Plus construction considerations.

Question

In order to realise the K-groups of a ring as the homotopy groups of some space associated to that ring, Quillen proposed the following (roughly-sketched) construction:

Recall that $K_1(R) = GL(R)/E(R)$, so we're, at least, looking for a space $X$ with $\pi_1(X) = K_1(R)$. The classifying space of $K_1(R)$ is obviously not a serious candidate, but we can start with the classifying space of $GL(R)$ (given the discrete topology), $BGL(R)$. Then, choosing representative loops whose classes generate $E(R) \subset \pi_1(BGL(R))$, and then attaching 2-cells using these loops on the boundaries, we end up with something that has a fundamental group of $K_1(R)$. Furthermore, now Quillen adjoins 3-cells essentially to correct the homology back to that of $BGL(R)$, which was messed up in the addition of those 2-cells. We end up with a space denoted $BGL(R)^+$, on which we define $K_i(R) := \pi_i(BGL(R)^+)$.

More specifically, Quillen sought to find a space $BGL(R)^+$ for which $(BGL(R), BGL(R)^+)$ was an acyclic pair (that is, the induced map $H_*(BGL(R), M) \to H_{*}(BGL(R)^+,M)$ is an isomorphism for all $K_1(R)$-modules $M$). My question is

In search of a space on which to define K-groups, why was it desirable to find something satisfying the above condition on homology?

My best guess is that it was observed that $K_1(R) = GL(R)/E(R) = GL(R)_{ab} = H_1(GL(R), \mathbb{Z})$, and $K_2(R) = H_2(E(R), \mathbb{Z}) = H_2([GL(R), GL(R)], \mathbb{Z})$, and so it seemed reasonable that all K-groups should be related to the homology of $GL(R)$ - and so the above is a stab at preserving that homology.

I am trying to learn about K-theory, but, as with most presentations of math, I'm sure, I'm coming across too many cleanly-shaven formulas and propositions, entirely divorced from any sorts of thought-processes, big-pictures, or mentions of what is trying to be done and to what ends. Please share with me what you think is going on; here, with the +-construction, that is. (And, if and only if it is not a completely unrelated question, what sort of K-theoretical phenomena suggested that these groups should be homotopy groups?)

Answer 1

Here are some thoughts, gathered from reading many texts about algebraic K-theory. Let me start with some historical remarks, then try to give a more revisionist motivation of the plus construction.

First of all, it's true as you say that the already-divined definitions of the lower K-groups made it seem like the higher K-groups, whatever they might be, bear the same relation to the homology of GL(R) as the homotopy groups of an H-space bear to its homology groups; thus one is already looking for an H-space K(R) whose homology agrees with that of GL(R), in order to define the higher algebraic K-groups as its homotopy groups. This line of thought is amplified by the observation that the known partial long exact sequences involving lower K-groups seemed like they could plausibly arise as the long exact sequences on homotopy groups associated to fibrations between these hypothetical H-spaces.

So it may already at this point be natural to try to turn GL(R) into a H-space while preserving its homology, which is what the + construction does. However, the facts that:

1) the + construction ignores the admittedly crucial K_0-group; and

2) in any case, at the time there were very few lower K-groups to extrapolate from in the first place

mean that perhaps the above is insufficient motivation for defining and investigating such a seemingly ad hoc construction as the + construction.

However, we can bear in mind that Quillen's definition of the + construction came on the heels of his work on the Adams conjecture, during the course of which --- using his expertise on the homology of finite groups --- he was able to produce a (mod l) homology equivalence

BGL(F) --> BU

when F is the algebraic closure of a finite field of characteristic different from l. Now, BU is a classifying space for complex K-theory (in positive degrees), so its homotopy groups provide natural definitions of the (topological) algebraic K-theory of the complex numbers. Furthermore, by analogy with the theory of etale cohomology (known to Quillen), it is not entirely unreasonable to guess that, from the (mod l) perspective, all algebraically closed fields of characteristic different from l should behave in the same way as the topological theory over C. (This was later borne out in work of Suslin.) Then the above (mod l) homology equivalence adds further weight to the idea that the hypothetical K-theoretic space K(F) we're searching for should have the same homology as BGL(F). But what's more, Quillen also calculated the homology of BGL(F) when F is a finite field, and found this to be consistent with the combination of the above and a ``Galois descent'' philosophy for going form the algebraic closure of F down to F.

That said, in the end, there is good reason why the plus construction of algebraic K-theory is difficult to motivate: it is, in fact, less natural than the other constructions of algebraic K-theory (group completion, Q construction, S-dot construction... applied to vector bundles, perfect complexes, etc.). This is partly because it has less a priori structure, partly because it ignores K_0, partly because it has narrower applicability, and partly because it is technically inconvenient (e.g. for producing the fiber sequences discussed above). Of course, Quillen realized this, which is why he spent so much time working on the other constructions. Probably the only claim to primacy the + construction has is historical: it was the first construction given, surely in no small part because of personal contingencies --- Quillen was an expert in group homology.

In fact, probably the best motivation for the + construction -- ahistorical though it may be -- comes by comparison with another construction, the group completion construction (developed by Segal in his paper "on categories and cohomology theories"). Indeed, Segal's construction is very well-motivated: it is the precise homotopy-theoretic analog of the classical procedure of going from isomorphism-classes of f.g. proj. modules to the Grothendieck group K_0 by formally turning direct sum into a group operation. To get this homotopy theoretic analog, one "simply" carries along the isomorphisms in this construction (c.f. also Grayson's article "higher algebraic k-theory II"). The connection with the plus construction comes from the ``group completion theorem'' (see the McDuff-Segal article on this subject), which, under very general conditions, allows to calculate the homology of such a homotopy-theoretic group completion in terms of the homology of the relevant isomorphism groups. If you look at the group completion theorem in the case of the space of f.g. proj. modules over a ring, you'll see the connection with the plus construction immediately.

Answer 2

I don't really know if this helps, but you can in effect give the plus-construction definition of $K$-groups without explicitly mentioning homotopy groups, and without ever doing the plus construction:

$H_1BGL(R)$ is $K_1(R)$. Map $BGL(R)$ to an Eilenberg-MacLane space $BK_1(R)$ and consider the homotopy fiber.

This space has trivial $H_1$. Its $H_2$ is $K_2(R)$. Map it to an Eilenberg-MacLane space $B^2K_2(R)$ and consider the homotopy fiber.

This space has trivial $H_2$. Its $H_3$ is $K_3(R)$. Map it to an Eilenberg-MacLane space $B^3K_3(R)$ and consider the homotopy fiber.

And so on.

Answer 3

For convenience (at least my own) and completeness, I want to give an explanation of Tom Goodwillie's answer, as it was not obvious to me how to prove the statement he makes. I wanted to leave it as a comment to his answer, but it became way too long. In summary, the point is that applying the plus construction to the spaces defined by Tom gives the connective covers (i.e. the Whitehead tower) of $BGL(R)^+$. This follows from the fact that the plus construction preserves certain fibre sequences.

Abbreviate $X=BGL(R)$. Let $F_1=X=BGL(R)$, and denote by $F_{i+1}$ the $i$-th space constructed by Tom, so that his claim becomes $H_i(F_i) = \pi_i(X^+) = K_i R$. Explicitly, the spaces $F_k$ are defined inductively by a fibre sequence $$ F_{i+1}\longrightarrow F_i\longrightarrow K(H_i(F_i),i) $$ for each $i\geq 1$, where the map on the right induces the "identity" on $H_i$. Applying the plus construction to this fibre sequence, we get a new sequence $$ (FS^+): \qquad\qquad (F_{i+1})^+ \longrightarrow (F_i)^+ \longrightarrow K(H_i(F_i),i)^+ \simeq K(H_i(F_i),i) \hphantom{\qquad\qquad\qquad} $$ where we have noted that the natural map $K(H_i(F_i),i) \to K(H_i(F_i),i)^+$ is a weak equivalence. Importantly, the new sequence $(FS^+)$ is again a fibre sequence. This result is an instance of proposition 3.D.3-(2) in page 74 of Dror Farjoun's book "Cellular spaces, null spaces and homotopy localization": applying the plus construction to a fibre sequence of path connected spaces gives a fibre sequence as long as the homotopy type of the base space is unchanged by the plus construction. In fact, the book states this for any nullification functor, and the plus construction is one such functor.

The fundamental group of $(F_1)^+=X^+$ is abelian (it is $K_1 R$, after all). From the fibration sequence $(FS^+)$ for $i=1$ we then conclude that $(F_2)^+$ is simply connected. Moreover, that fibre sequence is just $(F_2)^+\to X^+\to B(\pi_1(X^+))$, and it realizes $(F_2)^+$ as the universal cover of $X^+$.

We can proceed inductively in this manner. More precisely, knowing that $\pi_1((F_1)^+)=\pi_1(X^+)$ is abelian, we can show by induction that for all $i\geq 1$:

• The space $(F_i)^+$ is $(i-1)$-connected, and the Hurewicz theorem implies $\pi_i((F_i)^+)=H_i((F_i)^+)=H_i(F_i)$.

• Hence, the second map in the fibre sequence $(FS^+)$ above is an isomorphism on $\pi_i$. In other words, that fibre sequence is simply killing $\pi_i$ of $(F_i)^+$.

• Finally, there is a canonical equivalence $(F_i)^+ \simeq (X^+)^{\geq i}$ from $(F_i)^+$ to the $(i-1)$-connected cover of $X^+$ (also known as the $i$-th stage of the Whitehead tower of $X^+$).

Consequently, $H_i(F_i)=H_i((F_i)^+)=H_i((X^+)^{\geq i})=\pi_i((X^+)^{\geq i})=\pi_i(X^+)$ as desired.

Obviously, the above argument is not specific to $X=BGL(R)$. In fact, we only require that the pointed space $X$ is path connected, and $\pi_1(X^+)$ is abelian.