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If $X$ is a compact Hausdorff space, we can consider the Grothendieck ring of real vector bundles on $X$, $\mathit{KO}^0(X)$, and this extends to a generalized cohomology theory represented by a ring spectrum $\mathit{KO}$. Using complex vector bundles, we get another generalized cohomology theory, represented by $\mathit{KU}$. Using quaternionic vector bundles, we get a third one, represented by $\mathit{KSp} \simeq \Sigma^4\mathit{KO}$.

In the same way, one can define the Grothendieck ring of oriented vector bundles $\mathit{KSO}(X)$, spin vector bundles $\mathit{KSpin}(X)$, and so on for any $G$-structure. Do these constructions extend to spectra $\mathit{KSO}$, $\mathit{KSpin}$, and so on? If not, what issues arise?

The first concern I would imagine is a failure of Bott periodicity, but $\pi_k(\mathit{SO})$ and $\pi_k(\mathit{Spin})$ agree with $\pi_k(O)$ for $k\ge 2$, so in at least these cases some sort of construction might be possible.

An alternative approach sidestepping Bott periodicity would be to use algebraic $K$-theory: if $\mathsf{Vect}_{k}$ denotes the topological symmetric monoidal category of $k$-vector spaces, then $K(\mathsf{Vect}_{\mathbb R}) = \mathit{ko}$ and $K(\mathsf{Vect}_{\mathbb C}) = \mathit{ku}$. So if $\mathsf{Vect}^{\mathrm{or}}$ denotes the topological symmetric monoidal category of oriented real vector spaces, would it be reasonable to define $\mathit{kso}:= K(\mathsf{Vect}^{\mathrm{or}})$? Is this an interesting object? This could also generalize to $G$-structures.

Someone must have thought about this, but I can't find it written down anywhere.

Edit: Though I've accepted Denis' excellent answer, I would still be interested to learn of existing references which consider these kinds of cohomology theories/spectra.

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    $\begingroup$ $KSO(X)$ is just the kernel of the natural map from $KO(X)$ to $H^1(X,\mathbb Z/2)$. So probably if it can be defined, the spectrum is not far away from $KO$. $\endgroup$
    – Will Sawin
    Apr 18, 2017 at 2:04
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    $\begingroup$ Likewise, $KSpin$ is essentially only the $4$-connective cover of $KO$, only that we decided that we leave the $\pi_0$-unchanged. This happens in general if you take an $X$-structure classified by $BO\langle n\rangle BO$, just in case you want to consider $KString$ or so. What other interesting examples are there? $\endgroup$
    – Lennart Meier
    Apr 18, 2017 at 15:44

2 Answers 2

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I am not sure if this is going to be a real answer to the question. However I believe these observations might be interesting.

Let me briefly sketch a way to describe a $G$-structure in (excessively) wide generality. Consider a fibration of spaces $\theta:X\to \coprod_n BO_n$. Then a $\theta$-structure on a vector bundle $V\to B$ is just a lift along $\theta$ of the map $X\to \coprod_n BO_n$ classifying $V$. Examples are orientations (with $X=\coprod_n BSO_n$), spin structures ($X=\coprod_n BSpin_n$), complex structures ($X=\coprod_n BU_n$) etc.

Now you want to define the sum of vector bundles with $\theta$-structure. In order to do so we need to require a little bit more of the map $\theta$. I'm going to assume that $X$ is an $E_\infty$-space and that $\theta$ is equipped with the structure of map of $E_\infty$-spaces. Then the abelian group of vector bundles with $\theta$-structure on some base $B$ is easily seen to be $$ \pi_0\mathrm{Map}(B,X)$$ Note that $\mathrm{Map}(B,X)$ has a canonical $E_\infty$-structure inherited from $X$. So it is natural to define the connective K-theory of vector bundles with $\theta$-structure as $$ K\theta^i(B) = \pi_{-i}\mathrm{Map}(B,X)^+$$ where $(-)^+$ is the group completion of $E_\infty$-spaces.

When the group completion of $X$ is given by a filtered colimit over the divisibility poset of $\pi_0X$ (for example $X=\coprod_n BO_n$, $X=\coprod_n BU_n$ or $X=\coprod_n BSp_n$) and $B$ is a finite space (that is the homotopy type of a CW complex with finitely many cells) you can rewrite the right hand side as $$ K\theta^i(B) = \pi_{-i}\mathrm{Map}(B,X^+)$$ since filtered colimits in spaces commute with finite limits. As Marc Hoyois notes in the comments, this is always true if every connected component of $X$ is simply connected. So $K\theta^i(-)$ restricted to finite spaces is a cohomology theory represented by the connective spectrum $X^+$.

Note that you might want a little more of this theory: you might want your $X^+$ to be an $E_\infty$-ring and not just a connective ring spectrum. This corresponds to the assumption that $X$ is an $E_\infty$-ring space and that $\theta$ respects this structure.

I do not know of a way to generalize nonconnective K-theory in this situation (of course one could just $K(1)$-localize, but that is a rather ad hoc procedure).

Also, the map $\theta$ is obviously just along for the ride, but since it allows us to think of $K\theta$ as "K-theory of vector bundles with structure" I think it helps the intuition.

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  • $\begingroup$ This is nice! How do you see that you can pull the group completion inside when X is BO, BU and BSp? $\endgroup$
    – J Cameron
    Apr 18, 2017 at 5:16
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    $\begingroup$ @JamesCameron It follows from the fact that in the ∞-category of spaces finite limits commute with filtered colimits, and that in those cases the group completion can be given by a filtered colimit (since $\mathrm{Map}(B,-)$ is a finite limit). Now that I write it, I have the feeling that it should work for pretty much any $E_∞$-space $X$, let me see if I can straighten out the details.... $\endgroup$
    – Denis Nardin
    Apr 18, 2017 at 11:35
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    $\begingroup$ @DenisNardin I think that to express the group completion $X^+$ using filtered colimits you need to know that for every $x\in X$, there exists $m\geq 2$ such that the cyclic permutation of $x^m$ is homotopic to the identity (at least, I only know how to prove it in this case). These permutations live in $\pi_1$'s of $X$. In the $U_n$ and $Sp_n$ cases these are trivial, in the $O_n$ case we have $Z/2$ so any odd $m$ works. But this fails completely in other cases, eg when $X=\coprod_n BGL_n(R)$ for some nonzero commutative ring $R$. $\endgroup$
    – Marc Hoyois
    Apr 18, 2017 at 14:17
  • $\begingroup$ @MarcHoyois Hmm.. That is a good point. We can't expect to get algebraic K-theory spectra that easily. That's probably the best we can get in general. $\endgroup$
    – Denis Nardin
    Apr 18, 2017 at 14:35
  • $\begingroup$ This is really nifty. Thanks for writing it down! Time to play with some examples. $\endgroup$
    – Arun Debray
    Apr 23, 2017 at 4:49
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Regarding preexisting references, I found a few examples of people using Grothendieck groups of oriented or spin vector bundles, but no cohomology theories or spectra. I'll leave them here in case anyone else is interested.

For $\mathit{KSpin}$:

For $\mathit{KSO}$:

  • Pejsachowicz, “Bifurcation of Fredholm maps II: the dimension of the set of bifurcation points.”
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