An extension of the Sard–Smale Theorem to convex domains with an empty interior

Mayer-Vietoris property for relative symplectic cohomology

“…We have the map 𝒱 → 𝑀 × 𝑀 given by (ℎ, 𝑥) ↦ → (𝑥, 𝜑 1 ℎ+𝐻 (𝑥)). The Sard-Smale theory of transversality extends to this setting [1]. The previous lemma shows that this map is transverse to the diagonal in 𝑀 ×𝑀 in the appropriate sense, and finishes the proof.…”

“…be a 1-ray. The telescope 𝑡𝑒𝑙(𝒞) of such a diagram is defined to be the chain complex with the underlying 𝑅-module ⨁︀ 𝑖∈N 𝐶 𝑖 [1] ⊕ 𝐶 𝑖 and the differential as depicted below:…”

“…. .. Now define the 𝑅-modules at the vertices of 𝑡𝑒𝑙(𝒞) as the entrywise direct sum ⨁︀ 𝑖∈N 𝒞 𝑖 [1] ⊕ 𝒞 𝑖 . The maps in the (𝑛 − 1)-cube structure are depicted in:…”

“…Let 𝐹 : 𝑀 × [0, 1] → R be a Hamiltonian, and 𝛾 : [0, 1] → 𝑀 a flow line of 𝑋 𝐹𝑡 . Then, for any 𝑡 0 ∈ (0, 1), and neighborhood 𝑉 of 𝛾(𝑡 0 ); we have that for every 𝑣 ∈ 𝑇 𝛾(1) 𝑀 , there is a smooth one parameter family of functions 𝑓 𝑠 : 𝑀 × [0, 1] → R, 𝑠 ∈ [0, 𝜏 ), for some 𝜏 > 0, such that • 𝑓 0 = 0 • for some 𝜖 > 0, 𝑓 𝑠 (𝑥, 𝑡) = 0, for 𝑡 < 𝜖 and 𝑡 > 1 − 𝜖 and all 𝑠 • 𝑓 𝑠 ≥ 0, for all 𝑠 • 𝑠𝑢𝑝𝑝(𝑓 𝑠 ) ⊂ 𝑉 , for all 𝑠 • The tangent vector to the curve 𝑠 ↦ → 𝜑 1 𝐹 +𝑓𝑠 (𝛾(0)) at 𝑠 = 0 is equal to 𝑣.…”