ALGEBRAIC MODELS FOR TOPOLOGICAL

CYCLIC HOMOLOGY AND WHITEHEAD

SPECTRA OF SIMPLY-CONNECTED SPACES

Kathryn Hess and John Rognes

22.01.02

Introduction

((TBW))

1. Topological cyclic homology of spaces

Let X be a space and p a prime. The topological cyclic homology of X at p is a spectrum T C(X; p) constructed by Bökstedt, Hsiang and Madsen in [BHM]. It is the target of the so-called cyclotomic trace map trc_{X}: A(X) → T C(X; p) from Waldhausen’s algebraic K-theory of the space X [W1, §2.1], and provides a useful approximation to A(X).

By [BHM, §5] there is a homotopy Cartesian square of spectra

(1.1)  \xymatrix{T C(X; p) \ar[r]^{β} \ar[d]^{α} & ΛX_{+} \ar[d]^{1−Δ_{p}} \\ Σ(ES_{+}^{1} ∧_{S^{1}} ΛX_{+}) \ar[r]^{_{trf_{S^{1}}}} & ΛX_{+}}

(for a suitable choice of commuting homotopy for the square). Here, and elsewhere, we shall often think of a based space Y as a spectrum by way of its suspension spectrum Σ^{∞}Y, without showing this in the notation. This applies in particular to three of the corners in the diagram (1.1) above.

The notation is that ΛX = Map(S^{1}, X) is the free loop space of X, the circle S^{1} acts on ΛX by rotating the loops, ES^{1} is a free contractible S^{1}-space, and ES_{+}^{1} ∧_{S^{1}} ΛX_{+} ≅ (ES^{1} ×_{S^{1}} ΛX)_{+} is the S^{1}-equivariant Borel construction, or S^{1}- homotopy orbits, for this S^{1}-action on ΛX. The p-th power map Δ_{p}: ΛX → ΛX winds a loop p times around itself.

The stably defined map trf_{S^{1}} : Σ(ES_{+}^{1} ∧_{S^{1}} ΛX_{+}) → ΛX_{+} is the (dimensionshifting) S^{1}-transfer map associated to the principal S^{1}-bundle ES^{1} × ΛX → ES^{1} ×_{S^{1}} ΛX. We shall review a description of this map in terms of S^{1}-equivariant stable homotopy theory in the following section, as the map (2.1).

There is a unit map

X_{+} ⟶^{η_{X}} A(X) ⟶^{trc_{X}} T C(X; p)

Typeset by AMS-TEX

which extends to a T C-assembly map

T C(∗; p) ∧ X_{+} ⟶^{a_{X}^{T C}} T C(X; p) .

Choosing a base point ∗ in X, the two maps T C(∗; p) ≅ T C(∗; p)∧∗_{+} → T C(∗; p)∧ X_{+} and X_{+} ≅ ∗_{+} ∧ X_{+} → T C(∗; p) ∧ X_{+} agree on ∗_{+} = S^{0}, hence induce a map

T C(∗; p) ∨ X ≅ T C(∗; p) ∪_{S^{0}} X_{+} ⟶ T C(∗; p) ∧ X_{+} .

On the other hand, the projection X → ∗ induces T C(X; p) → T C(∗; p), and β: T C(X; p) → ΛX_{+} can be followed with the map ev_{1} : ΛX_{+} → X_{+} evaluating a free loop f: S^{1} → X at the base point 1 ∈ S^{1}, as well as the map X_{+} → X collapsing ∗_{+} to a point. The product of these two maps is a map T C(X; p) → T C(∗; p) × X, and the composite

(1.2)  T C(∗; p) ∨ X → T C(∗; p) ∧ X_{+} ⟶^{a_{X}^{T C}} T C(X; p) → T C(∗; p) × X

is a weak equivalence of spectra. Hence T C(∗; p)∨X splits off from T C(X; p), with the inclusion factoring through the T C-assembly map a_{X}^{T C}. We write

\frac{T C(X; p)}{T C(∗; p) ∨ X}

for the homotopy cofiber of this inclusion.

When X = ∗ the unit map S^{0} = ∗_{+} → T C(∗; p) splits β: T C(∗; p) → ∗_{+} = S^{0}, so T C(∗; p) ≃ S^{0} ∨ hofib(trf_{S^{1}}) (after implicit p-completion). In this case there is a homotopy equivalence hofib(trf_{S^{1}}) ≃ ΣℂP_{−1}^{∞}, where ℂP_{−1}^{∞} is a stunted complex projective spectrum with one cell in dimension 2n for each −1 ≤ n < ∞. (See (4.6) below.) We may thus write T C(∗; p) ≃ S^{0} ∨ ΣℂP_{−1}^{∞} and identify the homotopy cofiber of T C(∗; p) ∨ X → T C(∗; p) ∧ X_{+} as ΣℂP_{−1}^{∞} ∧ X.

2. The (dimension-shifting) G-transfer map

Quite generally, let G be a compact Lie group and let ad G be the adjoint representation on its tangent space at the identity. Adding a disjoint base point, G_{+} is a finite based G-CW complex, so its G-equivariant Spanier–Whitehead dual DG_{+} agrees with the functional dual F (G_{+}, S^{0}). Here S^{0} = Σ_{G}^{∞}S^{0} is the G-equivariant sphere spectrum, and F (Y, Z) denotes the G-equivariant mapping spectrum between two G-spectra Y and Z. By G-equivariant Spanier–Whitehead duality [LMS, III] ((check ?)), there is a G-equivariant homotopy equivalence DG_{+} ≃ G_{+}∧S^{− ad G}.

Consider the principal G-bundle π: G → G/G = ∗. Adding a disjoint base point and taking functional duals, we obtain a G-equivariant spectrum map

Dπ: S^{0} = D(∗_{+}) → D(G_{+}) ≃ G_{+} ∧ S^{− ad G} .

This is the fiberwise building block for the G-transfer.

Let p: E → B be a principal G-bundle. The (dimension-shifting) G-transfer associated to p is the map

trf_{G}: (E_{+} ∧ S^{ad G})/G ⟶ (E_{+} ∧ G_{+})/G ≅ E_{+}

defined as Σ^{ad G}(1∧Dπ)/G. The right-hand identification takes the G-orbit of e∧g to eg.

Specialized to the case G = S^{1} the adjoint representation is trivial, so S^{ad G} = S^{1} with trivial S^{1}-action. We have a principal S^{1}-bundle E = ES^{1} × ΛX → ES^{1} ×_{S^{1}} ΛX = B. The associated S^{1}-transfer is the map

(2.1) trf_{S^{1}} : Σ(ES_{+}^{1} ∧_{S^{1}} ΛX_{+}) = (ES_{+}^{1} ∧ ΛX_{+} ∧ S^{1})/S^{1}

⟶ ES_{+}^{1} ∧ ΛX_{+} ⟶_{≃}^{c∧1} ΛX_{+} .

The final homotopy equivalence uses the map c: EG_{+} → S^{0} collapsing EG to a point, in the case G = S^{1}.

To work with the S^{1}-transfer it will be convenient to use the Adams isomorphism to pass from S^{1}-orbit spectra, as above, to S^{1}-fixed point spectra, as in what follows. Working in the general context, this is a natural equivalence

τ̃: (EG_{+} ∧ Y ∧ S^{ad G})/G ⟶^{≃} [i_{∗}(EG_{+} ∧ Y)]^{G}

for naive G-spectra Y. See [GM, 5.3]. Here i_{∗} is the functor extending a naive G-spectrum to a genuine one.

Applied to the map (1 ∧ Dπ): Y → Y ∧ DG_{+} ≃ Y ∧ G_{+} ∧ S^{− ad G}, this yields a commutative square

\xymatrix{(EG_{+} ∧ Y ∧ S^{ad G})/G \ar[r]^{^{trf^{G}}} \ar[d]^{τ̃}_{≃} & (EG_{+} ∧ Y ∧ G_{+})/G \ar[d]^{τ̃}_{≃} \\ [i_{∗}(EG_{+} ∧ Y)]^{G} \ar[r]^{[1∧Dπ]^{G}} & [i_{∗}(EG_{+} ∧ Y ∧ DG_{+})]^{G} .}

Hence we are entitled to think of the G-transfer map in terms of the lower horizontal map. This in turn appears as the upper horizontal map in the following commutative square:

\xymatrix{[i_{∗}(EG_{+} ∧ Y)]^{G} \ar[r]^{[1∧Dπ]^{G}} \ar[d]^{[c∧1]^{G}} & [i_{∗}(EG_{+} ∧ Y ∧ DG_{+})]^{G} \ar[d]^{[c∧1]^{G}}_{≃} \\ [i_{∗}(Y)]^{G} \ar[r]^{[1∧Dπ]^{G}} & [i_{∗}(Y ∧ DG_{+})]^{G} .}

The right hand vertical map is a homotopy equivalence, since using the Spanier– Whitehead duality equivalence [Z ∧ DG_{+}]^{G} ≃ F (G_{+}, Z)^{G} ≅ Z it is equivalent to the map c ∧ 1: EG_{+} ∧ Y → Y, which clearly is a homotopy equivalence.

Therefore we may also think of the G-transfer map in terms of the composite around the left and lower sides of this diagram. Summarizing, we have proved the following:

Proposition 2.2. The (dimension-shifting) G-transfer map

trf_{G}: (EG_{+} ∧ Y ∧ S^{ad G})/G ⟶ (EG_{+} ∧ Y ∧ G_{+})/G ≃ Y

is naturally homotopy equivalent to the map obtained by applying [i_{∗}((−) ∧ Y)]^{G} to the composite map

Dπ ∘ c: EG_{+} → S^{0} → DG_{+} .

To study the homotopy cofiber of the S^{1}-transfer it is therefore natural to consider the homotopy cofiber of the G-equivariant spectrum map Dπ ∘ c: EG_{+} → DG_{+} for G = S^{1}.

3. Filtered Tate cohomology

Let ẼG = S^{0} ∪_{c} C(EG_{+}) be the homotopy cofiber of the collapse map c, so that there is a cofiber sequence

EG_{+} ⟶^{c} S^{0} ⟶ ẼG ⟶ ΣEG_{+} .

In [GM, 9.5] Greenlees and May consider a bi-infinite filtration {F^{n}}_{n} of ẼG, for n ∈ ℤ (with a change of notation). With d equal to the dimension of G, they let

F^{n} = \{ \begin{matrix}S^{0} ∪ C(EG_{+}^{(n−1)}) & for n > 0, \\ S^{0} & for −d ≤ n ≤ 0, \\ D(F^{−n−d}) & for n < −d. \end{matrix}

Here we are thinking of EG as a G-CW complex, and EG^{(n−1)} is its (n−1)-skeleton. Using a telescope construction, they may assume that this is a filtration of ẼG, in the sense that the natural maps F^{n−1} → F^{n} are cellular inclusions of G-CW spectra for every n, and F^{∞} = colim_{n} F^{n} ≃ S^{0} ∪_{c} C(EG_{+}) = ẼG. The filtration is indexed much like a skeleton filtration, in that the quotient F^{n}/F^{n−1} is homotopy equivalent to a wedge sum of copies of G_{+} ∧ S^{n}, for each n. For a ≤ b ≤ ∞ we let F_{a}^{b} = F^{b}/F^{a−1}, which we think of as the stunted version of ẼG having G-cells of dimensions n = a, …, b.

We may assume EG_{+}^{(0)} ≃ G_{+}, in which case there is a cofiber sequence

G_{+} ⟶^{π_{+}} S^{0} ⟶ F^{1}

with π: G → G/G = ∗ as before. The Spanier–Whitehead dual cofiber sequence is

F^{−1−d} ⟶ S^{0} ⟶^{Dπ} DG_{+} .

Hence there is a square of horizontal and vertical cofiber sequences

[figure: F^{−1−d}; F^{−1−d}; EG_{+}; c; S^{0}; Dπ; F^{∞}; EG_{+}; Dπ∘c; DG_{+}; F_{−d}^{∞} .]

Thus for any naive G-spectrum Y there is a cofiber sequence

(3.1)  [i_{∗}(EG_{+} ∧ Y)]^{G} ⟶^{trf_{G}} [i_{∗}(DG_{+} ∧ Y)]^{G} ⟶ [i_{∗}(F_{−d}^{∞} ∧ Y)]^{G}

providing us with a convenient model for the homotopy cofiber of the G-equivariant transfer map.

4. Equivariant cohomology

Now let G = S^{1} and d = 1.

In this case we can relate the cohomology of F_{−1}^{∞} ∧ Y to equivariant cohomology, following the presentation in [GM, §14]. Moreover, we can precisely relate the cofiber sequence

EG_{+} ∧ Y ⟶^{Dπ∘c∧1} DG_{+} ∧ Y ⟶ F_{−1}^{∞} ∧ Y ⟶ ΣEG_{+} ∧ Y

to the Connes exact sequence. In the case Y = ΛX_{+} this elaborates on the results of Jones [J] and others. Our presentation differs from that in [GM] in that we focus on truncating F^{∞} = ẼG below the (−1)-cell rather than below the 0-cell, so as to capture the S^{1}-transfer map as the attaching map of the positive-dimensional part of ẼG to the (−1)-cell, viewed as a bottom cell in F_{−1}^{∞}.

Briefly, [GM, 14.5] provides a not quite obvious free G-CW spectrum structure on F_{−1}^{∞} ∧ Y whose associated cellular chain complex is the total complex of the double complex ℤ[u] ⊗ C_{∗}(Y) used to define the cyclic homology of Y. Hence the G-equivariant homology of F_{−1}^{∞} ∧ Y can be computed using this cellular chain complex, or the total complex of the double complex, which is cyclic homology. Likewise for cohomology.

In more detail, let Y be a naive G-spectrum and give G = S^{1} the usual cell structure with the identity element as the only 0-cell, and a single 1-cell z. By [GM, 14.1] we may assume that Y is a naive CW spectrum with a cellular action f: G_{+} ∧ Y → Y by G. The (non-equivariant) cellular complex (C_{∗}(Y), d) has a degree 1 operation J: C_{∗}(Y) → C_{∗+1}(Y) induced by the cellular circle action, defined by J(y) = f_{∗}(z ⊗ y).

As in [GM, 9.6] we take the infinite unit sphere S(∞ℂ) as our model for ES^{1}, which yields a free G-CW spectrum structure on W = F_{−1}^{∞}, with skeleta W^{(2m−1)} = W^{(2m)} = S^{mℂ}/S^{−ℂ} for m ≥ −1. Its (G-equivariant) cellular complex is C_{∗}^{G}(F_{−1}^{∞}) ≅ Σ^{−1}ℤ[u], with generators Σ^{−1}u^{m} in dimension 2m − 1 for m ≥ 0. Its boundary map is zero, since all cells are in odd dimensions. (The algebra structure on ℤ[u] is only a notational convenience, without geometric meaning.)

By [GM, 14.5] applied to W = F_{−1}^{∞} we obtain a naive free G-CW spectrum structure on F_{−1}^{∞} ∧ Y with (G-equivariant) cellular complex

C_{∗}^{G}(F_{−1}^{∞} ∧ Y) ≅ C_{∗}^{G}(F_{−1}^{∞}) ⊗ C_{∗}(Y) .

The cellular boundary maps d are given geometrically by [GM, 14.6, 14.7], and are expressed algebraically as

d(Σ^{−1}u^{m} ⊗ y) = Σ^{−1}u^{m−1} ⊗ J(y) + Σ^{−1}u^{m} ⊗ d(y) .

Hence there are natural isomorphisms (with arbitrary coefficients for homology)

(4.1) H_{∗}([i_{∗}(F_{−1}^{∞} ∧ Y)]^{G}) ⟶_{≅}^{τ̃_{∗}} H_{∗}((F_{−1}^{∞} ∧ Y ∧ S^{ad G})/G)

≅ H_{∗}(ΣC_{∗}^{G}(F_{−1}^{∞} ∧ Y)) ≅ H_{∗}(ℤ[u] ⊗ C_{∗}(Y)) .

The middle map in the cofiber sequence

EG_{+} ⟶^{Dπ∘c} DG_{+} ⟶ F_{−1}^{∞} ⟶ ΣEG_{+}

is the cellular inclusion DG_{+} = F_{−1}^{−1} → F_{−1}^{∞}, so the cellular complexes of the three right hand spectra form a short exact sequence of chain complexes

(4.2)  0 → Σ^{−1}ℤ{1} ⟶ Σ^{−1}ℤ[u] ⟶ Σ^{−1}uℤ[u] → 0 .

Under the chain of isomorphisms (4.1), the long exact sequence in homology

(4.3) H_{∗}([i_{∗}(EG_{+} ∧ Y)]^{G}) ⟶^{(trf_{S^{1}} )_{∗}} H_{∗}([i_{∗}(DG_{+} ∧ Y)]^{G})

⟶ H_{∗}([i_{∗}(F_{−1}^{∞} ∧ Y)]^{G}) ⟶ H_{∗}([i_{∗}(ΣEG_{+} ∧ Y)]^{G})

associated to the cofiber sequence (3.1) is naturally isomorphic to the long exact sequence

H_{∗}(Σ^{−1}uℤ[u] ⊗ C_{∗}(Y)) ⟶ H_{∗}(ℤ ⊗ C_{∗}(Y)) ⟶ H_{∗}(ℤ[u] ⊗ C_{∗}(Y)) ⟶ H_{∗}(uℤ[u] ⊗ C_{∗}(Y))

induced by the short exact sequence (4.2). In terms of G-equivariant homology groups, as defined in [GM, 14.2] or [J], this can be written

ΣH_{∗}^{G}(C_{∗}(Y)) ⟶ H_{∗}(C_{∗}(Y)) ⟶ H_{∗}^{G}(C_{∗}(Y)) ⟶ Σ^{2}H_{∗}^{G}(C_{∗}(Y)) .

Dually, the long exact sequence in cohomology (with arbitrary coefficients)

(4.4) H^{∗}([i_{∗}(EG_{+} ∧ Y)]^{G}) ⟵^{(trf_{S^{1}} )^{∗}} H^{∗}([i_{∗}(DG_{+} ∧ Y)]^{G})

⟵ H^{∗}([i_{∗}(F_{−1}^{∞} ∧ Y)]^{G}) ⟵ H^{∗}([i_{∗}(ΣEG_{+} ∧ Y)]^{G})

is naturally isomorphic to the long exact sequence

(4.5) H^{∗}(Σ^{−1}uℤ[u] ⊗ C_{∗}(Y)) ⟵ H^{∗}(ℤ ⊗ C_{∗}(Y))

⟵ H^{∗}(ℤ[u] ⊗ C_{∗}(Y)) ⟵ H^{∗}(uℤ[u] ⊗ C_{∗}(Y))

induced by the short exact sequence (4.2). In terms of G-equivariant cohomology groups this can be written

ΣH_{G}^{∗}(C_{∗}(Y)) ⟵ H^{∗}(C_{∗}(Y)) ⟵ H_{G}^{∗}(C_{∗}(Y)) ⟵ Σ^{2}H_{G}^{∗}(C_{∗}(Y)) .

We can now identify the homotopy fiber of trf_{S^{1}} when X = ∗. Then Y = ΛX_{+} = S^{0} and hofib(trf_{S^{1}}) ≃ Σ^{−1}[i_{∗}(F_{−1}^{∞})]^{G} ≃ F_{−1}^{∞}/G ≃ (S^{∞ℂ}/S^{−ℂ})/G ≃ Σ(S(∞ℂ)_{+} ∧ S^{−ℂ})/G. This is the suspended Thom complex of minus the canonical complex line bundle over S(∞ℂ)/G = ℂP^{∞}, i.e., ΣℂP_{−1}^{∞}. Thus

(4.6)  hofib(trf_{S^{1}}) ≃ ΣℂP_{−1}^{∞} .

5. Cyclic (co-)homology

Let X be a connected, based space. When Y ≃ ΛX_{+} with the S^{1}-action rotating the loops, and X ≃ BG for a topological group G ≃ ΩX, the long exact sequences (4.3) and (4.4) above are naturally isomorphic to the Connes exact sequences (5.2) and (5.3) for the cyclic (co-)homology of the differential graded algebra C_{∗}(G).

To see this, let Y = ZG_{+} where ZG ≅ EG ×_{G} G^{ad} is the cyclic nerve of G. Here G^{ad} is G viewed as a G-space with the adjoint (conjugate) G-action. There is an S^{1}-equivariant map Y = ZG_{+} → ΛBG_{+} ≃ ΛX_{+} which is a homotopy equivalence, by e.g. [J, 6.2]. ((Check Goodwillie’s paper.)) By giving G a CW structure with a cellular product we can pass to cellular chains and obtain a chain equivalence

(5.1)  C_{∗}(Y) = C_{∗}(ZG) ≅ Z_{∗}(C_{∗}(G)) ≃ C_{∗}(ΛX) .

Here Z_{∗}(C_{∗}(G)) denotes the Hochschild complex of C_{∗}(G), whose homology by definition is the Hochschild homology of C_{∗}(G):

H_{∗}(C_{∗}(Y)) ≅ H_{∗}(Z_{∗}(C_{∗}(G))) = HH_{∗}(C_{∗}(G)) .

The J-operator on C_{∗}(ZG) translates to Connes’ B-operator on the Hochschild complex Z_{∗}(C_{∗}(G)), so ℤ[u] ⊗ Z_{∗}(C_{∗}(G)) is the total complex of the bicomplex whose homology by definition is the cyclic homology of C_{∗}(G):

H_{∗}(ℤ[u] ⊗ C_{∗}(Y)) ≅ H_{∗}(ℤ[u] ⊗ Z_{∗}(C_{∗}(G))) = HC_{∗}(C_{∗}(G)) .

The long exact sequence in homology (4.3) is thus isomorphic to Connes’ long exact sequence for the cyclic homology of C_{∗}(G):

(5.2) ΣHC_{∗}(C_{∗}(G)) ⟶^{B} HH_{∗}(C_{∗}(G)) ⟶^{I} HC_{∗}(C_{∗}(G)) ⟶^{S} Σ^{2}HC_{∗}(C_{∗}(G)) .

Dually, the long exact sequence in cohomology (4.4) is isomorphic to Connes’ long exact sequence for the cyclic cohomology of C_{∗}(G):

(5.3) ΣHC^{∗}(C_{∗}(G)) ⟵^{B} HH^{∗}(C_{∗}(G)) ⟵^{I} HC^{∗}(C_{∗}(G)) ⟵^{S} Σ^{2}HC^{∗}(C_{∗}(G)) .

If C_{∗}(G) has finite type, this sequence may be rewritten as

(5.4) ΣHC_{∗}^{−}(C^{∗}(G)) ⟵^{B} HH_{∗}(C^{∗}(G)) ⟵^{I} HC_{∗}^{−}(C^{∗}(G)) ⟵^{S} Σ^{2}HC_{∗}^{−}(C^{∗}(G))

by using the equivalences Hom(Z_{∗}(C_{∗}(G)), ℤ) ≅ Z_{∗}(C^{∗}(G)) and Hom(ℤ[u] ⊗ Z_{∗}(C_{∗}(G)), ℤ) ≅ ℤ[v] ⊗ Z_{∗}(C^{∗}(G)) with ℤ[v] = Hom(ℤ[u], ℤ). (The algebra structure on ℤ[v] is topologically meaningful, modeling the cup product on the group cohomology of G.)

The map I is induced by the dual of the chain map Z_{∗}(C_{∗}(G)) → ℤ[u] ⊗ Z_{∗}(C_{∗}(G)). When C_{∗}(G) has finite type, this is the cochain map

ℤ[v] ⊗ Z_{∗}(C^{∗}(G)) ⟶^{ϵ⊗1} ℤ ⊗ Z_{∗}(C^{∗}(G)) ≅ Z_{∗}(C^{∗}(G))

mapping v^{0} = 1 to 1 and v^{m} to 0 for m > 0.

Returning to topological cyclic homology of spaces, we obtain from the homotopy Cartesian square (1.1) a map of long exact sequences in cohomology (with arbitrary coefficients) (5.5)

[figure: H^{∗}(T C(X; p)); HH^{∗}(C_{∗}(G)); β^{∗}; HC^{∗}(C_{∗}(G)); ΣH^{∗}(T C(X; p)); ΣHC^{∗}(C_{∗}(G)); α^{∗}; HH^{∗}(C_{∗}(G)); 1−Δ_{p}^{∗}; B; HC^{∗}(C_{∗}(G)); I; Σ^{2}HC^{∗}(C_{∗}(G)) .; α^{∗}; S]

Here 1 − Δ_{p}^{∗} is the composite homomorphism

HH^{∗}(C_{∗}(G)) ≅ H^{∗}(ΛX_{+}) ⟶^{(1−Δ_{p})^{∗}} H^{∗}(ΛX_{+}) ≅ HH^{∗}(C_{∗}(G))

induced by 1 − Δ_{p} via the identifications in (5.1) above.

In order to obtain a model for T C(X; p) we need to identify the chain map inducing 1−Δ_{p}^{∗}. To achieve this, we restrict to the case when X is simply-connected.

6. Algebraic quasi-models

Let X be a based simply-connected CW complex of finite type. We fix a ground field k; modules and algebras are defined over k, tensor products are formed over k and cellular chain and cochain complexes have coefficients in k.

An Adams–Hilton k-model for X is a differential graded algebra (T W, ∂) with a chain equivalence T W ≃ C_{∗}(ΩX) to the cellular chain complex of ΩX, such that T W is the tensor algebra on the free graded module W = s^{−1}C̄_{∗}(X), with one generator in degree n − 1 for each n-cell of X. The model can taken to be enriched in the sense of Dupont and Hess [DH2, §1], meaning that it comes equipped with a homotopy-coassociative and homotopy-cocommutative coproduct ψ: T W → T W ⊗ T W that models the diagonal map on C_{∗}(ΩX) in a precise sense.

Let (C, ˆ∂) be the k-dual of (T W, ∂). This is a differential graded coalgebra with a cochain equivalence C ≃ C^{∗}(ΩX). It has a homotopy-associative and homotopycommutative product modeling the cup product on C^{∗}(ΩX). We can form the cobar construction Ω(C, ˆ∂) = (T (s^{−1}C̄), D_{Ω}) on (C, ˆ∂), which is a differential graded algebra chain homotopy equivalent to C^{∗}(X).

Dupont and Hess [DH2, 4.1] prove that there is a twisted algebra extension Ω(C, ˆ∂) ⊙ (C, ˆ∂) = (Ω(C) ⊙ C, D̄) that is chain homotopy equivalent to C^{∗}(ΛX). There is a diagram of quasi-algebra maps [DH1, §1]

\xymatrix{Ω(C) \ar[r] \ar[d]^{≃} & Ω(C) ⊙ C \ar[r] \ar[d]^{≃} & C \ar[d]^{≃} \\ C^{∗}(X) \ar[r]^{ev_{1}^{∗}} & C^{∗}(ΛX) \ar[r] & C^{∗}(ΩX)}

where the vertical maps are quasi-algebra quasi-isomorphisms and ev_{1} : ΛX → X is the fibration evaluating a free loop at the base point 1 ∈ S^{1}.

The circle action on ΛX ≃ ZG induces an operation

J^{∗}: Z_{∗}(C^{∗}(G)) → ΣZ_{∗}(C^{∗}(G)) ,

which is modeled by an operator J^{∗}: Ω(C) ⊙ C → ΣΩ(C) ⊙ C. ((Explain !)) Let k[v] = Hom(k[u], k), with v^{m} k-dual to u^{m}. The cochain complex k[v] ⊗ Ω(C) ⊙ C has coboundary map given as

(6.1)  δ(v^{m} ⊗ x) = v^{m+1} ⊗ J^{∗}(x) + v^{m} ⊗ D̄(x) .

The homotopy cofiber map ι: ΛX_{+} → hocofib(trf_{S^{1}}) of the S^{1}-transfer map is thus modeled by the cochain map

k[v] ⊗ Ω(C) ⊙ C ⟶^{ϵ} k ⊗ Ω(C) ⊙ C ≅ Ω(C) ⊙ C

mapping v^{0} to 1 and v^{m} to 0 for m > 0, inducing I on cohomology.

The p-th power map Δ_{p}: ΛX → ΛX factors as the the p-fold fibered diagonal

ΛX ⟶^{ψ_{p}} ΛX ×_{X} ⋯ ×_{X} ΛX

followed by the p-fold fibered composition

ΛX ×_{X} ⋯ ×_{X} ΛX ⟶^{ϕ_{p}} ΛX .

These maps fiber over X, as in the commutative diagram

[figure: ΩX; ψ_{p}; ΩX × ⋯ × ΩX; ϕ_{p}; ΩX; ΛX; ψ_{p}; ev_{1}; ΛX ×_{X} ⋯ ×_{X} ΛX; ϕ_{p}; ΛX; ev_{1}; X; X; X .]

The corresponding diagram of cochain algebras is quasi-algebra quasi-isomorphic ((Explain terminology ?)) to

[figure: C; C ⊗ ⋯ ⊗ C; ϕ_{p}; C; ψ_{p}; Ω(C) ⊙ C; Ω(C) ⊙ (C ⊗ ⋯ ⊗ C); ϕ_{p}; Ω(C) ⊙ C; ψ_{p}; Ω(C); (ev_{1})^{∗}; Ω(C); Ω(C); (ev_{1})^{∗}]

Here we are using the isomorphism

(Ω(C) ⊙ C) ⊗_{Ω(C)} ⋯ ⊗_{Ω(C)} (Ω(C) ⊙ C) ≅ Ω(C) ⊙ (C ⊗ ⋯ ⊗ C)

to model the space in the middle.

((Check the Ω(C)-bimodule structures used.)) ((Explain how the model enrichment gives the cup product maps ϕ_{p}.))

Hence we have the following commutative diagram:

(6.2)  \xymatrix{Ω(C) ⊙ C \ar[d]^{≃} & Ω(C) ⊙ C \ar[l]_{1⊙(1−ϕ_{p}ψ_{p})} \ar[d]^{≃} \\ C^{∗}(ΛX) & C^{∗}(ΛX) \ar[l]_{(1−Δ_{p})^{∗}}}

and 1 ⊙ (1 − ϕ_{p}ψ_{p}) provides a quasi-model for the map 1 − Δ_{p} occurring in (1.1) and (5.5) above.

Let C̄ ⊂ C be the kernel of the augmentation ϵ: C → k. There is a map of horizontal extensions

\xymatrix{Ω(C) ⊙ C̄ \ar[r] \ar[d]^{1⊙(1−ϕ_{p}ψ_{p})} & Ω(C) ⊙ C \ar[r]^{1⊗ϵ} \ar[d]^{1⊙(1−ϕ_{p}ψ_{p})} & Ω(C) \ar[d]^{0} \\ Ω(C) ⊙ C̄ \ar[r] & Ω(C) ⊙ C \ar[r]^{1⊗ϵ} & Ω(C)}

since 1 − ϕ_{p}ψ_{p} is zero on the unit of C. Here we have identified Ω(C) ⊙ k ≅ Ω(C). The map 1 − ϕ_{p}ψ_{p}: C̄ → C̄ models the stable map

1 − Δ_{p}: Σ^{∞}ΩX ⟶ Σ^{∞}ΩX

which is a homotopy equivalence after p-completion by the argument following [6A, 3.10]. (This uses that X is simply-connected.) Hence when k has characteristic p the map

(6.3)  1 ⊙ (1 − ϕ_{p}ψ_{p}): Ω(C) ⊙ C̄ ⟶ Ω(C) ⊙ C̄

is a chain equivalence.

7. Algebraic models for topological cyclic homology

Let X be simply-connected and k a field of characteristic p.

We view T C(X; p) as the homotopy fiber of the composite map

ΛX_{+} ⟶^{1−Δ_{p}} ΛX_{+} ⟶^{ι} hocofib(trf_{S^{1}}) .

Compare (1.1). By (4.5) and (6.2) it is thus modeled by the algebraic mapping cone of the composite cochain map

(7.1) k[v] ⊗ Ω(C) ⊙ C ⟶^{ϵ} k ⊗ Ω(C) ⊙ C ≅ Ω(C) ⊙ C ⟶^{1⊙(1−ϕ_{p}ψ_{p})} Ω(C) ⊙ C .

((Discuss coboundary on this model for T C(X; p) ?))

In the case X = ∗, we can take C = k and Ω(C) = k, and the model for T C(∗; p) is the algebraic mapping cone of the composite map

k[v] ⟶^{ϵ} k ⟶^{0} k ,

i.e., k ⊕ Σ^{−1}k[v]. By the Künneth theorem a model for the source T C(∗; p) ∧ X_{+} of the T C-assembly map is the algebraic mapping cone of the composite map

k[v] ⊗ Ω(C) ⟶^{ϵ⊗1} k ⊗ Ω(C) ⟶^{0} k ⊗ Ω(C) ,

i.e., (k ⊕ Σ^{−1}k[v]) ⊗ Ω(C). The assembly map itself is modeled by the map of algebraic mapping cones induced by the augmentation maps k[v] ⊗ Ω(C) ⊙ C → k[v] ⊗ Ω(C) and Ω(C) ⊙ C → Ω(C) for C. Hence there are two maps of horizontal extensions

(7.2)  \xymatrix{k[v] ⊗ Ω(C) ⊙ C̄ \ar[r] \ar[d]^{ϵ⊗1} & k[v] ⊗ Ω(C) ⊙ C \ar[r]^{1⊙ϵ} \ar[d]^{ϵ⊗1} & k[v] ⊗ Ω(C) \ar[d]^{ϵ⊗1} \\ Ω(C) ⊙ C̄ \ar[r] \ar[d]^{1⊙(1−ϕ_{p}ψ_{p})}_{≃} & Ω(C) ⊙ C \ar[r]^{1⊙ϵ} \ar[d]^{1⊙(1−ϕ_{p}ψ_{p})} & Ω(C) \ar[d]^{0} \\ Ω(C) ⊙ C̄ \ar[r] & Ω(C) ⊙ C \ar[r]^{1⊙ϵ} & Ω(C)}

such that the algebraic mapping cones of the vertical composites and the induced maps between them are ((quasi-?)) models for the cofiber sequence

(7.3)  hocofib(a_{X}^{T C}) ⟵ T C(X; p) ⟵^{a_{X}^{T C}} T C(∗; p) ∧ X_{+} .

Since the left hand map 1 ⊙ (1 − ϕ_{p}ψ_{p}) is a mod p chain equivalence by (6.3), it induces a chain equivalence from the algebraic mapping cone of

(7.4)  k[v] ⊗ Ω(C) ⊙ C̄ ⟶^{ϵ⊗1} Ω(C) ⊙ C̄

to the first model for hocofib(a_{X}^{T C}). Hence this is a second model for hocofib(a_{X}^{T C}). Since ϵ is surjective, we may replace the mapping cone by the desuspended kernel. So

(7.5)  Σ^{−1}k̄[v] ⊗ Ω(C) ⊙ C̄

is a third model for hocofib(a_{X}^{T C}). Here k̄[v] is the kernel of the augmentation ϵ: k[v] → k. The coboundary δ on this model for hocofib(a_{X}^{T C}) is that inherited as a desuspended sub-cocomplex of k[v] ⊗ Ω(C) ⊙ C given in (6.1).

The induced maps of algebraic mapping cones of the vertical composites in

[figure: k[v] ⊗ Ω(C) ⊙ C; 1⊙ϵ; ϵ⊗1; k[v] ⊗ Ω(C); 1⊗ϵ; ϵ⊗1; k[v]; ϵ; Ω(C) ⊙ C; 1⊙ϵ; 1⊙(1−ϕ_{p}ψ_{p}); Ω(C); ϵ; 0; k; 0; Ω(C) ⊙ C; 1⊙ϵ; Ω(C); Ω(C)]

model the two maps

T C(X; p) ⟵^{a_{X}^{T C}} T C(∗; p) ∧ X_{+} ⟵ T C(∗; p) ∨ X

whose composite is split injective up to homotopy, cf. (1.2). The horizontal cochain maps are surjective, so their kernels model the homotopy cofibers of these spectrum maps. Hence the induced maps of algebraic mapping cones of the vertical composites in

[figure: k[v] ⊗ Ω(C) ⊙ C̄; ϵ⊗1; k[v] ⊗ \overline{Ω(C) ⊙ C}; ϵ⊗1; k[v] ⊗ \overline{Ω(C)}; ϵ⊗1; Ω(C) ⊙ C̄; 1⊙(1−ϕ_{p}ψ_{p}); ≃; \overline{Ω(C) ⊙ C}; 1⊙(1−ϕ_{p}ψ_{p}); \overline{Ω(C)}; Ω(C) ⊙ C̄; Ω(C) ⊙ C̄; 0]

(with some obvious notations) model the cofiber sequence

hocofib(a_{X}^{T C}) ⟵ \frac{T C(X; p)}{T C(∗; p) ∨ X} ⟵ ΣℂP_{−1}^{∞} ∧ X .

Passing from this first model to the third model (7.5) for hocofib(a_{X}^{T C}), we obtain a model for T C(X; p)/(T C(∗; p)∨X) as the algebraic mapping cone of the connecting cochain map

(7.6)  k[v] ⊗ \overline{Ω(C)} ⟶^{∂} Σ^{−1}k̄[v] ⊗ Ω(C) ⊙ C̄

that models the connecting spectrum map

Σ^{2}ℂP_{−1}^{∞} ∧ X ⟵^{∂} hocofib(a_{X}^{T C}) .

((This involves the equivalence of the three models for hocofib(a_{X}^{T C}).)) ((Make ∂ explicit.)) ((Can we assume ∂ is injective ?))

8. Algebraic models for Whitehead spectra

The smooth and topological Whitehead spectra W h^{d}(X) and W h^{t}(X) are defined so that there are natural cofiber sequences of spectra

X_{+} ⟶^{η_{X}} A(X) ⟶ W h^{d}(X)

and

A(∗) ∧ X_{+} ⟶^{a_{X}^{A}} A(X) ⟶ W h^{t}(X)

respectively, where η_{X} is the unit map and a_{X}^{A} is the A-theory assembly map [W1, 3.3.1].

By Waldhausen’s stable parametrized h-cobordism theorem [W2], there are homotopy equivalences

H^{d}(M) ≃ ΩΩ^{∞}W h^{d}(M)

and

H^{t}(M) ≃ ΩΩ^{∞}W h^{t}(M)

when M is a smooth, resp. topological, compact manifold. Here H^{d}(M) is the stable smooth h-cobordism space of M, which in a stable range carries information about the homotopy type of the topological group Diff(M) of self-diffeomorphisms of M. Likewise H^{t}(M) is the stable topological h-cobordism space of M, which in a stable range carries information about the topological group Homeo(M) of self-homeomorphisms of M.

We are therefore interested in finding models for these smooth and topological (= PL) Whitehead spectra, starting with the smooth case.

For X simply-connected, the natural square

(8.1)  \xymatrix{A(∗) \ar[r] \ar[d]^{_{trc_{∗}}} & A(X) \ar[d]^{_{trc_{X}}} \\ T C(∗; p) \ar[r] & T C(X; p)}

is homotopy Cartesian after p-completion, by [6A]. We map the trivially homotopy Cartesian square

[figure: ∗_{+}; X_{+}; _{∗_{+}}; X_{+}]

to the square (8.1) above using the unit map. Then the square of homotopy cofibers

\xymatrix{W h^{d}(∗) \ar[r] \ar[d] & W h^{d}(X) \ar[d] \\ T C(∗; p)/∗_{+} \ar[r] & T C(X; p)/X_{+}}

is also homotopy Cartesian after p-completion. This says that there is a (co-)fiber sequence of spectra

W h^{d}(∗) ⟶ W h^{d}(X) ⟶ \frac{T C(X; p)}{T C(∗; p) ∨ X}

which is obviously split. Hence there is a homotopy equivalence

(8.2)  W h^{d}(X) ≃ W h^{d}(∗) ∨ \frac{T C(X; p)}{T C(∗; p) ∨ X} .

For p = 2 the mod 2 cohomology of the left hand summand W h^{d}(∗) has been completely described as a module over the Steenrod algebra in [R1]. For odd regular primes p the mod p cohomology of W h^{d}(∗) has been given a similar description in [R2], assuming the Lichtenbaum–Quillen conjecture holds for K(ℤ[1/p]) at p. Our model (7.6) for the right hand summand T C(X; p)/(T C(∗; p) ∨ X), given as the algebraic mapping cone of the connecting map

∂: k[v] ⊗ \overline{Ω(C)} ⟶ Σ^{−1}k̄[v] ⊗ Ω(C) ⊙ C̄

thus facilitates the computation of the relative term, measuring the difference between W h^{d}(X) and W h^{d}(∗).

Turning to the topological case, we consider the commutative diagram

\xymatrix{A(∗) ∨ X \ar[r] \ar[d]^{trc_{∗} ∨1} & A(∗) ∧ X_{+} \ar[r]^{a_{X}^{A}} \ar[d]^{trc_{∗} ∧1} & A(X) \ar[d]^{_{trc_{X}}} \\ T C(∗; p) ∨ X \ar[r] & T C(∗; p) ∧ X_{+} \ar[r]^{a_{X}^{T C}} & T C(X; p) .}

The lower row was constructed in section 1 and the upper row is similar. The outer square is homotopy Cartesian, by [6A] again. We obtain a map of horizontal cofiber sequences of homotopy cofibers

(8.2)  \xymatrix{W h^{d}(∗) ∧ X \ar[r] \ar[d] & A(X)/(A(∗) ∨ X) \ar[r] \ar[d]^{≃} & W h^{t}(X) \ar[d] \\ ΣℂP_{−1}^{∞} ∧ X \ar[r] & T C(X; p)/(T C(∗; p) ∨ X) \ar[r] & hocofib(a_{X}^{T C}) .}

Hence there is a cofiber sequence

W h^{d}(∗) ∧ X ⟶ \frac{T C(X; p)}{T C(∗; p) ∨ X} ⟶ W h^{t}(X) .

We have given a model (7.6) for T C(X; p)/(T C(∗; p) ∨ X), and given a model for W h^{d}(∗) it can be tensored with \overline{Ω(C)} to give an algebraic model for W h^{d}(∗) ∧ X. A model for W h^{t}(X) can then be obtained as the desuspended algebraic mapping cone for a map between these models. ((Can we be more explicit about the map ?))

There is also a cofiber sequence

W h^{t}(X) ⟶ hocofib(a_{X}^{T C}) ⟶ Σ^{2} hofib(trc_{∗}) ∧ X

obtained from the same map of cofiber sequences (8.2). We gave three models for hocofib(a_{X}^{T C}) in section 7 above, and given a model for hofib(trc_{∗}) it can clearly be tensored with Σ^{2}\overline{Ω(C)} to give a model for Σ^{2} hofib(trc_{∗}) ∧ X. The algebraic mapping cone for an appropriate map between these models then gives a model for W h^{t}(X).

The p-complete homotopy type of hofib(trc_{∗}) was obtained in [R1] for p = 2, and in [R2] for odd regular primes p under the assumption that the Lichtenbaum– Quillen conjecture holds for K(ℤ[1/p]) at p.

9. Examples

((Do the cases X = S^{2n+1}, S^{2n}, ℂP^{n} for n ≥ 1 ?))

References

[6A] M. Bökstedt, G. Carlsson, R. Cohen, T. Goodwillie, W.C. Hsiang and I. Madsen, On the algebraic K-theory of simply connected spaces, Duke Math. J. 84 (1996), 541–563.

[BHM] M. Bökstedt, W.C. Hsiang and I. Madsen, The cyclotomic trace and algebraic K-theory of spaces, Invent. Math. 111 (1993), 465–539.

[DH1] N. Dupont and K. Hess, Noncommutative algebraic models for fiber squares, Math. Ann. 314 (1999), 449–467.

[DH2] N. Dupont and K. Hess, How to model the free loop space algebraically, Math. Ann. 314 (1999), 469–490.

[GM] J.P.C. Greenlees and J.P. May, Generalized Tate cohomology, Mem. Am. Math. Soc., vol. 543, 1995.

[J] J.D.S. Jones, Cyclic homology and equivariant homology, Invent. Math. 87 (1987), 403– 423.

[LMS] L.G. Lewis, jr., J.P. May and M. Steinberger, Equivariant stable homotopy theory, Lecture Notes in Mathematics, vol. 1213, Springer-Verlag, Berlin etc., 1986.

[R1] J. Rognes, Two-primary algebraic K-theory of pointed spaces, Topology (to appear).

[R2] J. Rognes, The smooth Whitehead spectrum of a point at odd regular primes, preprint (2001).

[W1] F. Waldhausen, Algebraic K-theory of spaces, Algebraic and geometric topology (Proc. Conf., New Brunswick/USA 1983), Lect. Notes Math., vol. 1126, 1985, pp. 318–419.

[W2] F. Waldhausen, W. Vogell, B. Jahren and J. Rognes, Spaces of PL manifolds and categories of simple maps, in preparation.

Ecole Polytechnique Fédérale de Lausanne, Switzerland

   E-mail address: kathryn.hess@epfl.ch

Department of Mathematics, University of Oslo, Norway

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