Terminology question for real K-theory

Question

This is a terminology question. Answers will help me satisfy a referee but I'm also genuinely interested. Consider the following two things that you could define for a topological space X:

(1) The Grothendieck ring of the category of real vector bundles over X;

(2) the set $[X, \mathbb{Z} \times BO]$ of homotopy classes of maps from $X$ to $\mathbb{Z} \times BO$ where $BO$ is the classifying space for the infinite orthogonal group.

It is well known that (1) and (2) are in bijection when $X$ is, say, a finite CW-complex. This can fail when $X$ is not compact though, and I think it also fails when $X$ is a nested sequence of circles in the plane with the subspace topology. Anyway, the two are different in general.

Now consider the following names and notation :

(A) the real K-theory of $X$;

(B) the real topological K-theory of $X$;

(C) the K-theory of real vector bundles over $X$;

(D) $K(X)$;

(E) $KO(X)$;

(F) $K_\mathbb{R}(X)$.

Which would you pair to which? and what is your favorite for (1) and for (2)?

For example I don't expect anyone to answer that C2 is reasonable. I'm curious whether E1 or E2 will appear. Some other notation may also be prefered, I think I saw $\mathbf{KO}(X)$ once, meaning (1), while $KO(X)$ was for (2). Or perhaps the other way around. There are also alternative names like "the Atiyah real K-theory", not sure meaning what.

Thanks! There may not be any universal convention, but at least we can have a vote of sorts.

Answer 1

One standard notational convention is that (2) = (E) for unbased spaces $X$, where the brackets mean unbased homotopy classes of maps. This is consistent with the classification theorem that equivalence classes of $n$-plane bundles over $X$ are classified as $[X,BO(n)]$. The classification works for general CW complexes $X$, not just finite ones. The restriction to finite CW complexes to prove that (1) = (2) comes in showing that the Grothendieck group, restricted to elements of virtual dimension zero, is isomorphic to $[X,BO]$.

However (E) is perhaps more usually used for the entire generalized cohomology theory whose zeroth term is (2). Then, in modern algebraic topology, it has become standard to focus on reduced cohomology theories defined on based spaces $X$, and the bracket $[X,Y]$ is then understood to mean homotopy classes of based maps. In early literature, $[X,Y]_*$ meant based homotopy classes, but the modern literature prefers confusion, leaving to context which is meant. Thus for unbased spaces, $$KO(X) = KO^0(X) = [X,BO\times \mathbf{Z}] = [X_+, BO\times \mathbf{Z}]_*.$$ The complex analogue is $$K(X) = K^0(X) = [X, BU\times \mathbf{Z}] = [X_+, BU\times \mathbf{Z}]_*,$$ so you absolutely must not use (D). Here $X_+$ is not the one-point compactification of Paul's answer in general, but rather the union of $X$ and a disjoint basepoint. This is the general way to describe unreduced cohomology in terms of reduced cohomology. It has gradually become more standard in algebraic topology to write $KO(X)$ rather than the historical $\widetilde{KO}(X)$ for $[X,BO\times \mathbf{Z}]_*$ when $X$ is based. One point crucial to modern algebraic topology is that generalized cohomology theories are represented by $\Omega$-spectra $E$, which are sequences of based spaces $E_n$ and based (weak) homotopy equivalences $E_n\longrightarrow \Omega E_{n+1}$; of course loop spaces only make sense in the based context.

Adams used $K_{\mathbf R}(X)$ and $K_{\mathbf C}(X)$ as synonyms for $KO(X)$ and $K(X) = KU(X)$. That is the context of your (F), but it never really caught on. One can use (B) instead of (A) for emphasis when necessary, to distinguish from algebraic $K$-theory. By abuse, (C) is also sometimes used as synonymous with (A), (B), (E), and (F), although it ought more logically be paired with (1). People sometimes use 'orthogonal' rather than `real' to avoid confusion with Atiyah's Real $K$-theory $KR$ as in Mark's answer. Atiyah's real vector bundles are often called Real vector bundles to avoid confusion, which works Really badly in talks.

In answer to Tom and Pierre, the nice paper in mind is ``Vector bundles over classifying spaces of compact Lie groups'' by Stefan Jackowski and Bob Oliver. It exploits the Sullivan conjecture to study (1) when $X=BG$ for a compact Lie group $G$.