# Augmentations of wrapped Floer cochains

Let $$M$$ be a closed, simply-connected spin manifold and let $$F_b \subset T^*M$$ be the cotangent fiber over a point $$b \in M$$. Let $$CW^*(L,L)$$ be the $$A_{\infty}$$-algebra of wrapped Floer cochains over a field k and let $$C_{-*}(\Omega_b M)$$ be the dg-algebra of chains on the based loop space.

Abouzaid proved that there is an equivalence of $$A_{\infty}$$-algebras $$CW^*(L,L) \simeq C_{-*}(\Omega_b M)$$.

Observe now that the collapse map $$M \mapsto b$$ induces an augmentation $$C_{-*} (\Omega_b M) \to C_{-*} ({pt}) \simeq k$$. Precomposing with Abouzaid's equivalence, we conclude that $$CW^*(L,L)$$ is an augmented $$A_{\infty}$$-algebra (i.e. it admits a morphism of $$A_{\infty}$$-algebras to $$k$$).

Question: can this augmentation be seen at the level of wrapped Floer homology, i.e. without going through Abouzaid's equivalence? A natural thing to do would be to intersect with the zero section. However, this does not appear define a morphism of $$A_{\infty}$$-algebras in an obvious way.

• Intersecting with the zero section defines a 1-dimensional module, i.e. a $A_\infty$-homomorphism to $End_k(k)\cong k$, which is the same as an augmentation. – Zack Feb 7 at 23:45