Conventions and identities

Actions

All group actions are written additively. For \(g\in G\), \(\alpha_g\) denotes the induced automorphism. We use

\[ U_h aU_h^{\ast}=\alpha_h(a). \]

Changing to the inverse covariance convention changes some displayed curvatures by an overall sign but does not change any vanishing result.

For a \(\mathbb Z\)-module \(M\) with automorphism \(\alpha\),

\[ M_\alpha=M/(\alpha-1)M, \qquad M^\alpha=\ker(\alpha-1). \]

For \(M=C(\Sigma,\mathbb Z)\),

\[ K_0(C(\Sigma)\rtimes_\alpha\mathbb Z)\cong M_\alpha, \qquad K_1(C(\Sigma)\rtimes_\alpha\mathbb Z)\cong M^\alpha. \]

Our transports satisfy

\[ W_i^{\ast}W_i=\alpha_i(P), \qquad W_iW_i^{\ast}=P. \]

Thus \(W_i\alpha_i\) acts from the module \(PB_h^n\) back to itself.

\[ \Omega_{ij} =W_i\alpha_i(W_j) \bigl(W_j\alpha_j(W_i)\bigr)^{\ast}. \]

It is the identity precisely when the two semilinear generator transports commute.

For \(z_i\in U(PM_n(B_h)P)\), set \(W_i'=z_iW_i\) and

\[ \beta_i(a)=W_i\alpha_i(a)W_i^{\ast}. \]

Then

\[ \Omega_{ij}' =z_i\beta_i(z_j)\Omega_{ij} \beta_j(z_i)^{\ast}z_j^{\ast}. \]

For a full group cochain \(d_g\), the Busby–Smith cocycle changes by

\[ \nu'(g,h) =d_g\bar\beta_g(d_h)\nu(g,h)d_{g+h}^{\ast}. \]

These formulas fix the cohomology convention used throughout the book.