The plus construction

In this section we collect the facts we need about Quillen’s plus construction made into a functorial construction, using Bousfield and Kan’s integral completion functor [30]. For a long time, the best sources on the plus construction were the paper by Hausmann and Husemoller [96] and Berrick’s textbook [12], but Quillen’s original account which had circulated for a very long time finally made it into the appendix of [68].

1.1.1 Acyclic maps

Recall from I.1.6.2 that a map of pointed connected spaces is calledacyclic if the integral homology of the homotopy fiber vanishes. We need some facts about acyclic maps.

If Y is a connected space, we may form its universal cover Y˜ as follows. From sin|Y|, form the spaceB by identifying two simplicesu, v ∈sin|Y|q whenever, considered as maps

∆[q]→sin|Y|, they agree on the one-skeleton of ∆[q]. Then sin|Y|։B is a fibration of fibrant spaces [155, 8.2], and Y˜ is defined by the pullback diagram

Y˜ −−−→ B^∆[1]



y y Y −−−→ B

and we note that Y˜ →Y is a fibration with fiber equivalent to the discrete set π1Y. Lemma 1.1.2 Let f: X →Y be a map of connected spaces, and Y˜ the universal cover of Y. Then f is acyclic if and only if

H∗(X×_Y Y˜)→H∗( ˜Y) is an isomorphism.

Proof: We may assume that X →Y is a fibration with fiberF. Then X×_Y Y˜ →Y˜ also is a fibration with fiber F, and the Serre spectral sequence [78, IV.5.1]

Hp( ˜Y;Hq(F))⇒Hp+q(X×_Y Y˜)

gives that ifH˜∗(F) = 0, then the edge homomorphism (which is induced byX×_Y Y˜ →Y˜) is an isomorphism as claimed.

1. DEGREEWISE K-THEORY 103 Conversely, if H∗(X ×Y Y˜) → H∗( ˜Y) is an isomorphism. Then it is easy to check directly that H˜q(F) = 0 for q≤1. Assume we have shown that H˜q(F) = 0 forq < k for a k ≥2. Then the spectral sequence gives an exact sequence

Hk+1(X×_Y Y˜) −→^∼= Hk+1( ˜Y) −→ Hk(F) −→ Hk(X×_Y Y˜) −→^∼= Hk( ˜Y) −→ 0 which implies that Hk(F) = 0 as well.

The lemma can be reformulated using homology with local coefficients: H∗( ˜Y) = H∗(Y;Z[π1Y]) and H∗(X ×_Y Y˜) ∼= H∗(Z[ ˜X]⊗_Z[π1X] Z[π1Y]) = H∗(X;f^∗Z[π1Y]), so f is acyclic if and only if it induces an isomorphism

H∗(X;f^∗Z[π1Y])∼=H∗(Y;Z[π1Y]) This can be stated in more general coefficients:

Corollary 1.1.3 A map f:X →Y of connected spaces is acyclic if and only if it for any local coefficient system G on Y, f induces an isomorphism

H∗(X;f^∗G)∼=H∗(Y;G)

Proof: By the lemma we only need to verify one implication. If i: F → Y is the fiber of f, the Serre spectral sequence gives

H_p(Y;H_q(F;i^∗f^∗G))⇒H_p+q(X;f^∗G).

However, i^∗f^∗G is a trivial coefficient system, so if H˜∗(F) = 0, the edge homomorphism must be an isomorphism.

This reformulation of acyclicity is useful, for instance when proving the following lemma.

Lemma 1.1.4 Let

X −−−→^f Y

g



y ^g′

 y Z −−−→^f′ S

be a pushout cube of connected spaces with f acyclic, and either f or g a cofibration. Then f^′ is acyclic.

Proof: Let G be a local coefficient system in S. Using the characterization 1.1.3 of acyclic maps as maps inducing isomorphism in homology with arbitrary coefficients, we get by excision that

H∗(S, Z;G)∼=H∗(Y, X; (g^′)^∗G) = 0 implying that f^′ is acyclic.

Lemma 1.1.5 Letf: X→ Y be a map of connected spaces. Thenf is a weak equivalence if and only if it is acyclic and induces an isomorphism of the fundamental groups.

Proof: Let F be the homotopy fiber of f. If f induces an isomorphism π1X ∼= π1Y on fundamental groups, then π1F is abelian. If f is acyclic, then π1F is perfect. Only the trivial group is both abelian and perfect, soπ₁F = 0. AsH˜∗F = 0the Whitehead theorem tells us that F is contractible.

1.1.6 The functorial construction

We now give a functorial construction of the plus construction, following the approach of Bousfield and Kan [30, p. 218].

If X is any set, we may consider the free abelian group generated by X, and call it Z[X]. If X is pointed we let Z[X] =˜ Z[X]/Z[∗]. This defines a functor Ens∗ → Ab which is adjoint to the forgetful functor U: Ab → Ens∗, and extends degreewise to all spaces. The transformation given by the inclusion of the generators X → Z[X]˜ (where we symptomatically have forgotten to write the forgetful functor) induces the Hurewicz homomorphism π∗(X)→π∗( ˜Z[X]) = ˜H∗(X).

As explained in appendix A.0.12, the fact that Z˜ is a left adjoint implies that it gives rise to a cosimplicial space Z

˜ via

Z˜[X] ={[n]7→Z˜ⁿ⁺¹[X]},

where the superscript n+ 1means that we have used the functorZ˜ n+ 1times. The total (see section A.1.8) of this cosimplicial space is called the integral completion of X and is denoted Z_∞X.

Bousfield and Kan define the integral completion in a slightly different, but isomorphic, manner, which has the advantage of removing the seeming dependence on a base point. Let eX: Z[X] → Z be the homomorphism that sends P

nixi to P

ni. Instead of considering the abelian groupZ[X], Bousfield and Kan consider the set˜ e⁻¹_X (1). The compositee⁻¹_X (1)⊆ Z[X]։Z[X]˜ is a bijection, and one may define the integral completion for nonbased spaces usingX 7→e⁻¹_X (1)instead. A disadvantage is that this no longer gives a cosimplicial abelian group.

If f:X →Y is a function of sets, define Z[X]˙ →Y by Z[X] =˙ e⁻¹_X (1)∩ `

y∈Y

Z[f⁻¹(y)] ={P

n_ix_i|f(x₁) =· · ·=f(x_n),P

n_i = 1}

Pnixi7→f(x1)

−−−−−−−−→ Y .

Note that if Y is a one-point space, then Z[X] =˙ e⁻¹_X (1), but usually Z[X]˙ will not be an abelian group. This construction is natural in f, and we may extend to spaces, giving a cosimplicial subspace of Z

˜[X], whose total is called the fiberwise integral completion of X (or rather, of f).

IfX is a space, there is a natural fibrationsin|X| →sin|X|/P given by “killing, in each component, πi(X) for i > 1 and the maximal perfect subgroup P π1(X) ⊆ π1(X)”. More precisely, let sin|X|/P be the space obtained from sin|X| by identifying two simplices

1. DEGREEWISE K-THEORY 105 u, v ∈sin|X|q whenever, for every injective mapφ ∈∆([1],[q]), we havediφ^∗u=diφ^∗v for i= 0,1, and

[φ^∗u]⁻¹∗[φ^∗v] = 0 ∈π₁(X, d₀φ^∗u)/P π₁(X, d₀φ^∗u) The projection sin|Y| →sin|X|/P is a fibration.

Definition 1.1.7 The plus construction X 7→ X⁺ is the functor given by the fiberwise integral completion ofsin|X| →sin|X|/P (called thepartial integral completionin I.1.6.1), andqX: X →X⁺ is the natural transformation coming from the inclusionX ⊆Z[sin˙ |X|].

That this is the desired definition follows from [30, p. 219], where they use the alter-native description of corollary 1.1.3 for an acyclic map:

Proposition 1.1.8 If X is a pointed connected space, then qX: X →X⁺

is an acyclic map killing the maximal perfect subgroup of the fundamental group.

We note thatqX is always a cofibration (=inclusion).

1.1.9 Uniqueness of the plus construction

Now, Quillen provides the theorem we need: X⁺ is characterized up to homotopy under X by this property

Theorem 1.1.10 Consider the (solid) diagram of connected spaces X ^{f //}

qX

Y

X⁺

h {==

{{ {

If Y is fibrant andP π1X ⊆ker{π₁X →π1Y} then there exists a dotted maph making the resulting diagram commutative. Furthermore, the map is unique up to homotopy, and is a weak equivalence if f is acyclic.

Proof: Let S =X⁺`

XY and consider the solid diagram X ^{f //}

qX

Y

g

Y

X^{+ f}

′

// S

H ??





// ∗

By lemma 1.1.4, we know that g is acyclic. The van Kampen theorem [78, III.1.4] tells us that π1S is the “free product” π1X⁺∗_π1X π1Y, and the hypothesis imply that π1Y →π1S must be an isomorphism.

By lemma 1.1.5, this means that g is a weak equivalence. Furthermore, as qX is a cofibration, so is g. Thus, as Y is fibrant, there exist a dotted H making the diagram commutative, and we may choose h = Hf^′. By the universal property of S, any h must factor throughf^′, and the uniqueness follows by the uniqueness of H.

Iff is acyclic, then bothf =hqX andqX are acyclic, and sohmust be acyclic. Further-more, asf is acyclicker{π₁X →π₁Y}must be perfect, but as P π₁X ⊆ker{π₁X →π₁Y} we must have P π1X =ker{π1X →π1Y}. So,h is acyclic and induces an isomorphism on the fundamental group, and by 1.1.5 h is an equivalence.

Lemma 1.1.11 Let X → Y be a k-connected map of connected spaces. Then X⁺ → Y⁺ is also k-connected

Proof: Either one uses the characterization of acyclic maps by homology with local coefficients, and check by hand that the lemma is right in low dimensions, or one can use our choice of construction and refer it away: [30, p. 113 and p. 42].

1.1.12 The plus construction on simplicial spaces

The plus construction on the diagonal of a simplicial space (bisimplicial set) may be per-formed degreewise in the following sense. Remember, I.1.2.1, that a quasi-perfect group is a groupG in which the maximal perfect subgroup is the commutator: P G= [G, G].

Lemma 1.1.13 Let {[s] 7→X_s} be a simplicial space such that X_s is connected for every s≥0. Let X⁺={[s]7→X_s⁺} be the “degreewise plus-construction”. Consider the diagram

diag^∗X −−−→ diag^∗(X⁺)



y y

(diag^∗X)⁺ −−−→ (diag^∗(X⁺))⁺ ,

where the upper horizontal map is induced by the plus construction qXs: Xs → X_s⁺, and the lower horizontal map is plus of the upper horizontal map.

The lower horizontal map is always an equivalence, and the right vertical map is an equivalence if and only if π1diag^∗(X⁺) has no nontrivial perfect subgroup. This is true if, for instance, π1(X₀⁺) is abelian, which follows if π1(X0) is quasi-perfect.

Proof: LetA(Xs) =f iber{qXs: Xs →X_s⁺}, and consider the sequence A(X) ={[s]7→A(X_s)} −−−→ X −−−→^qX X⁺

of simplicial spaces. As Xs and X_s⁺ are connected, theorem A.5.0.4 gives that diag^∗A(X) −−−→ diag^∗X −−−−→^diag∗qX diag^∗(X⁺)

is a fiber sequence. But as each A(Xs) is acyclic, the spectral sequence A.5.0.6 cal-culating the homology of a bisimplicial set gives that H˜∗(diag^∗A(X)) = 0, and so the

1. DEGREEWISE K-THEORY 107 map diag^∗qX: diag^∗X → diag^∗(X⁺) is acyclic. The lower horizontal map (diag^∗X)⁺ → (diag^∗(X⁺))⁺ in the displayed square is thus the plus of an acyclic map, and hence acyclic itself. However, P π₁((diag^∗X)⁺) is trivial, so this map must be an equivalence.

The right vertical map is the plus construction applied to diag^∗(X⁺), and so is an equivalence if and only if it induces an equivalence on π1, i.e., if P π1(diag^∗(X⁺)) = ∗. If π₁(X₀) is quasi-perfect, then π₁(X₀⁺) =π₁(X₀)/P π₁(X₀) =H₁(X₀) is abelian, and so the quotient π1diag^∗(X⁺) is also abelian, and hence has no perfect (nontrivial) subgroups.

Remark 1.1.14 Note that some condition is needed to ensure thatπ1diag^∗(X⁺)is without nontrivial perfect subgroups, for let Xq = BFq where F ^{∼ //}P is a free resolution of a perfect group P. Then diag^∗(X⁺)≃BP 6≃BP⁺.

1.1.15 Nilpotent fibrations and the plus construction

Let π and G be groups, and let π act on G. The action is nilpotentif there exists a finite filtration

∗=G_n+1 ⊆G_n⊆ · · · ⊆G₂ ⊆G₁ =G

respected by the action, such that each Gi+1 ⊂Gi is a normal subgroup and such that the quotients G_i/G_i+1 are abelian with induced trivial action.

A groupGis said to benilpotentif the self-action via inner automorphisms is nilpotent.

Definition 1.1.16 Iff: E →B is a fibration of connected spaces with connected fiberF, then π1(E) acts on each πi(F) (see A.4.1), and we say that f is nilpotent if these actions are nilpotent.

Generally, we will say that a map of connected spaces X → Y is nilpotent if the associated fibration is.

Lemma 1.1.17 F → E → B is any fiber sequence of connected spaces where π1E acts trivially on π∗F, then the fibration is nilpotent.

Proof: Since πqF is abelian for q > 1, a trivial action is by definition nilpotent, and the only thing we have to show is that the action of π1E on π1F is nilpotent. Let A^′ = ker{π1F →π1E}and A^′′ =ker{π1E →π1B}. Since π1E acts trivially onA^′ and A^′′, and both are abelian (the former as it is the cokernel of π2E → π2B, and the latter as it is in the center ofπ1E),π1E acts nilpotently on π1F.

Lemma 1.1.18 Let f:X →Y be a map of connected spaces. If either

1. f fits in a fiber sequence X ^{f //}Y ^//Z where Z is connected and P π1(Z) = ∗, or 2. f is a nilpotent,

then

X −−−→^f Y

qX



y ^qY

 y X⁺ −−−→^f+ Y⁺ is (homotopy) cartesian.

Proof: Part 1. Consider the map of fiber sequences X −−−→ Y −−−→ Z



y ^qYy ^qZy F −−−→ Y⁺ −−−→ Z⁺

in the homotopy category. Since P π1Z is trivial, qZ: Z → Z⁺ is an equivalence, and so the homotopy fibers of X →F and qY are equivalent. Hence the map X →F is acyclic.

To see that X →F is equivalent to qX we must show that π1X → π1F is surjective and that π1F is without nontrivial perfect subgroups. Surjectivity follows by chasing the map of long exact sequences of fibrations. That a perfect subgroup P ⊆ π₁F must be trivial follows since π1Y⁺ is without nontrivial perfect subgroups, and so P must be a subgroup of the abelian subgroup ker{π₁F →π1Y⁺} ∼= coker{π₂Y⁺ →π2Z⁺}.

Part 2. That f is nilpotent is equivalent, up to homotopy, to the statement that f factors as a tower of fibrations

Y =Y0 f1

←−−− Y1 f2

←−−− . . . ←−−−^fk Yk=X where each fi fits in a fiber sequence

Yi fi

−−−→ Yi−1 −−−→ K(Gi, ni)

with ni >1 (see e.g., [30, page 61]). But statement 1tells us that this implies that Yi −−−→ Y_i⁺



y y Yi−1 −−−→ Y_i−1⁺ is cartesian, and by induction on k, the statement follows.

In document The local structure of algebraic K-theory (sider 102-108)