(my question is also meaningful for complex K-theory, but since Kn(X) is always isomorphic to K-n(X), it's less interesting)
I start by recalling the analytic definition of KO-theory:
The following ingredients are needed:
Clifford algebras:
For n∈ℤ, the Clifford algebra Cliff(n) is the following ℤ/2-graded C*-algebra:
• $\langle e_1,\ldots, e_n\;|\; e_i \text{ is odd}, e_i^2=1, e_ie_j=-e_je_i, e_i^*=e_i\rangle$ if n ≥ 0.
• $\langle e_1,\ldots, e_{-n}\;|\; e_i \text{ is odd}, e_i^2=-1, e_ie_j=-e_je_i, e_i^*=-e_i\rangle$ if n ≤ 0.
Note: The above definition might seem a bit weird with its two cases n ≥ 0 and n ≤ 0, but actually it's quite ok: the map n $\mapsto$ Cliff(n) can be extended to a homomorphism from the associative group (ℤ,+) to the monoidal 2-category of ℤ/2-graded C${}^*$-algebras, bimodules, and bimodule maps.
Fredholm operators:
If H is a Hilbert space, then
an operator F : H→ H is called Fredholm if boths its kernel and cokernel are finite dimensional.
The definition:
Let X be a topological space. A class in KOn(X) is represented by:
Option #1:
• A bundle of ℤ/2-graded real Hilbert spaces over X.
• Actions of Cliff(n) on the fibers of the above bundle.
• Fiberwise Fredholm operators that are odd, Cliff-linear, and skew-adjoint.
Option #2:
• A bundle of ℤ/2-graded real Hilbert spaces over X.
• Actions of Cliff(-n) on the fibers of the above bundle.
• Fiberwise Fredholm operators that are odd, Cliff-linear, and self-adjoint.
Dually, a class in KO-n(X) can be represented by:
Option #1:
• A bundle of ℤ/2-graded real Hilbert spaces over X.
• Actions of Cliff(n) on the fibers of the above bundle.
• Fiberwise Fredholm operators that are odd, Cliff-linear, and self-adjoint.
Option #2:
• A bundle of ℤ/2-graded real Hilbert spaces over X.
• Actions of Cliff(-n) on the fibers of the above bundle.
• Fiberwise Fredholm operators that are odd, Cliff-linear, and skew-adjoint.
If the bundles are finite dimensional, then the Fredholm operators can be taken to be zero. As a consequence, we get two maps
Bundles of finite dimensional Cliff(n)-modules over X${}\to KO^n(X)$
Bundles of finite dimensional Cliff(n)-modules over X${}\to KO^{-n}(X)$
In other words, a bundle of finite dimensional Cliff(n)-modules represents both an element in KOn(X) and in KO-n(X)!
Is there a homotopy-theoretical explanation of the above phenomenon?
Are there other cohomology theories E that have functors mapping to both En(X) and E-n(X)?
Maybe for E=TMF?