LECTURES ON THE STABLE

PARAMETRIZED H-COBORDISM THEOREM

John Rognes

February 8th 2006

We first present work of Hatcher, Waldhausen and others, on the relation of automorphism groups of manifolds to concordance theory, h-cobordism spaces and algebraic K-theory of spaces. See [Ha78], [Wa87b] and [WW01] for more detailed surveys. Then we proceed to present the relation of the algebraic K-theory of spaces to algebraic K-theory of rings and to topological cyclic homology.

These are informal lecture notes. Corrections and comments are welcome.

1. High-dimensional geometric topology

1.1. Automorphisms of manifolds.

We consider the three geometric categories of manifolds, CAT = TOP (topological), PL (piecewise linear) or DIFF (smooth). There is a forgetful functor from DIFF to PD (piecewise differentiable), which is equivalent to PL, and another one from PL to TOP.

We are interested in the CAT automorphism groups CAT (M) = T OP (M), P L(M) or DIF F (M) of compact manifolds. When M is not closed, we assume that the automorphism is the identity on the boundary. These groups can be interpreted as a compact-open mapping space, a simplicial group and a C^{∞} mapping space, respectively, or uniformly as simplicial groups. The simplicial group [q] ↦ CAT_{q}(M) has q-simplices the set of diagrams

\xymatrix{M × Δ^{q} \ar[rr]^{ϕ} \ar[dr] &  & M × Δ^{q} \ar[dl] \\  & Δ^{q}}

where ϕ is a CAT isomorphism. So CAT_{0}(M) is the set of CAT automorphisms of M, while CAT_{1}(M) is the set of CAT isotopies between the automorphisms at the two ends. Let G(M) be the (simplicial) grouplike monoid of self-homotopy equivalences of M. We get homomorphisms

DIF F (M) → P L(M) → T OP (M) → G(M) ,

Typeset by AMS-TEX

starting at the category where M is a manifold.

We are interested in the moduli space of all CAT manifolds, which classifies (locally trivial) CAT manifold bundles. It is homotopy equivalent to

∐_{[M]} BCAT (M)

where M ranges over the isomorphism classes of CAT manifolds. Compare with how the Grassmann space of all finite-dimensional real vector spaces is homotopy equivalent to ∐_{n≥0} BO(n).

1.2. Surgery.

Surgery theory classifies CAT manifolds of a given homotopy type, focusing on a fixed manifold dimension n, but in a weaker sense than the moduli space. Let { \atop CAT}(M) be the CAT block automorphism group. It is the simplicial group with q-simplices the CAT isomorphisms

M × Δ^{q} ⟶^{ϕ} M × Δ^{q}

that restrict to a map (and thus a CAT isomorphism) ϕ|: M × F → M × F for each simplicial face F ⊂ Δ^{q}. So { \atop CAT}_{0}(M) is still the set of CAT automorphisms of M, but { \atop CAT}_{1}(M) is the set of CAT concordances, or pseudoisotopies, between the automorphisms at the two ends. There is an inclusion CAT (M) ⊂ { \atop CAT}(M). In the case G(M) ⊂ G̃(M), it is a homotopy equivalence. Surgery theory classifies manifolds up to block isomorphism, rather than up to isomorphism

Let CAT (n) = CAT (ℝ^{n}, 0) be the simplicial group of (germs of) CAT automorphisms of ℝ^{n} fixing the origin. Let G(n) be the simplicial grouplike monoid of homotopy equivalences of ℝ^{n} \ 0, or equivalently, of the unit sphere S^{n−1}. There are group homomorphisms

DIF F (n) → P L(n) → T OP (n) → G(n) ,

where the last map restricts a homeomorphism to the complement of 0. There are stabilization maps CAT (n) → CAT (n+1), taking an automorphisms to its product with the identity on ℝ, and we let CAT = colim_{n} CAT (n). The inclusions O(n) ⊂ DIF F (n) and O ⊂ DIF F are homotopy equivalences. Stably, G is homotopy equivalent to the ±1 components of Q(S^{0}) = Ω^{∞}(S) = colim_{n} Ω^{n}S^{n}.

Let G/CAT be the homotopy fiber of BCAT → BG. For each CAT manifold M with stable normal (micro-)bundle ν: M → BCAT the set of possible lifts ν′ of the underlying stable spherical fibration M → BG to a CAT (micro-)bundle is in bijection with the set of homotopy classes [M, G/CAT]. The parametrized form of this set of surgery problems is the mapping space Map(M, G/CAT). The surgery obstruction to realizing ν′ as the stable normal bundle of a CAT n-manifold, up to normal cobordism, is in the Wall L-group L_{n}^{s}(M) = L_{n}^{s}(π, w), which only depends on the fundamental group π = π_{1}(M) and the orientation character w = w_{1} : π → ℤ/2 of M. In its parametrized form, it is a map θ to Quinn’s surgery space L^{s}(M).

The Browder–Novikov–Sullivan–Wall surgery exact sequence, see [Wl70, 17.A], is then the long exact sequence associated to a fiber sequence

\frac{G(M)}{{ \atop CAT}(M)} ⟶ Map(M, G/CAT) ⟶^{θ} L^{s}(M) ,

where the left hand term is defined as the fiber of B{ \atop CAT}(M) → BG̃(M) ≃ BG(M). Here G/CAT and the L-groups are quite well understood. For example, by Sullivan and Kirby–Siebenmann [KS77], G/T OP is a product of Eilenberg–Mac Lane spaces when localized at 2 and homotopy equivalent to BSO when localized at an odd prime. For simply-connected M, the L-groups L_{n}^{s}(1) are ℤ for n = 4k, ℤ/2 for n = 4k + 2 and 0 otherwise, and these are also the homotopy groups of G/T OP. In this sense, surgery theory computes the space G(M)/{ \atop CAT}(M) of manifolds homotopy equivalent to M, up to concordance, or in other words, up to block isomorphism.

When the homotopy type of M is well understood, e.g., if M is aspherical (= a K(π, 1)), the homotopy type of BG(M) can also be worked out. In view of the fiber sequence

\frac{G(M)}{{ \atop CAT}(M)} → B{ \atop CAT}(M) → BG(M)

we will also grant that surgery theory identifies the component of M in the block moduli space

∐_{[M]} B{ \atop CAT}(M) .

To get at the honest moduli space, or the honest automorphisms groups of M, we wish to identify the following homotopy fiber

\frac{{ \atop CAT}(M)}{CAT (M)} → BCAT (M) → B{ \atop CAT}(M) ,

which measures the difference between blocked and unblocked automorphisms. For example, the difference between the isotopies in CAT_{1}(M) and the pseudoisotopies in { \atop CAT}_{1}(M) contributes to the lowest homotopy group of the fiber, namely π_{1}.

1.3. Concordances.

This difference is controlled by the concordance space C(M) = C^{CAT} (M), also known as the pseudoisotopy space. It has various homotopy equivalent definitions. One is

C(M) = CAT (M × I rel M × 0) ,

i.e., the CAT automorphisms ψ of M × I that restrict to the identity on (a neighborhood of) M × 0. A homotopy equivalent definition is

C(M) = CAT (M × I rel M × 0 ∪ ∂M × I) .

When M is not closed, the latter is properly contained in the former, but the two groups are homotopy equivalent, by conjugation with a chosen CAT isomorphism of pairs

(M × I, M × 0) ≅ (M × I, M × 0 ∪ ∂M × I) .

A third definition is

C(M) = \frac{{ \atop CAT}_{1}(M)}{CAT_{1}(M)}

where the free pseudoisotopies { \atop CAT}_{1}(M) = CAT (M × I, M × 0, M × 1) consists of the automorphisms of M × I that restrict to automorphisms of M × 0 and M × 1, respectively, and the free isotopies CAT_{1}(M) = CAT (M × I ↓ I) consists of the automorphisms of M × I that fiber over I, i.e., that commute with the projection to I. (Note: In this formulation of the definition, CAT_{1}(M) and { \atop CAT}_{1}(M) should be interpreted as the CAT automorphism spaces indicated, not as discrete sets.) The first definition maps isomorphically to { \atop CAT}_{1}(M)/CAT (M), and CAT (M) ⊂ CAT_{1}(M) is a homotopy equivalence.

By triangulation theory and Kirby–Siebenmann’s result that T OP (n)/P L(n) ≃ T OP/P L for n ≥ 5, the inclusion C^{P L}(M) ⊂ C^{T OP} (M) is a homotopy equivalence, for M a PL n-manifold for n ≥ 5. However, C^{DIFF} (M) is usually different from the other two concordance spaces.

There is a Hatcher spectral sequence

E_{s,t}^{1} = π_{t−1}(C^{CAT} (M × I^{s})) ⟹ π_{s+t}(\frac{{ \atop CAT}(M)}{CAT (M)}) ,

which is concentrated in the region s ≥ 0, t ≥ 1, see [Ha78]. We shall say more about its E^{2}-term below. Modulo differentials and extensions in this spectral sequence, the concordance spaces C(M × I^{s}) determine the difference between CAT (M) and

{ \atop CAT}(M).

We briefly explain the filtration. The group

π_{n}({ \atop CAT}(M)/CAT (M)) ≅ π_{n}({ \atop CAT}(M), CAT (M))

is realized as the CAT automorphisms ψ of M × I^{n} that restrict to automorphisms of M × ∂I^{n} that fiber over ∂I^{n}, modulo those that fiber over I^{n}. Let Ā_{n} be this space. It is contained in the larger group C̄_{n−1} of CAT automorphisms ψ that are only required to fiber over I^{n−1} × 0 ∪ ∂I^{n−1} × I. This larger group is homotopy equivalent to the concordance space C(M × I^{n−1}), in view of the third definition above. Restriction to M × I^{n−1} × 1 ≅ M × I^{n−1} identifies the quotient space C̄_{n−1}/Ā_{n} with (some path components of) Ā_{n−1}, so there is a Puppe fiber sequence

ΩĀ_{n−1} → Ā_{n} → C̄_{n−1} → Ā_{n−1} .

Roughly, we can identify loops in Ā_{n−1} with elements in Ā_{n} that fiber over I in the sense that they commute with the composite projection to I^{n} → I^{n}/I^{n−1} ≅ I. So these are the block automorphisms that can be “unblocked” in one of the parameters. Varying n one gets an exact couple and a spectral sequence, essentially filtering Ā_{n} by the extent to which ψ fibers over a quotient I^{n}/I^{s} ≅ I^{n−s}.

There is a stabilization (= suspension) map

σ: C(M) → C(M × J) ,

where J = [0, 1], like I, which in terms of the first definition of C(M) takes an automorphism ψ of M × I that is the identity on M × 0 to ψ × id on M × I × J, now fixing M × 0 × J. Identifying M × I × J ≅ M × J × I, we have an element of C(M × J). In terms of the second definition of C(M), the isomorphism of pairs mentioned yields a more curved image.

Iterating the stabilization, we obtain the stable concordance space 𝒞(M) = 𝒞^{CAT} (M), defined as the homotopy colimit

𝒞(M) = colim_{s} C(M × J^{s}) .

In the smooth category, CAT = DIF F, Igusa’s stability theorem for smooth pseudoisotopies (= concordances) asserts that the stabilization map σ: C(M) → C(M × J) is k-connected if

dim(M) ≥ max(2k + 7, 3k + 4) .

See [Ig88]. In other words, if n = dim(M) then σ is at least [(n−4)/3]-connected, for n ≥ 10, and at least [(n−7)/2]-connected for n < 10. It follows that the stabilization map σ^{∞}: C(M) → 𝒞(M) is k-connected if dim(M) ≥ max(2k + 7, 3k + 4). We call the range of degrees in which σ^{∞} is an isomorphism the concordance stable range of M.

It was shown by Burghelea and Lashof [BL77] that a PL stability theorem for concordances implies a DIFF stability theorem, with the known stable range doubled. This was relevant, as Hatcher was thought to have proved such a PL stability theorem as part of his parametrized h-cobordism theorem, in [Ha75]. Without Hatcher’s result, presumably it is possible to turn the argument around and deduce a PL stability theorem from Igusa’s DIFF theorem, but there does not seem to be a reference for this.

There is a canonical involution χ on C(M), which is relevant to the spectral sequence above. In terms of the third definition, it conjugates an isomorphism in CAT (M × I, M × 0, M × 1) by the flip

ρ: M × I ⟶^{≅} M × I

that reflects I about its mid-point. In symbols, χ(ψ) = ρψρ^{−1}. This preserves the subgroup CAT (M × I ↓ I).

In terms of the second definition, it maps ψ in CAT (M × I rel M × 0 ∪ ∂M × I) to its conjugate with the same flip ρ, followed by ϕ^{−1} × id, where ϕ is the image of ψ under the map r that restricts an automorphism to M × 1 ≅ M, and which appears in the fiber sequence

CAT (M × I) → C^{CAT} (M) ⟶^{r} CAT (M) .

The involution anti-commutes with stabilization, up to homotopy, so it induces an involution on 𝒞(M), at least at the level of homotopy groups, given as the colimit of (−1)^{s} times the involution on C(M × I^{s}). So C(M) → 𝒞(M) respects the involution.

((Describe d_{s,t}^{1} on an element in E_{s,t}^{1} = π_{t−1}C(M × I^{s}) of the form σ(ψ). It equals ψ + χ_{s−1}(ψ), where χ_{s−1} denotes the involution in C(M × I^{s−1}). In the stable range, E_{s,t}^{1} = π_{t−1}𝒞(M) and d_{s,t}^{1} takes σ(ψ) to ψ + (−1)^{s−1}χ(ψ), which stably equals σ(ψ) + (−1)^{s}χ(σ(ψ)). So the d^{1}-differential takes x to x + (−1)^{s}χ(x) for x ∈ E_{s,t}^{1}.))

In the concordance stable range (3t + 4 ≤ dim(M) + s or so, for CAT = DIF F),

E_{s,t}^{2} = H_{s}^{gp}(ℤ/2; π_{t−1}𝒞(M))

where ℤ/2 acts on π_{∗}𝒞(M) via minus the given involution. ((Should double-check the signs!))

If we localize away from 2, i.e., study homotopy groups with 2 inverted, then the spectral sequence collapses to the vertical axis, in the stable range, with

E_{0,t}^{2} = π_{t−1}(𝒞^{CAT} (M))[\frac{1}{2}]^{−} ≅ π_{t}(\frac{{ \atop CAT}(M)}{CAT (M)})[\frac{1}{2}] .

Here (−)^{−} indicates the coinvariants for minus the canonical involution, i.e., the (−1)-eigenspace for the involution.

1.4. h-Cobordisms.

A cobordism on a closed n-manifold M is a compact (n + 1)-manifold W with ∂W = M ⊔ M′, a disjoint union. It is an h-cobordism if the two inclusions

M ⊂ W ⊃ M′

are both homotopy equivalences. For example, W = M ×I is a trivial h-cobordism. ((Discuss the non-closed case.)) For connected M with fundamental group π = π_{1}(M), the Whitehead group

Wh(π) = Wh_{1}(π) is defined as the quotient group

Wh_{1}(π) = K_{1}(ℤ[π])/(±π)

of the first algebraic K-group

K_{1}(R) = GL_{∞}(R)^{ab} = \frac{GL_{∞}(R)}{[GL_{∞}(R), GL_{∞}(R)]}

of the integral group ring R = ℤ[π]. The Whitehead torsion of (W, M) is an element τ(W, M) ∈ Wh_{1}(π), see [Mi66], and the s-cobordism theorem says that the pair (W, M) is isomorphic to the trivial h-cobordism (M × I, M × 0) if and only if the Whitehead torsion τ(W, M) is zero, in which case we say that W is an s-cobordism. See [Mi65] for the simply-connected DIFF case.

In dimensions n ≥ 5 all elements of the group Wh_{1}(π) are realized by the torsion of h-cobordisms, so given an h-cobordism W from M to some M′, as above, there exists an inverse h-cobordism W′ from M′ to some M′′ with the negative torsion, so that the combined h-cobordism W ∪_{M′} W′ is an s-cobordism, and thus isomorphic to M ×I. So W can be embedded as a codimension 0 submanifold of M ×I, taking M to M × 0.

Let H(M) = H^{CAT} (M) be the space of h-cobordisms on M. As a set, we can let this consist of the set of codimension 0 submanifolds W of M × I that are h-cobordisms on M × 0. Instead of giving this a topology, we realize it as a simplicial set, with q-simplices H_{q}(M) the set of CAT bundles of h-cobordisms over Δ^{q}, contained as codimension 0 submanifolds of M × I × Δ^{q}. By a CAT bundle we mean a family that is locally trivial in the CAT sense.

There is a fiber sequence

C(M) → K(M) ⟶^{π} H(M)

where K(M) is the space of collars on M × 0 in M × I, i.e., of embeddings k: M × I → M × I relative to M × 0, and π takes a collar k to its image W = k(M × I), viewed as an (s-)cobordism on M. The fiber at the trivial h-cobordism is the group of automorphisms of M × I rel M × 0, i.e., the concordance group C(M).

The existence and essential uniqueness of collars (like tubular neighborhoods), asserts that the space of collars K(M) is contractible. Hence there is a homotopy equivalence

C(M) ≃ ΩH(M)

and the connected delooping BC(M), given by the bar construction on the simplicial group, realizes the base point component H^{s}(M) of H(M), i.e., the space of s-cobordisms on M, at least for n ≥ 5.

Again, there is a stabilization map

σ: H(M) → H(M × J)

taking W ⊂ M × I to something like

W × J ⊂ (M × I) × J ≅ (M × J) × I .

To be more precise, M × J is here a non-closed manifold, so some adjustment is needed to arrange that W × J is made standard near M × ∂J × I. The details are presented in Waldhausen’s “manifold approach” paper, [Wa82].

Iterating the stabilization, we obtain the stable h-cobordism space ℋ(M) = ℋ^{CAT} (M), defined as the homotopy colimit

ℋ(M) = colim_{s} H(M × J^{s}) .

The whole fiber sequence above stabilizes, so there is a homotopy equivalence

𝒞(M) ≃ Ωℋ(M)

and B𝒞(M) is a path component ℋ^{s}(M) of ℋ(M). Igusa’s DIFF stability theorem now says that the stabilization map σ: H(M) → H(M × J) is k-connected if

dim(M) ≥ max(2k + 5, 3k + 1) ,

and likewise for σ^{∞}: H(M) → ℋ(M).

The involution on C(M) carries over to one on BC(M), and on all of H(M), and corresponds very roughly to turning an h-cobordism up-side down.

The Hatcher spectral sequence can be rewritten as

E_{s,t}^{1} = π_{t}H^{s}(M × I^{s}) ⟹ π_{s+t}(\frac{{ \atop CAT}(M)}{CAT (M)})

for s ≥ 0, t ≥ 1, with

E_{s,t}^{2} = H_{s}^{gp}(ℤ/2; π_{t}ℋ^{s}(M))

in the stable range, where ℤ/2 acts on π_{∗}ℋ(M) by minus the canonical involution. Note that the groups π_{0}H(M × I^{s}) ≅ Wh_{1}(π) do not appear in the spectral sequence. (The decoration s is not related to the spectral sequence index s.) Inverting 2, the spectral sequence again collapses to the isomorphism

π_{∗}(ℋ^{s}(M))[\frac{1}{2}]^{−} ≅ π_{∗}(\frac{{ \atop CAT}(M)}{CAT (M)})[\frac{1}{2}] ,

in the concordance stable range.

Weiss and Williams [WW88] show that there is a CAT Whitehead spectrum Wh^{s}(M) with ℤ/2-action, such that Ω Wh^{s}(M) has underlying infinite loop space ℋ^{s}(M), and a map

\frac{{ \atop CAT}(M)}{CAT (M)} → Ω^{∞}(Eℤ/2_{+} ∧_{ℤ/2} Ω Wh^{s}(M))

that is as highly connected as the CAT concordance stable range, plus 1. The homotopy orbit spectral sequence for the target, with

E_{s,t}^{2} = H_{s}^{gp}(ℤ/2; π_{t}Ω Wh^{s}(M)) ,

agrees with the Hatcher spectral sequence in the stable range. We shall return to the Whitehead spaces in the following section.

2. The stable parametrized h-cobordism theorem

2.1. Simple maps of polyhedra.

A compact polyhedron K is the underlying topological space |X| of a finite simplicial complex X. The affine linear structure on each simplex determines a piecewise linear (= PL) structure on the polyhedron, i.e., a notion of what constitutes a PL map on K, and any subdivision of the simplicial complex determines the same PL structure, by definition. We call X a triangulation of K. More precisely, a polyhedron is a topological space equipped with such a PL structure.

A map f: K → L of compact polyhedra is piecewise linear (= PL) if there exist triangulations X and Y of K and L, respectively, in their PL structures, and a simplicial map X → Y whose geometric realization is f. A PL map f: K → L of compact polyhedra is simple if its point inverses are contractible, i.e., if for each p ∈ L the preimage f^{−1}(p) has the homotopy type of a point.

When L′ = L ∪_{e^{n}} e^{n+1} is obtained from L by attaching a PL (n + 1)-cell along an n-cell in its boundary, there is by definition an elementary expansion from L to L′ and an elementary collapse from L′ to L. Two compact polyhedra are said to be simple homotopy equivalent, in the sense of J.H.C. Whitehead, if they can be connected by a finite chain of elementary collapses and expansions. This is, in general, a more restrictive relation than that of homotopy equivalence. The elementary collapse can be realized by a simple map L′ → L, contracting e^{n+1} onto the e^{n} in its boundary. So simple/simply homotopy equivalent polyhedra can be connected by a chain of simple maps. Conversely, Marshall Cohen [Co67] proved that any simple map is a simple homotopy equivalence. Hence simple maps and elementary collapses/expansions generate the same equivalence relation on compact polyhedra, namely simple homotopy equivalence.

Let ℰ be the category of compact polyhedra L and PL maps f: L → L′. More generally, for a fixed compact polyhedron K let ℰ(K) be the category of compact polyhedra L containing K as a subcomplex, and morphisms from L to L′ the PL maps f: L → L′ that restrict to the identity on K. Let sℰ^{h}(K) ⊂ ℰ(K) be the subcategory with objects such that the inclusion K ⊂ L is a homotopy equivalence, and morphisms the simple maps f: L → L′ fixing K.

[figure: K; ≃; ≃; L; f; ≃_{s}; L′]

The set of simple homotopy types of compact polyhedra L containing K as a deformation retract can now be expressed as π_{0}(sℰ^{h}(K)). The operation (L, L′) ↦ L∪_{K} L′, union along K, makes this set an abelian group. The basic result of simple homotopy theory is that this group is isomorphic to the Whitehead group of the fundamental group of K, for connected K, so

π_{0}(sℰ^{h}(K)) ≅ Wh_{1}(π_{1}(K)) .

Note that when M is a compact PL manifold, each PL h-cobordism (W, M) defines an object of sℰ^{h}(M), and each PL homeomorphism W → W′ of h-cobordisms on M defines a morphism in this category. This transformation forgets the PL manifold structure, and only retains the underlying polyhedron. The Whitehead torsion of (W, M) is the image in Wh_{1}(π_{1}(M)) of the path component of this polyhedral object, under the isomorphism above (for M = K). This invariant suffices for the classical h- or s-cobordism theorem. To obtain a parametrized result, we wish to accommodate for the simplicial direction in the h-cobordism space H(M) = H^{P L}(M), so we also define simplicial categories ℰ_{•}, ℰ_{•}(K) and sℰ_{•}^{h}(K).

In simplicial degree q, the objects of ℰ_{q} are PL bundles p: P ↓ Δ^{q} of compact PL manifolds over Δ^{q}. Again, by a PL bundle we mean a family that is locally trivial in the PL sense. The morphisms of ℰ_{q} are PL maps of bundles over Δ^{q}.

\xymatrix{P \ar[rr] \ar[dr]_{p} &  & P′ \ar[dl]^{p′} \\  & Δ^{q}}

The objects of ℰ_{q}(K) are compact PL manifold bundles p: P ↓ Δ^{q} that contain the trivial bundle K ×Δ^{q} ↓ Δ^{q} as a PL subbundle. So these are commutative diagrams

[figure: K × Δ^{q}; P; p; Δ^{q}]

such that the pair (P, K × Δ^{q}) is locally trivial over Δ^{q} in the PL sense. The morphisms are PL bundle maps f: P → P′ over Δ^{q} that restrict to the identity on K × Δ^{q}. Let sℰ_{q}^{h}(K) ⊂ ℰ_{q}(K) be the subcategory where the objects satisfy that the inclusion K × Δ^{q} ⊂ P is a homotopy equivalence, and the morphisms satisfy that f: P → P′ is a simple map.

[figure: K × Δ^{q}; ≃; ≃; P; f; ≃_{s}; p; P′; p′; Δ^{q}]

The composite map K × Δ^{q} → Δ^{q} is the projection.

Here is then the manifold part of the stable parametrized h-cobordism theorem, in its PL form. As noted above, ℋ^{P L}(M) ≃ ℋ^{T OP} (M) for any PL manifold M. We always think of a category, like sℰ_{•}^{h}(M), as a space by way of its nerve.

Theorem. For each compact PL manifold M there is a homotopy equivalence

ℋ^{P L}(M) ≃ sℰ_{•}^{h}(M) .

More precisely, the forgetful maps H^{P L}(M × J^{s}) → sℰ_{•}^{h}(M × J^{s}) induce a natural chain of homotopy equivalences

ℋ^{P L}(M) = colim_{s} H^{P L}(M × J^{s}) ⟶^{≃} colim_{s} sℰ_{•}^{h}(M × J^{s}) ⟵^{≃} sℰ_{•}^{h}(M) .

We shall present a proof of this theorem in the second part of these lectures. In the published version of this theorem [Ha75] there is no mention of a simplicial direction in sℰ^{h}(M), but we do not know if such a claim can be proved.

2.2. Simple maps of simplicial sets.

Let X be a finite simplicial set. Its geometric realization can be triangulated, i.e., it is homeomorphic to the compact polyhedron underlying some finite simplicial complex, but not in any natural way. Conversely, each compact polyhedron can be triangulated by some finite simplicial complex, which is a finite simplicial set, but again, the association cannot be made natural. Some work is therefore required to continue the passage from spaces of PL manifolds to categories of polyhedra across to categories of simplicial sets.

A map f: X → Y of finite simplicial sets is a weak homotopy equivalence if the induced map |f|: |X| → |Y | of geometric realizations is a homotopy equivalence, and it is a simple map if the induced map |f|: |X| → |Y | has contractible point inverses, so that |f|^{−1}(p) is contractible for each p ∈ |Y |. Simple maps are weak homotopy equivalences.

Fix a finite simplicial set X. Let 𝒞(X) be the category of finite cofibrations X ⊂ Y, i.e., finite simplicial sets Y that contain X, and let the morphisms from Y to Y′ be the simplicial maps f: Y → Y′ that are the identity on X. Let s𝒞^{h}(X) ⊂ 𝒞(X) be the subcategory where the objects satisfy that X ⊂ Y is a weak homotopy equivalence, and the morphisms are the simple maps.

[figure: X; ≃; ≃; Y; f; ≃_{s}; Y′]

The non-manifold part of the stable parametrized h-cobordism theorem follows.

Theorem. Let X be a finite simplicial complex. There is a homotopy equivalence

s𝒞^{h}(X) ≃ sℰ_{•}^{h}(|X|)

induced by a natural chain of homotopy equivalences

s𝒞^{h}(X) ⟵^{≃} s𝒟^{h}(X) ⟶^{≃} s𝒟_{•}^{h}(X) ⟶^{≃} sℰ_{•}^{h}(|X|) .

Here s𝒟^{h}(X) ⊂ s𝒞^{h}(X) denotes a full subcategory of non-singular finite simplicial sets Y containing X, and s𝒟_{•}^{h}(X) is a simplicial version involving PL bundles. A simplicial set is non-singular if each non-degenerate simplex is embedded (not just its interior), and the geometric realization of such a simplicial set has a canonical PL structure.

We shall discuss its (nearly complete) proof in the third part of these lectures. The union along X defines a sum operation (Y_{1}, Y_{2}) ↦ Y_{1} ∪_{X} Y_{2} on 𝒞(X) and the

subcategory s𝒞^{h}(X). Following [Se74] and [Wa85, §1.8], one can deloop the nerve (= classifying space) of such categories by forming a simplicial category N_{•}𝒞(X) (resp. sN_{•}𝒞^{h}(X)) of sum diagrams. In simplicial degree q, N_{q}𝒞(X) is the category of q-tuples

(Y_{1}, Y_{2}, …, Y_{q})

in 𝒞(X), together with suitably compatible choices of sums

Y_{{i_{1},...,i_{s}}} ≅ Y_{i_{1}} ∪_{X} ⋯ ∪_{X} Y_{i_{s}}

for each subset {i_{1}, …, i_{s}} ⊂ {1, …, q}. For 0 < i < q the i-th face map replaces Y_{i} and Y_{i+1} with their sum Y_{i} ∪_{X} Y_{i+1}. For i = 0 it deletes Y_{1}, and for i = q it deletes Y_{q}. Likewise, sN_{q}𝒞^{h}(X) has as objects q-tuples of objects in 𝒞^{h}(X), plus sums, and the morphisms are q-tuples of simple maps Y_{i} → Y′_{i}, for 1 ≤ i ≤ q.

In low degrees, N_{0}𝒞(X) = ∗ is the trivial category with one object and one morphism, and N_{1}𝒞(X) ≅ 𝒞(X). So each object Y of 𝒞(X) corresponds to a 1simplex in (the geometric realization of) N_{•}𝒞(X) with ends at the base point ∗, i.e., it corresponds to a closed loop [Y]. There is a canonical map

ι: s𝒞^{h}(X) → ΩsN_{•}𝒞^{h}(X)

and Segal proves that it is a group completion map, making the monoid π_{0} a group. But π_{0}(s𝒞^{h}(X)) ≅ Wh_{1}(π_{1}(X)), the Whitehead group, and the sum operation ∪_{X} corresponds to the group operation in the Whitehead group, so s𝒞^{h}(X) is already group complete, and this ι is a homotopy equivalence [Wa85, 3.1.1].

Definition. Let X be a finite simplicial set. Its PL Whitehead space is defined as

Wh^{P L}(X) = sN_{•}𝒞^{h}(X) .

When X is a finite combinatorial manifold, so |X| is a compact PL manifold, we have obtained a chain of homotopy equivalences

ℋ^{P L}(|X|) ≃ Ω Wh^{P L}(X) .

2.3. Algebraic K-theory of spaces.

With X a finite simplicial set, as above, let ℛ(X) be the category of finite retractive spaces over X, i.e., finite simplicial sets Y containing X that come equipped with a structural retraction r: Y → X. A morphism from (Y, r) to (Y′, r′) is a map f: Y → Y′ that is the identity on X and commutes with the retractions: r′f = r.

[figure: X; Y; f; r; Y′; r′; X]

The composite map X → X is the identity.

For example, when X = ∗, ℛ(∗) is the category of finite based simplicial sets. For infinite X, the correct generalization is to ask that X ⊂ Y is a finite cofibration, i.e., Y contains only finitely many non-degenerate simplices that are not in X.

As usual, we are mostly interested in subcategories of ℛ(X). Let ℛ^{h}(X) be the full subcategory with objects (Y, r) such that the inclusion X ⊂ Y is a weak homotopy equivalence. Let hℛ(X) be the subcategory with morphisms f that are weak homotopy equivalences. Let sℛ(X) be the subcategory with morphisms f that are simple maps. Let sℛ^{h}(X) = sℛ(X) ∩ ℛ^{h}(X).

Forgetting the structural retractions defines a functor ℛ(X) → 𝒞(X). In a sufficiently fibrant situation, the restricted functor ℛ^{h}(X) → 𝒞^{h}(X) is a homotopy equivalence, because the space of possible retractions r in a pair X ⊂ Y is the fiber at id_{X} of the fibration Map(Y, X) → Map(X, X). When X ≃ Y, the fibration is a homotopy equivalence so the fiber is contractible. However, in the present simplicial setting, this argument does not immediately apply.

A morphism f: Y → Y′ is a cofibration if it is injective, and cofibrations are closed under base change (pushout), so this makes ℛ(X) a category with cofibrations in the sense of [Wa85, §1.1]. Similarly for ℛ^{h}(X). The morphisms of the subcategory hℛ(X) of ℛ(X) satisfy a gluing lemma for base change along a cofibration, so hℛ(X) ⊂ ℛ(X) is a category with cofibrations and weak equivalences in the sense of [Wa85, §1.2]. Similarly for sℛ(X) ⊂ ℛ(X) and sℛ^{h}(X) ⊂ ℛ^{h}(X).

The S_{•}-construction S_{•}ℛ(X) of [Wa85, §1.3] is then defined as a simplicial category. In degree q, the category S_{q}ℛ(X) has as objects the sequences of cofibrations

[figure: X; Y_{1}; …; Y_{q}]

in ℛ(X), together with choices of pushouts Y_{i,j} = X ∪_{Y_{i}} Y_{j} for all 1 ≤ i < j ≤ q. Here r_{i}: Y_{i} → X is the structural retraction, and Y_{i} → Y_{j} is a composite of cofibrations from the sequence above. For 0 < i ≤ q the i-th simplicial face map deletes Y_{i} from the sequence. For i = 0, the face is

[figure: X; Y_{1,2}; …; Y_{1,q} .]

In low degrees, S_{0}ℛ(X) = ∗ is the trivial category with one object and one morphism, and S_{1}ℛ(X) ≅ ℛ(X). Next, S_{2}ℛ(X) is the category of diagrams

[figure: X; Y_{1}; Y_{2}; X; Y_{1,2}; X]

where the square is a pushout and each composite maps X → X is the identity. So each object (Y, r) in ℛ(X) produces a closed loop [Y] in (the geometric realization of) S_{•}ℛ(X), since it is a 1-simplex with both ends at the base point. Furthermore, each extension Y_{1} → Y_{2} → Y_{1,2}, in the sense of retractive spaces over X, produces a planar triangle in S_{•}ℛ(X) that connects the loops associated to Y_{1}, Y_{2} and Y_{1,2}. So up to homotopy, the loop [Y_{2}] is homotopic to the loop sum of [Y_{1}] and [Y_{1,2}].

Let hS_{•}ℛ(X) be the simplicial subcategory of S_{•}ℛ(X) with morphisms in simplicial degree q the commutative diagrams

[figure: X; =; Y_{1}; ≃; …; Y_{q}; ≃; X; Y′_{1}; …; Y′_{q}]

in ℛ(X), where the vertical maps are weak homotopy equivalences. If f: Y → Y′ is a weak homotopy equivalence in ℛ(X), then the loops [Y] and [Y′] are homotopic in hS_{•}ℛ(X).

The algebraic K-theory of the space X, denoted A(X), is now defined [Wa85, §2.1] as the loop space of the geometric realization of the simplicial category hS_{•}ℛ(X):

A(X) = ΩhS_{•}ℛ(X) .

The identification hℛ(X) ≅ hS_{1}ℛ(X) induces a map

ι: hℛ(X) ⟶ A(X)

that takes the point corresponding to an object Y to the loop [Y]. This map converts each extension in ℛ(X) to a loop sum in A(X), up to preferred homotopy. In this sense, A(X) is a suitably universal receptacle for homotopy invariants of finite retractive spaces over X, that are homotopy additive in the sense that extensions go to loop sums. To make this more precise would involve discussing coherences for the preferred homotopies, which essentially comes down to recovering the definition of the S_{•}-construction.

2.4. The fiber sequence relating A(X) and Wh^{P L}(X).

The right fiber at X = Y of the inclusion functor sℛ(X) → hℛ(X) is the subcategory sℛ^{h}(X). The composite functor

sℛ^{h}(X) → sℛ(X) → hℛ(X)

is naturally equivalent to the constant functor at X = Y, but this diagram is not in general a homotopy fiber sequence. However, the generic fibration theorem [Wa85, §1.6] asserts that after applying the S_{•}-construction, there is indeed a homotopy fiber sequence

sS_{•}ℛ^{h}(X) → sS_{•}ℛ(X) → hS_{•}ℛ(X) .

Only the right hand term is a homotopy functor in X, so we apply this degreewise to the simplicial space X^{Δ^{•}}, given in simplicial degree q as X^{Δ^{q}} = Map(Δ^{q}, X). Then there is a homotopy fiber sequence

(∗)  sS_{•}ℛ^{h}(X^{Δ^{•}}) → sS_{•}ℛ(X^{Δ^{•}}) → hS_{•}ℛ(X^{Δ^{•}}) .

At the left of (∗), Waldhausen shows [Wa85, 3.1.7] that there is a chain of homotopy equivalences

sN_{•}𝒞^{h}(X) ⟶^{≃} sN_{•}𝒞^{h}(X^{Δ^{•}}) ⟵^{≃} sN_{•}ℛ^{h}(X^{Δ^{•}}) ⟶^{≃} sS_{•}ℛ^{h}(X^{Δ^{•}}) .

The first equivalence uses that s𝒞^{h}(X) is a homotopy functor in X. The second equivalence forgets the structural retractions. The third equivalence takes a q-tuple (Y_{1}, Y_{2}, …, Y_{q}) in the N_{•}-construction to the the cofibration sequence

[figure: X; Y_{1}; Y_{1} ∪_{X} Y_{2}; …; Y_{1} ∪_{X} ⋯ ∪_{X} Y_{q}]

in the S_{•}-construction.

In the middle of (∗), Waldhausen shows [Wa85, 3.2.1] that the functor

X ↦ sS_{•}ℛ(X^{Δ^{•}})

is a homology theory. The excision property follows from the hereditary behavior of simple maps, under restriction to subspaces of the target. It is nontrivial to verify that this property is preserved by the construction imposing homotopy invariance. There are many other incarnations of this functor in the theory, but the other contexts do not appear to be flexible enough to allow a proof of this very important property.

At the right of (∗), the homotopy invariance of A(X) shows that there is a homotopy equivalence

hS_{•}ℛ(X) ⟶^{≃} hS_{•}ℛ(X^{Δ^{•}}) .

In the special case X = ∗, it is easy to show that sS_{•}ℛ^{h}(∗) ≃ ∗, so the coefficient spectrum of the middle homology theory is sS_{•}ℛ(∗) ≃ hS_{•}ℛ(∗). Hence there is a homotopy fiber sequence [Wa85, §3.3]

Ω Wh^{P L}(X) → h(X; A(∗)) ⟶^{α} A(X) → Wh^{P L}(X)

where h(X; A(∗)) denotes the unreduced homology theory of X, with coefficient spectrum A(∗).

Theorem. For compact PL manifolds M there is a natural fiber sequence

ℋ^{P L}(M) → h(M; A(∗)) ⟶^{α} A(M) ,

where h(M; A(∗)) = Ω^{∞}(A(∗) ∧ M_{+}) and α is the assembly map.

There are involutions on A(M) and A(∗), depending on the tangent (micro- )bundle on M, so that the maps in this fiber sequence commute with the involutions. See [Vo85]. In this sense, the study of the canonical involution on C(M) or H(M), in the stable range, is translated into a problem about the involution in the algebraic K-theory of spaces.

2.5. A manifold approach.

In [Wa82], Waldhausen constructs a fiber sequence

ℋ^{CAT} (M) → h^{CAT} (M) → A(M) → Wh^{CAT} (M)

in each geometric category, which is homotopy equivalent to the one above in the PL case.

Let 𝒫(M) = 𝒫^{CAT} (M) be the simplicial set of CAT partitions of M × I, i.e., codimension 0 submanifolds W ⊂ M ×I that contain M ×0, but do not meet M ×1. If M has boundary, W is to be a product near ∂M. As usual, a q-simplex is a CAT bundle of such partitions over Δ^{q}. Let h𝒫(M) be the simplicial category with morphisms W → W′ the codimension 0 inclusions such that W′ is obtained from W by adjoining an h-cobordism. (The precise definition is a little more flexible, allowing for identity morphisms.)

For each m ≥ 0, let h𝒫^{m}(M) ⊂ h𝒫(M) be the connected component containing the object W obtained from M ×[0, ϵ] by attaching finitely many trivial m-handles, i.e., unknotted handles of index m, and let 𝒫^{m}(M) be the simplicial set of objects in h𝒫^{m}(M). It is possible to define lower stabilization maps

\underline{σ}: h𝒫^{m}(M) → h𝒫^{m}(M × J)

that increase the manifold dimension, and upper stabilization maps

\overline{σ}: h𝒫^{m}(M) → h𝒫^{m+1}(M × J)

that also increase the handle index. There is a homotopy fiber sequence

colim_{m,s} H^{CAT} (M × J^{s}) → colim_{m,s} 𝒫^{m}(M × J^{s}) → colim_{m,s} h𝒫^{m}(M × J^{s})

which after group completion (in the middle and on the right) yields the fiber sequence

ℋ^{CAT} (M) → h^{CAT} (M) → A(M) .

In this model, Wh^{CAT} (M) is defined as a delooping of ℋ(M), with respect to the (partially defined) sum of h-cobordisms. By the fiber sequence in Section 2.4, we know that h^{P L}(M) is the homology theory h(M; A(∗)).

The forgetful map from DIFF to PL induces a map of horizontal fiber sequences

\xymatrix{ℋ^{DIFF} (M) \ar[r] \ar[d] & h^{DIFF} (M) \ar[r]^{η} \ar[d] & A(M) \ar[r] \ar[d]^{≃} & Wh^{DIFF} (M) \ar[d] \\ ℋ^{P L}(M) \ar[r] & h(M; A(∗)) \ar[r]^{α} & A(M) \ar[r] & Wh^{P L}(M)}

where we now elucidate on the structure of h^{DIFF} (M). By smoothing theory the fiber of ℋ^{DIFF} (M) → ℋ^{P L}(M) is a homology theory in M, as is h(M; A(∗)), hence so is h^{DIFF} (M). The stabilization (= Goodwillie derivative at ∗) of ℋ^{DIFF} (M) is trivial, by Morlet’s disjunction lemma and the lamp-bulb trick [Ha78], while the stabilization of A(M) is Q(M_{+}), by the “vanishing of the mystery homology theory” [Wa87a], or alternatively by “relative K-theory equals relative T C” [Du97] and a calculation of the stabilization of T C, so in fact the homology theory h^{DIFF} (M), which equals its stabilization, must be Q(M_{+}) = h(M; S) = Ω^{∞}Σ^{∞}(M_{+}).

The stabilization map A(M) → Q(M_{+}), or the trace map [Wa79]

A(M) ⟶^{tr} Q(ΛM_{+}) → Q(M_{+}) ,

splits η, so in the DIFF case the fiber sequence becomes a product decomposition.

Theorem. For compact DIFF manifolds M there is a natural fiber sequence

ℋ^{DIFF} (M) → Q(M_{+}) ⟶^{η} A(M) ,

where the unit map η is split injective, and a homotopy equivalence

A(M) ≃ Q(M_{+}) × Wh^{DIFF} (M) .

Sverre Lunøe-Nielsen’s Master’s thesis [LN00] provides a definition of these manifold models that is strictly involutive, so that the stabilized fiber sequences are ℤ/2-equivariant.

3. Algebraic K-theory of structured ring spectra

3.1. A-theory as K-theory.

Suppose that X is a based and connected simplicial set. Let G = ΩX be the Kan loop group of X, a simplicial group with BG ≃ X, see [Wa96]. Let ℛ(∗, G) be the category with cofibrations of finite G-free based simplicial sets, i.e., those simplicial sets with G-action Z that can be obtained from ∗ by attaching G_{+} ∧ Δ^{n} along a G-map from G_{+} ∧ ∂Δ^{n}, finitely often. Let hℛ(∗, G) be the subcategory of weak homotopy equivalences, i.e., based G-maps Z → Z′ that are weak homotopy equivalences when considered as non-equivariant maps.

There is a pair of exact functors relating hℛ(X) and hℛ(∗, G), which induce a homotopy equivalence

A(X) = ΩhS_{•}ℛ(X) ≃ ΩhS_{•}ℛ(∗, G) = A(∗, G) .

See [Wa85, 2.1.3]. If G → EG → BG is a principal fiber bundle, with EG weakly contractible and X ≃ BG, then one functor takes a retractive space (Y, r) over BG to the pullback EG ×_{X} Y, viewed as a G-free retractive space over EG, and then collapses EG, to view (EG ×_{X} Y)/EG as a G-free retractive space over ∗, i.e., a G-free based space. The other functor takes a G-free based space Z to the Borel construction EG ×^{G} Z (often denoted EG ×_{G} Z), viewed as a retractive space over BG ≃ X. More care than usual is required about the finiteness conditions. See also [KR02, §8] for some variations.

The three functors id, C and Σ in the natural cofiber sequence

Z → CZ → ΣZ

induce self-maps id, C and Σ of A(∗, G), and by the additivity theorem [Wa85, 1.4], id + Σ ≃ C (loop sum). The natural homotopy equivalence CZ → ∗ makes C null-homotopic, so the suspension functor Σ induces minus the identity on A(∗, G). A similar argument can be made for A(X), interpreting cone and suspension in the category ℛ(X). Thus A(∗, G) is homotopy equivalent to the colimit (telescope) of

A(∗, G) ⟶^{Σ} A(∗, G) ⟶^{Σ} A(∗, G) ⟶^{Σ} …

The latter is also the algebraic K-theory of the colimit of the categories with cofibrations and weak equivalences

hℛ(∗, G) ⟶^{Σ} hℛ(∗, G) ⟶^{Σ} hℛ(∗, G) ⟶^{Σ} ….

This is essentially the localization with respect to suspension of the category of finite G-CW complexes, i.e., the category of finite G-CW spectra. Let S[G] = Σ^{∞}(G_{+}) be the suspension spectrum on G_{+}. The multiplication on G induces a pairing S[G]∧S[G] → S[G]. If S[G] is interpreted in a structured sense, e.g. as a symmetric spectrum, then this pairing makes it a symmetric ring spectrum.

The colimit of categories above is then equivalent to the category 𝒞_{S[G]} of finite cell S[G]-module spectra, i.e., the S[G]-module spectra that can be obtained from ∗ by attaching S[G]∧D^{n} along an S[G]-module map from S[G]∧S^{n−1}, finitely many times. More generally, for any symmetric ring spectrum (or S-algebra) B there is such a category 𝒞_{B} of finite cell B-modules. This is a category with cofibrations (the cellular inclusions) and weak equivalences (the stable equivalences, h), and the algebraic K-theory of B is defined as

K(B) = ΩhS_{•}𝒞_{B} .

Mandell [EKMM, VI] verifies that this definition agrees with the plus construction

K(B) ≃ ℤ × BGL_{∞}(B)^{+}

and K(HR) = K(R), where HR is the Eilenberg–Mac Lane spectrum of a ring R, and K(R) is Quillen’s algebraic K-theory.

Returning to the case B = S[ΩX], the approximation theorem [Wa85, 1.6] proves that

A(X) ≃ A(∗, ΩX) ≃ K(S[ΩX]) .

This is explained in [Wa84]. For example, with X = ∗ we have A(∗) ≃ K(S), the arithmetic of the sphere spectrum.

For DIFF manifolds M, Tore A. Kro’s Ph.D. thesis [Kr05, 4.3.13] constructs a strict involution on S[ΩM], as a map of structured ring spectra, that takes the contribution from the tangent bundle of M into account. Presumably Vogell’s involution on A(M) and the involution on K(S[ΩM]) induced by Kro’s involution agree.

3.2. The linearization map and rational calculations.

From the plus construction models, it follows that a k-connected map A → B of S-algebras, such that π_{0}(A) ≅ π_{0}(B), induces (k+1)-connected maps BGL_{∞}(A) → BGL_{∞}(B) and K(A) → K(B). So for X based and connected, with fundamental group π = π_{1}(X), the 2-connected map X → Bπ induces a 1-connected map S[ΩX] → ℤ[π], so the linearization map

L: A(X) = K(S[ΩX]) → K(ℤ[π])

is 2-connected.

When X = Bπ is aspherical, so that the universal cover X̃ is contractible, then the map S[ΩX] → ℤ[π] is a π_{0}-isomorphism and a rational equivalence, from which it follows that L: A(Bπ) → K(ℤ[π]) is also a rational equivalence.

In the simply-connected case,

L: A(∗) = K(S) → K(ℤ)

is a rational equivalence. By Borel’s rational calculation [Bo74]

K_{i}(ℤ) ⊗ ℚ ≅ \{ \begin{matrix}ℚ & if i = 0 or i = 4k + 1, k ≥ 1 \\ 0 & otherwise, \end{matrix}

so the same formula gives π_{i}A(∗) ⊗ ℚ. Thus

π_{i}H^{DIFF} (D^{s}) ⊗ ℚ ≅ \{ \begin{matrix}ℚ & if i = 4k, k ≥ 1 \\ 0 & otherwise \end{matrix}

for i in the DIFF concordance stable range for D^{s}, or one better. In this case { \atop DIFF}(D^{s}) is rationally trivial, and the involution on π_{i}ℋ^{DIFF} (∗)⊗ℚ has eigenvalue +1, so Farrell and Hsiang [FH78] could deduce that

π_{i}DIFF (D^{s}) ⊗ ℚ ≅ \{ \begin{matrix}ℚ & if i = 4k − 1, k ≥ 1 and s odd \\ 0 & otherwise \end{matrix}

for i in the DIFF concordance stable range for D^{s} i.e., for s ≥ max(2i + 7, 3i + 4) or so. This is notably different from the TOP and PL cases, since T OP (D^{s}) ≃ P L(D^{s}) ≃ ∗ for all s, by the Alexander (coning, or conning) trick.

Hsiang and Jahren [HJ82], [HJ83] obtain similar results for π_{i}DIFF (M) ⊗ ℚ when M is a lens space, or more generally, a spherical space form, in which case the maps M → Bπ and A(M) → A(Bπ) are as highly connected as the dimension of M.

By [Dw80], [Be86], A(X) has finite type (each π_{i}A(X) is a finitely generated abelian group) if X has finite type and π_{1}(X) is finite.

3.3. The cyclotomic trace map.

For each S-algebra B and prime p, there is a topological cyclic homology spectrum T C(B; p) and a natural map

trc_{B}: K(B) → T C(B; p)

called the cyclotomic trace map [BHM93]. Very briefly, T C(B; p) is built as a homotopy limit from the topological Hochschild homology T HH(B) and its cyclic fixed point spectra T HH(B)^{C_{p^{n}}}, over the category generated by restriction and Frobenius maps

R, F: T HH(B)^{C_{p^{n}}} → T HH(B)^{C_{p^{n−1}}} .

The trace map tr: K(B) → T HH(B) constructed by Bökstedt admits a preferred lift to the homotopy limit T C(B; p).

This target is a very fine relative invariant of algebraic K-theory, by [Du97] (building on previous work of Goodwillie and McCarthy). For any map A → B of connective S-algebras such that π_{0}(A) → π_{0}(B) is a nilpotent extension, i.e., a surjection with nilpotent kernel, the commutative square

\xymatrix{K(A) \ar[r] \ar[d] & K(B) \ar[d] \\ T C(A; p) \ar[r] & T C(B; p)}

becomes homotopy Cartesian after p-adic completion. In fact there is an integral model for T C that makes the square homotopy Cartesian before completion; this is the main result of [BGM].

For example, this result applies with A = S[ΩX] and B = ℤ[π], with π = π_{1}(X) as above, so that there is a homotopy Cartesian square

\xymatrix{A(X) \ar[r]^{L} \ar[d]_{trc_{X}} & K(ℤ[π]) \ar[d]^{trc_{ℤ[π]}} \\ T C(X; p) \ar[r]^{L} & T C(ℤ[π]; p)}

after p-adic completion. Here T C(X; p) is notation for T C(S[ΩX]; p).

Writing T HH(X) for T HH(S[ΩX]), there is a homotopy equivalence

T HH(X) ≃ Σ^{∞}(ΛX)_{+} ,

where ΛX = Map(S^{1}, X) is the free loop space on X, now thought of as a topological space rather than as a simplicial set. The circle S^{1} acts on both sides, by the Connes cyclic structure on the topological Hochschild construction, and by rotating the loops in ΛX. The homotopy equivalence above is C_{p^{n}}-equivariant for each finite subgroup C_{p^{n}} ⊂ S^{1}, when Σ^{∞} is formed in a complete S^{1}-universe. From this, [BHM93, §5] (see also [Ro02, 1.16]) deduce that there is a homotopy Cartesian square

\xymatrix{T C(X; p) \ar[r]^{α} \ar[d]_{β} & Σ^{∞}Σ_{+}(ES^{1} ×_{S^{1}} ΛX) \ar[d]^{trf_{S^{1}}} \\ Σ^{∞}(ΛX)_{+} \ar[r]^{1−Δ_{p}} & Σ^{∞}(ΛX)_{+}}

after p-adic completion. Here trf_{S^{1}} is the S^{1}-transfer map for the S^{1}-bundle ES^{1} × ΛX → ES^{1} ×_{S^{1}} ΛX (which is only defined up to homotopy), and Δ_{p}: ΛX → ΛX winds a free loop p times around itself. (More precisely, given a choice of map trf_{S^{1}} there exists a commuting homotopy for the square so that the induced map of vertical fibers is a homotopy equivalence.) The composite β ∘ trc is the trace map A(X) → Σ^{∞}(ΛX)_{+}, previously constructed by Waldhausen [Wa79].

In the special case X = ∗, the map β has the section S → A(∗) → T C(∗; p), and the homotopy fiber of trf_{S^{1}} : Σ^{∞}Σ_{+}ℂP^{∞} → S is the “stunted projective” spectrum ΣℂP_{−1}^{∞}, so there is a splitting

T C(∗; p) ≃ S ∨ ΣℂP_{−1}^{∞} .

after p-adic completion. Similarly, there is a splitting

A(∗) ≃ S ∨ Wh^{DIFF} (∗) .

Hence there are two homotopy Cartesian squares

\xymatrix{Wh^{DIFF} (∗) \ar[d]_{t̃rc} & A(∗) \ar[l] \ar[r]^{L} \ar[d]_{trc_{∗}} & K(ℤ) \ar[d]^{trc_{ℤ}} \\ ΣℂP_{−1}^{∞} & T C(∗; p) \ar[l]_{α} \ar[r]^{L} & T C(ℤ; p)}

and a (co-)fiber sequence

hofib(trc_{ℤ}) → Wh^{DIFF} (∗) \underrightarrow{t̃rc} ΣℂP_{−1}^{∞} .

3.4. Primary calculations.

We now analyze the outer corners of the last square, completed at a prime p. We deal with the cases p = 2, and p an odd regular prime.

The homotopy type of ℂP_{−1}^{∞} is about as complicated as that of the sphere spectrum, but some detailed information is available. In particular, its mod p cohomology is easily described as a module over the Steenrod algebra A.

H^{∗}(ℂP_{−1}^{∞}; ℤ/p) = ℤ/p{y^{n} | i ≥ −1}

with P^{i}(y^{n}) = \binom{n}{i}y^{n+(p−1)i} and β(y^{n}) = 0. For p = 2, we read P^{i} as Sq^{2i} and β as Sq^{1}, as usual. In this case the cohomology is a cyclic A-module:

H^{∗}(ℂP_{−1}^{∞}; ℤ/2) ≅ Σ^{−2}A/C

where C ⊂ A is the ideal generated by the admissible Sq^{I} of length ≥ 2, together with the Sq^{2i+1} of odd degree.

The homotopy type of K(ℤ) is predicted by the Lichtenbaum–Quillen conjecture, in the strong form given by Dwyer and Friedlander [DF85], comparing algebraic Ktheory to étale K-theory. These conjectures were verified at p = 2 in [RW00], using Voevodsky’s proof of the Milnor conjecture and the Bloch–Lichtenbaum spectral sequence. At odd primes these conjectures follow from the Bloch–Kato conjecture, which is now (apparently) proven by Rost and Voevodsky.

Completed at p = 2, there is a well-understood fiber sequence

\xymatrix{Σbo \ar[r] & K(ℤ) \ar[d] \\  & ju \ar@{-->}[ul]}

where ju is the connective complex image-of-J spectrum and bo is the connected real K-theory spectrum.

Recall that a prime p is regular if it does not divide the order h_{p} of the ideal class group Cl(ℚ(ζ_{p})) of the p-th cyclotomic field ℚ(ζ_{p}), or equivalently, if it does not divide the numerator of any divided Bernoulli number B_{n}/n. For irregular primes p, the p-torsion in Cl(ℚ(ζ_{p})) is quite well understood by Iwasawa’s Main Conjecture, which was proved in this case by Mazur and Wiles, but the precise homotopy type of the étale K-theory of ℤ[1/p] is not yet explicitly understood (as far as I know), so the same goes for the p-completed homotopy type of K(ℤ).

Completed at an odd regular prime p the situation is easier; there is a splitting

K(ℤ) ≃ j ∨ Σ^{5}ko

where j is the connective image-of-J spectrum and ko the connective real K-theory spectrum.

The homotopy type of T C(ℤ; p) was computed for p odd by Bökstedt and Madsen in [BM94] and [BM95], and for p = 2 in [Ro99a] and [Ro99b]. Its connective cover is homotopy equivalent to K(ℤ_{p})_{p}. For p = 2 there is a tower of maximally non-trivial fiber sequences

\xymatrix{Σju \ar[r] & K^{red}(ℤ_{2}) \ar[r] \ar[d] & K(ℤ_{2}) \ar[d] \\  & Σ^{3}ku \ar@{-->}[ul] & ju \ar@{-->}[ul]}

where ku is the connective complex K-theory spectrum. At odd primes there is a splitting

K(ℤ_{p}) ≃ j ∨ Σj ∨ Σ^{3}ku .

For p = 2, the calculation [Ro99b] contains enough information to compute trc: K(ℤ) → T C(ℤ; 2) and its homotopy fiber. There is an explicit fiber sequence

\xymatrix{Σ^{3}ko \ar[r] & hofib(trc_{ℤ}) \ar[d] \\  & Σ^{−2}ku \ar@{-->}[ul]}

obtained in [Ro02, 3.13]. At odd regular primes, [Ro03, 3.3] uses Tate–Poitou duality to identify trc: K(ℤ) → T C(ℤ; p) and its homotopy fiber, which splits as

hofib(trc_{ℤ}) ≃ j ∨ Σ^{−2}ko .

For p = 2, we can proceed to determine the mod 2 cohomology of Wh^{DIFF} (∗) as a module over the Steenrod algebra A. Recall the left ideal C ⊂ A from above. It contains the ideal A{Sq^{1}, Sq^{3}} ⊂ A.

Theorem [Ro02, 4.5]. The mod 2 spectrum cohomology of Wh^{DIFF} (∗) is the unique non-trivial extension of A-modules

0 → Σ^{−2}C/A{Sq^{1}, Sq^{3}} → H^{∗}(Wh^{DIFF} (∗); ℤ/2) → Σ^{3}A/A{Sq^{1}, Sq^{2}} → 0 .

In [Ro02, §5] the Adams spectral sequence

E_{2}^{s,t} = Ext_{A}^{s,t}(H^{∗}(Wh^{DIFF} (∗); ℤ/2), ℤ/2) ⟹ π_{t−s} Wh^{DIFF} (∗)_{2}

is determined in degrees t − s ≤ 21, up to some extensions in degrees ≥ 19, in part by comparison with the Adams spectral sequence for ℂP_{−1}^{∞}.

As a corollary, a geometrically defined map of spaces G/O → Ω Wh^{DIFF} (∗) is found to be precisely 8-connected, after 2-completion.

For odd regular primes p, we show that there is a commutative diagram

\xymatrix{Σc \ar[r]^{Σf} \ar[d]_{g} & S^{1} \ar[d]^{i_{1}} \\ Wh^{DIFF} (∗) \ar[r] & ΣℂP_{−1}^{∞}}

where i_{1} is inclusion of the 1-cell in ΣℂP_{−1}^{∞}, and c is the cokernel-of-J spectrum defined by the fiber sequence

c ⟶^{f} S ⟶^{e} j

where e induces the Adams e-invariant. The lift g is split injective:

Wh^{DIFF} (∗) ≃ Σc ∨ (Wh^{DIFF} (∗)/Σc)

Let t: ΣℂP^{∞} → S be the restricted S^{1}-transfer, i.e., the composite of ΣℂP^{∞} ⊂ ΣℂP_{+}^{∞} and trf_{S^{1}}. Splitting the suspended c → S → j fiber sequence off from that containing t̃rc leaves a fiber sequence

Σ^{−2}ko → (Wh^{DIFF} (∗)/Σc) → hofib(t) .

Here the right hand map induces an isomorphism from the p-torsion in the homotopy of (Wh^{DIFF} (∗)/Σc) to the p-torsion in the homotopy of hofib(t).

Theorem [Ro03, 3.8]. When completed at an odd regular prime p, Wh^{DIFF} (∗) splits off a suspended copy of the cokernel-of-J spectrum, and the torsion homotopy of the remainder is isomorphic to the torsion homotopy of the fiber of the restricted S^{1}-transfer map

t: ΣℂP^{∞} → S .

In symbols,

π_{∗}(Wh^{DIFF} (∗)) ≅ π_{∗}(Σc) ⊕ π_{∗}(Wh^{DIFF} (∗)/Σc)

and

tors π_{∗}(Wh^{DIFF} (∗)/Σc) ≅ tors π_{∗}(hofib(t)) .

In fact, we can also split the suspended infinite quaternionic projective space ΣℍP^{∞} off from (Wh^{DIFF} (∗)/Σc), but this makes for a more complicated statement.

Corollary [Ro03, 4.9]. The p-torsion in π_{∗} Wh^{DIFF} (∗) begins:

For p = 3, ℤ/3{Σβ_{1}} in degree 11 and ℤ/3{Σα_{1}β_{1}} in degree 14.

For p ≥ 5 regular, a ℤ/p in degrees ∗ = 2n for m(p−1) < n < mp and 1 < m < p, except in degree 2p^{2} − 2p − 2, followed by ℤ/p{Σβ_{1}} in degree 2p^{2} − 2p − 1 and a group of order p^{2} in degree 2p^{2} − 2p + 2.

Corollary [Ro03, 6.4].

For p = 3 and s ≥ 34, π_{10}H^{DIFF} (D^{s}) and either π_{9}DIFF (D^{s}) or π_{9}DIFF (D^{s+1}) contain a copy of ℤ/p.

For p ≥ 5 regular and s ≥ 12p − 5, π_{4p−3}H^{DIFF} (D^{s}) and either π_{4p−4}DIFF (D^{s}) or π_{4p−4}DIFF (D^{s+1}) contain a copy of ℤ/p.

A more complete calculation, incorporating the results from surgery theory on { \atop DIFF}(D^{s}), should be possible.

One can also make a cohomological analysis. For p = 3 there is a splitting of A-modules

H^{∗}(Wh^{DIFF} (∗); ℤ/3) ≅ H^{∗}(Σc; ℤ/3) ⊕ H^{∗}(ΣℍP^{∞}; ℤ/3) ⊕ Σ^{−2}C/A{β, Q_{1}}

where C ⊂ A is the annihilator ideal of y^{−1}, spanned by all admissible monomials in A except 1 and the P^{i} for i ≥ 1.

For p ≥ 5 there is a non-trivial extension with more terms. See [Ro03, §5.]

4. Some open or unresolved problems

(1) Establish a PL concordance stable range (Burghelea).

(2) Study concordance spaces in a meta-stable range (Goodwillie, Meng, Weiss), i.e., study Ω^{s} of the homotopy fiber of fiber of σ: C(M ×J^{s}) → C(M ×J^{s+1}) as s grows.

(3) Compare Vogell’s involution on A(M) with Kro’s involution on K(S[ΩM]), [Vo85], [Kr05].

(4) Identify the involution on T C(X), in the model from [BHM93], making A(X) → T C(X) involutive (Kro, Dundas).

(5) Can one recover or improve on the explicit results of Hatcher–Wagoner on π_{0}C(M) and Igusa on π_{1}C(M) in the DIFF category, derived using Cerf theory, see [Ha78, §3], using T C(X; p)?

(6) Compute A(S^{1}) = K(S[t, t^{−1}]) (Hüttemann, Klein, Vogell, Waldhausen, Williams, Madsen, Hesselholt, Schlichtkrull). By Farrell–Jones [FJ91], this determines A(M) for all compact Riemannian M with everywhere nonpositive sectional curvature.

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Max-Planck-Institut für Mathematik, Bonn, Germany

E-mail address: rognes@math.uio.no