Counterexamples in scale calculus

Abstract

We construct counterexamples to classical calculus facts such as the Inverse and Implicit Function Theorems in Scale Calculus -a generalization of Multivariable Calculus to infinite dimensional vector spaces in which the reparameterization maps relevant to Symplectic Geometry are smooth. Scale Calculus is a cornerstone of Polyfold Theory, which was introduced by Hofer-Wysocki-Zehnder as a broadly applicable tool for regularizing moduli spaces of pseudoholomorphic curves. We show how the novel nonlinear scale-Fredholm notion in Polyfold Theory overcomes the lack of Implicit Function Theorems, by formally establishing an often implicitly used fact: The differentials of basic germs -the local models for scale-Fredholm maps -vary continuously in the space of bounded operators when the base point changes. We moreover demonstrate that this continuity holds only in specific coordinates, by constructing an example of a scale-diffeomorphism and scale-Fredholm map with discontinuous differentials. This justifies the high technical complexity in the foundations of Polyfold Theory.

Quotient theorems in polyfold theory and S1‐equivariant transversality

“…The notation of an sc-Fredholm section was introduced in [21] to replace the naive definition of Fredholm section such that implicit function theorem holds. The novel extra condition in the sc-Fredholm notion is necessary as the naive definition does not imply implicit function theorem [10]. Therefore, preserving sc-Fredholm property is the most important property in any abstract constructions of polyfolds.…”

“…By Lemma A.2, there exists θ 1 ∈ Mor(F a (w a ), F a (w a )), θ 2 ∈ Mor(G a (w a ), G a (w a )), such that If we write 10) then h close to g and z close to x we have…”

“…x0 , that is, 10) and the inverse to Id x0 is itself, we can require the neighborhoodW is small enough such that we can invert (3.10) to get…”

“…By Theorem 5.10, there is a S 1 -equivariant sc + -perturbation γ supported in N such that s + γ is transverse to 0 on Z S 1 . By [10,Corollary 3.3], there is a S 1 -invariant neighborhood U ⊂ Z 3 of (Z S 1 ) 3 , such that s + τ is proper and transverse to 0 on U ⊂ Z 3 . Let C := U \Z S 1 , then C is a closed subset in Z 3 \Z S 1 .…”

“…Before proving Proposition B.5, we first state [10,Corollary 3.3], which will be useful for our proof. By Proposition B.3, we can find trivial sub-bundle F 1 extending F 1 hence covers coker D λ s over 1 .…”