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I have some basic questions about real K-theory (I mean $KO$-theory).

Question 1: I have seen the table $$ KO^{-i}(\mathrm{pt})= \begin{cases} \mathbb{Z},& i=0\\ \mathbb{Z}_2,& i=1\\ \mathbb{Z}_2,& i=2\\ 0,& i=3\\ \mathbb{Z},& i=4\\ 0,& i=5\\ 0,& i=6\\ 0,& i=7\\ \end{cases} $$ in various places, for example on p. 15 of this paper, but haven't been able to find a reference where this is computed explicitly. Where could I find such a reference?

Question 2: What is the reason for using the negative indices $-i$, as opposed to $i$, for keeping track of the $KO$ groups?

Question 3: Can $KO^0(\mathrm{pt})$ be thought of as the path components of Fredholm operators, i.e. $\pi_0(\mathcal{F})$, where $\mathcal{F}$ are the Fredholm operators on a real Hilbert space? If so, is there a similar meaning in these terms for higher $i$?

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    $\begingroup$ Bott-periodicity $\endgroup$
    – F Zaldivar
    Commented Dec 11, 2021 at 0:02
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    $\begingroup$ To explain that previous comment a little more, which is for Question 1, it's to do with the homotopy groups of $\mathbb{Z}\times BO(\infty) = \mathbb{Z}\times \mathrm{colim}_nB O(n)$, which, IIRC, is a classifying space for $KO$. $\endgroup$
    – David Roberts ♦
    Commented Dec 11, 2021 at 0:18
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    $\begingroup$ Q1: Atiyah's paper "Clifford modules" (maths.ed.ac.uk/~v1ranick/papers/abs.pdf) is a good reference. I also explained the calculation in Section 7.2 of my thesis (neil-strickland.staff.shef.ac.uk/research/thesis.pdf). There is nothing significantly original there, but you might or might not find the exposition useful. $\endgroup$
    – Neil Strickland
    Commented Dec 11, 2021 at 0:33
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    $\begingroup$ Behrens's paper "A new proof of the Bott periodicity theorem" is another source for Q1. www3.nd.edu/~mbehren1/papers/bott.pdf . $\endgroup$
    – Gregory Arone
    Commented Dec 11, 2021 at 10:03
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    $\begingroup$ @GregoryArone For real Bott periodicity he has a follow up paper clarifying and simplifying certain sections: www3.nd.edu/~mbehren1/papers/addendum.pdf $\endgroup$
    – Connor Malin
    Commented Dec 11, 2021 at 22:51

1 Answer 1

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I have only answers for the first two questions.

Starting from the second one. Naturally you can define only $KO^{-i}(X)$. Namely, $KO^{-i}(X) \overset{\mathrm{def}}{=} KO^0(\Sigma^i X)$. And the standard way to define $KO^{i}(X)$ is the Bott periodicity.

As a reference to $KO^{-i}(pt)$ there is a nice book ``Algebraic topology -- homotopy and homology'' by R. Switzer. Chapter 11 contains the required computations.

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