟶^{π} H^{CAT} (M)

with contractible total space, and thus a homotopy equivalence

ΩH^{CAT} (M) ≃ C^{CAT} (M) .

Proposition. The space of h-cobordisms H^{CAT} (M) is a (possibly non-connective) delooping of the concordance space C^{CAT} (M).

A path in H^{CAT} (M) is a family of h-cobordisms parametrized over Δ^{1} ≅ I, i.e., a concordance of h-cobordisms. Then the ends are CAT isomorphic, and it follows that the set of path components of H^{CAT} (M) is in bijection with the Whitehead group W h(π_{1}(M)) under the Whitehead torsion map.

Two abstract h-cobordisms (W; M, M′) and (W′; M′, M′′) can be composed by forming W′′ = W ∪_{M′} W′. It is not obvious how to define a pairing on H^{CAT} (M) in this way, but the Whitehead torsion map is additive with respect to this composition.

If two h-cobordisms W, W′ ∈ H^{CAT} (M) both agree with M × [0, t] near ∂M × I (same t), and W ∩ W′ = M × [0, t], we can try to add the h-cobordisms by forming W′′ = W ∪ W (union inside M × I). This gives a partially defined pairing on H^{CAT} (M), additively compatible with Whitehead torsion. After stabilizing M with respect to dimension (passing to colimits over M × I^{j}), the resulting stable h-cobordism space H^{CAT} (M) becomes a Γ-space and thus the underlying space of an infinite loop space, in this way. Its first delooping is called the CAT Whitehead space W h^{CAT} (M). It is connected, and π_{1}W h^{CAT} (M) ≅ W h(π_{1}(M)) in each category.

9. Block automorphism spaces

10. Classifying spaces

(BO(n), BCAT (n), B{ \atop CAT} (n).)

22 JOHN ROGNES

11. Vector bundles

(Tangent bundles, normal bundles, tubular neighborhood theorem, τ_{M} ≅ ν_{Δ} for Δ: M → M × M.)

12. Sphere bundles 13. Microbundles

(Milnor’s definition, stable normal bundles, TOP vs. Euclidean bundles, Kister-Mazur theorem, PL vs. Block bundles, Rourke-Sanderson, DIFF vs. vector bundles, BO ≃ BDIF F.)

14. Structure sets and structure spaces

(Simplicial category of families of (simple) homotopy equivalences h: N → M, with N DIFF or PL, M a TOP manifold.)

15. Triangulation theory

(Kirby-Siebenmann.)

16. Smoothing theory

(Cairns-Hirsch, Milnor, Hirsch-Mazur, Lashof-Rothenberg.)

17. Surgery theory

(Browder-Novikov-Sullivan-Wall.)

18. Some important homotopy types

(As in QS^{0}, G, BG, O, BO, C → SG → J, G/T OP, T OP/P L ≃ K(ℤ/2, 3), C → G/O → BSO, P L/O.)