Algebraic construction and properties of Hermitian analogs of K-theory over rings with involution from the viewpoint of Hamiltonian formalism. Applications to differential topology and the theory of characteristic classes

Abstract. The present paper is an immediate continuation of the author’s paper [22], Except in the last section, it is implicitly assumed here, as in [22], that the underlying ring contains 1/2 and all the theorems relate to the theory U ⊗ Z[1/2] without further comment. 7. Proof of the theorems stated in § 6 We begin with a proof of the technically most difficult theorem. Theorem 6.3, which says B1U B̄ 1 U = ±1. The first stage of the proof consists of clarifying the projective class of the lagrangian plane which represents an element from B1U B̄ 1 U (α), where α ∈ U1j (A) is the lagrangian plane pL = L = (aX + bP) in the hamiltonian space Hn = (x1,..., xn, p1,..., pn) and where L ∗ = xL is the lagrangian plane L ∗ = (cX + dP), written in terms of the basis (for the free case), and is considered to be chosen explicitly. We recall that B̄1U (α) is a form φ = z −1φ−1+φ0+zφ1 with basis E, X̃, P ̃ and with the relations (z − 1)E = aX ̃ + bP ̃ , 〈X̃i, P̃j 〉 = (z − 1)δij (X ̃ = P ̃ ∗), 〈X̃, X̃ 〉 = 〈P ̃ , P ̃ 〉 = 0. For definiteness we will write all the formulas in the hermitian case so that φ becomes skew-hermitian. In the other case they will be the same, apart from the necessary change of sign. Recall that the projective class of the required lagrangian plane B1U (φ) is given as B(detφ) — see § 6. Here we will make our calculations in terms of the basis (e) of module E and (v) = cX ̃ + dP ̃ ∼ = L∗z = xL generating the module B̄1U (α) = F. We have the relations x ̃ = d̄(z − 1)e+ b̄v, p ̃ = c̄(z − 1)e+ āv. The module F is free. Let us denote by v ∗ and e ∗ submodules in F such that 〈v∗, e 〉 = 〈e∗, v 〉 = 0 in the sense of the form φ; thus e ∗ is dual to E[z, z−1], and v ∗ is dual to V ∗ ∼ = L[z, z−1]. If E and V (∼ = Lz, L∗z) are free modules then it is convenient to choose canonical dual bases in (e∗) and (v∗). In all cases we choose “bases ” in the modules (e∗) and (v∗) over A[z, z−1] which are dual to the bases of E and V in the sense that decomposition into the sum