(my question is also meaningful for complex *K*-theory, but since *K*^{n}(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

Xbe a topological space. A class inKO(^{n}X) is represented by:

Option#1:

• A bundle of ℤ/2-graded real Hilbert spaces overX.

• Actions ofCliff(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 overX.

• Actions ofCliff(-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 overX.

• Actions ofCliff(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 overX.

• Actions ofCliff(-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 *KO ^{n}*(

*X*) and in

*KO*(

^{-n}*X*)

*!*

Is there a homotopy-theoretical explanation of the above phenomenon?

Are there other cohomology theoriesEthat have functors mapping to bothE(^{n}X) andE(^{-n}X)?

Maybe forE=TMF?

in the opposite order! Weird. – Charles Rezk Apr 22 '11 at 23:01