Hochschild chain model for the evaluation map at half

Question

Let M be a manifold and $\Lambda(M)$ its free loop space, $A= C^*(M)$ denotes the cochain algebra of $M$. We know that Hochschild chain model for the evaluation $ ev_0: \Lambda(M) \rightarrow M$ is given by $ A \hookrightarrow A \otimes T(s \bar{A})$, where $T(s \bar{A})$ denotes the free coalgebra generated by the graded vector space $s \bar{A}$ with $\bar{A}= \{A^{i}\}_{i \geq 1}$ and $s \bar{A}^i= A^{i+1}$. I would like to know if there is a Hochschild chain model for the map $ ev_{\frac{1}{2}}: \Lambda(M) \rightarrow M, \alpha \rightarrow \alpha(\frac{1}{2}) $.

I would also like to know a model for the inclusion $M\hookrightarrow \Lambda(M)$.

Thanks!

Answer 1

You can't model this map using Hochschild chains since this information is too fine for this algebraic model to capture. However, you can model the homotopies between the evaluation maps.

You could, similarly, ask to model the map $ev_{1/2}: PM \to M$, where $PM$ is the free path space, as a map from $A=C^*(M)$ to the two sided bar construction on $A$, but again this information is still too geometric to capture in terms of these models. However, you can model the homotopic maps $ev_0: PM \to M$ and $ev_1: PM \to M$ and the homotopy between them. The map $ev_0$ is modeled by the composition

$A \to A \otimes B(A,A,A)\to B(A,A,A)$

where $B(A,A,A)$ is the two sided bar construction on $A$, the first map is the inclusion (using the unit), and the second map comes from the left $A$ module structure of $B(A,A,A)$. Similarly $ev_1$ is modeled using the right $A$ module structure of $B(A,A,A)$. Its not hard to find an explicit formula for the chain homotopy between these two maps.

The inclusion of $M$ into $PM$ as constant paths is modeled by the quasi isomorphism $B(A,A,A) \to A$ that sends $a \otimes a_1 \otimes ... \otimes a_k \otimes b$ to the product $ab$ if $k=0$ and to $0$ otherwise.

My favorite way to see that this maps are modeling the right maps of spaces is through Chen iterated integrals.