2
$\begingroup$

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.

$\endgroup$
  • 2
    $\begingroup$ 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. $\endgroup$ – Zack Feb 7 at 23:45

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.