E0035 proposition notation E0020 → E0001

Inverse pushforward

The symmetric pushforward \(D_+(\phi; x)\) E0027 is invertible whenever the first derivative is, by a finite Neumann sum that separates the inverse of the linear part from the higher-order corrections.

(1) Proposition. Let \(\phi\) be \(C^k\) near \(x\) with invertible differential \(D(\phi; x) \in \KL(E, F)\). Split the pushforward by partition type:

\[ D_+(\phi; x) = S + N, \qquad S := \bigoplus_{r=0}^{k} \SYM^r(D(\phi; x)), \]

where \(S\) collects the singleton partitions and \(N\) the partitions containing a block of size \(\geq 2\).

  • \(N\) strictly lowers the degree (a partition with a block of size \(\geq 2\) has fewer blocks than elements).
  • \(S\) is degreewise invertible with \(S^{-1} = \bigoplus_{r=0}^{k} \SYM^r(D(\phi; x)^{-1})\).
  • \(S^{-1} N\) is nilpotent: \((S^{-1} N)^k = 0\) on \(ST_k\).

  • \(D_+(\phi; x)\) is invertible with

    \[ D_+(\phi; x)^{-1} = \left(\sum_{m=0}^{k-1} (-S^{-1} N)^m\right) S^{-1}. \]

  • If \(\phi\) has a \(C^k\) local inverse at \(x\), then \(D_+(\phi^{-1}; y) = D_+(\phi; x)^{-1}\) by smooth functoriality E0028.

Proof. A partition of \([r]\) with a block of size \(\geq 2\) has at most \(r - 1\) blocks, so its term in the pushforward E0027 lands in degree \(\leq r - 1\): \(N\) strictly lowers degree and vanishes in degrees \(\leq 1\), while \(S\) is degreewise invertible with the stated inverse. Hence \(S^{-1} N\) strictly lowers degree, \((S^{-1} N)^k = 0\) on \(ST_k(X; x)\), and the finite geometric sum inverts \(1 + S^{-1} N\). Then \(D_+(\phi; x) = S(1 + S^{-1} N)\) is invertible with the stated sum. No smallness condition enters — the inversion is purely algebraic.

For the last claim, apply functoriality E0028 to \(\phi^{-1} \circ \phi = \id\) and \(\phi \circ \phi^{-1} = \id\).

(2) Remark. In degree two, the degree-one component of the inverse gives \(D(\phi^{-1}; y; w_1, w_2) = -A\, D(\phi; x; Aw_1, Aw_2)\) where \(A = D(\phi; x)^{-1}\). Expanded, the Neumann sum is the tree expansion of the classical inverse Faà di Bruno formula.