# Cyclic theory for commutative differential graded algebras and s–cohomology.

Dan Burghelea  Dept. of Mathematics, The Ohio State University, 231 West
Avenue, Columbus, OH 43210, USA<EMAIL_ADDRESS>
###### Abstract

In this paper one considers three homotopy functors on the category of
manifolds , $hH^{\ast},cH^{\ast},sH^{\ast},$ and parallel them with other
three homotopy functors on the category of connected commutative differential
graded algebras, $HH^{\ast},CH^{\ast},SH^{\ast}.$ If $P$ is a smooth
1-connected manifold and the algebra is the de-Rham algebra of $P$ the two
pairs of functors agree but in general do not. The functors $HH^{\ast}$ and
$CH^{\ast}$ can be also derived as Hochschild resp. cyclic homology of
commutative differential graded algebra, but this is not the way they are
introduced here. The third $SH^{\ast},$ although inspired from negative cyclic
homology, can not be identified with any sort of cyclic homology of any
algebra. The functor $sH^{\ast}$ might play some role in topology. Important
tools in the construction of the functors $HH^{\ast},CH^{\ast}$and
$SH^{\ast},$ in addition to the linear algebra suggested by cyclic theory, are
Sullivan minimal model theorem and the ”free loop” construction described in
this paper.

(dedicated to A. Connes for his 60-th birthday)

###### Contents

  1. 1 Introduction
  2. 2 Mixed complexes, a formalism inspired from Connes’ cyclic theory
  3. 3 Mixed commutative differential graded algebras
  4. 4 De-Rham Theory in the presence of a smooth vector field
  5. 5 The free loop space and s-cohomology
  6. 6 The free loop construction for CDGA
  7. 7 Minimal models and the proof of Theorem 3

##  1 Introduction

This paper deals with commutative graded algebras and the results are of
significance in “commutative” geometry/topology. However they were inspired
largely by the linear algebra underlying Connes’ cyclic theory. The
topological results formulated here, Theorem 2 and Theorem 3, were first
established as a consequence of the identification of the cohomology resp.
$S^{1}-$equivariant cohomology of the free loop spaces of a 1-connected smooth
manifold with the Hochschild resp. cyclic homology of its de-Rham algebra, cf.
[J], [BFG], [B2]. In this paper this identification is circumvented. Still,
the results illustrate the powerful influence of Connes’ mathematics in areas
outside the non commutative geometry.

In this paper, inspired by the relationship between Hochschild, cyclic and
negative cyclic homology of a unital algebra, one considers two systems of
graded vector space valued homotopy functors
$hH^{\ast},cH^{\ast},sH^{\ast}$and $HH^{\ast},CH^{\ast},SH^{\ast}$ and
investigate their relationship. The first three functors are defined on the
category of smooth manifolds and smooth maps via the free loop space
$P^{S^{1}}$ of a smooth manifold $P,$ which is a smooth $S^{1}$-manifold of
infinite dimension. The next three functors are defined on the category of
connected commutative differential graded algebras via an algebraic analogue
of “free loop” construction and via Sullivan minimal model theorem, Theorem 1.
The relationship between them is suggested by the general nonsense- diagram
Fig 2 in section 2.

When applied to the De–Rham algebra of a 1-connected smooth manifold the last
three functors take the same values as the first three. This is not the case
when the smooth manifold is not 1-connected; the exact relationship will be
addressed in a future work.

The first three functors are based on a formalism (manipulation with
differential forms) which can be considered for any smooth (finite or infinite
dimensional) manifold $M$ and any smooth vector field $X$ on $M.$ However it
seems to be of relevance when the vector field $X$ comes from a smooth
$S^{1}-$action on $M.$ This is of mild interest if the manifold is of finite
dimension but more interesting when the manifold is of infinite dimension. In
particular, it is quite interesting when $M=P^{S^{1}},$ the free loop space of
$P,$ and the action is the canonical $S^{1}-$action on $P^{S^{1}}.$
Manipulation with differential forms on $P^{S^{1}}$ leads to the graded vector
spaces $hH^{\ast}(P)$, $cH^{\ast}(P),$ $sH^{\ast}(P)$ with the first two being
the cohomology resp. the $S^{1}-$equivariant cohomology of $P^{S^{1}}$ but
$sH^{\ast}$ being a new homotopy functor, referred here as s–cohomology.

This functor was first introduced in [B1], [B2] but so far not seriously
investigated. The functor $sH^{\ast}$ relates, at least in the case of a
1-connected manifold $P$, the Waldhaussen algebraic $K-$theory of $P$ and the
Atiyah–Hirtzebruch complex $K-$theory (based on complex vector bundles) of
$P.$ It has a rather simple description in terms of infinite sequence of
smooth invariant differential forms on $P^{S^{1}}.$

The additional structures on $P^{S^{1}},$ the power maps $\psi_{k}$
$k=1,2,\cdots,$ and the involution $\tau=\psi_{-1},$ provide endomorphisms of
$hH^{\ast}(P)$, $cH^{\ast}(P),sH^{\ast}(P)$ whose eigenvalues and eigenspaces
are interesting issues. They are clarified only when $P$ is 1-connected. This
is done in view of the relationship with the functors
$HH^{\ast},CH^{\ast},SH^{\ast}.$

It might be only a coincidence but still a appealing observation that the
symmetric resp. antisymmetric part of $sH^{\ast}(P)$ w.r. to the canonical
involution $\tau$ calculates, for a 1-connected manifold $P$ and in the
stability range, the vector space $Hom(\pi_{\ast}(H/\textit{Diff}(P),\
\kappa),$ $\kappa=\mathbb{R},\mathbb{C};$ the symmetric part when $\dim P$ is
even the antisymmetric part when $\dim P$ is odd, cf. [Bu], [B1]. Here
$H/\textit{Diff}(P)$ denotes the (homotopy) quotient of the space of homotopy
equivalences of $P$ by the group of diffeomorphisms with the
$C^{\infty}-$topology.

The functors $HH^{\ast},CH^{\ast},SH^{\ast}$ are the algebraic version of
$hH^{\ast},cH^{\ast},sH^{\ast}$ and are defined on the category of
(homologically connected) commutative differential graded algebras. Their
definition uses the “free loop” construction, an algebraic analogue of the
free loop space, described in this paper only for free connected commutative
differential graded algebras $(\Lambda[V],d_{V}).$ A priori these functors are
defined only for free connected commutative differential graded algebras.
Since they are homotopy functors they extend to all connected commutative
differential graded algebras via Sullivan minimal model theorem, Theorem 1.

Using the definition presented here one can take full advantage of the simple
form the algebraic analogue of the power maps take on the free loop
construction. As a consequence one obtains a simple description of the
eigenvalues and eigenspaces of the endomorphisms induced from the algebraic
power maps on $HH^{\ast}$ and $CH^{\ast}$ and implicitly understand their
additional structure.

The extension of the results of Sullivan–Vigué, cf. [VS], to incorporate
$S^{1}-$action and the power maps in the minimal model of $P^{S^{1}}$
summarized in Section 7 leads finally to results about
$hH^{\ast},cH^{\ast},sH^{\ast}$ when $P$ is 1-connected, cf. Theorem 3.

In addition to the algebraic definition of $HH^{\ast},CH^{\ast},SH^{\ast}$
this paper contains the proof of the homotopy invariance of $sH^{\ast}.$

##  2 Mixed complexes, a formalism inspired from Connes’ cyclic theory

A mixed complex $(C^{\ast},\delta^{\ast},\beta_{\ast})$ consists of a graded
vector space $C^{\ast}$ ( $\ast$ a non negative integer) and linear maps,
$\delta^{\ast}:C^{\ast}\to C^{\ast+1},$ $\beta_{\ast+1}:C^{\ast+1}\to
C^{\ast}$ which satisfy:

$\displaystyle\delta^{\ast+1}\cdot\delta^{\ast}=$ $\displaystyle 0$
$\displaystyle\beta_{\ast}\cdot\beta_{\ast+1}=$ $\displaystyle 0$
$\displaystyle\beta_{\ast+1}\cdot\delta^{\ast}+\delta^{\ast-1}\cdot\beta_{\ast}=$
$\displaystyle 0.$

When no risk of confusion the index $\ast$ will be deleted and we write
$(C,\delta,\beta)$ instead of $(C^{\ast},\delta^{\ast},\beta_{\ast})$. Using
the terminology of [B2], [BV2] a mixed complex can be viewed either as a
cochain complex $(C^{\ast},d^{\ast})$ equipped with an $S^{1}-$action
$\beta_{\ast}$, or as a chain complex $(C^{\ast},\beta_{\ast})$ equipped with
an algebraic $S^{1}-$action $\delta^{\ast}.$

To a mixed complex $(C^{\ast},\delta^{\ast},\beta_{\ast})$ one associates a
number of cochain, chain and $2-$periodic cochain complexes, and then their
cohomologies, homologies and $2-$periodic cohomologies111 We will use the word
”homology” for a functor derived from a chain complex and “cohomology” for one
derived from a cochain complex. The $2-$periodic chain and cochain complexes
can be identified., as follows.

First denote by

$\displaystyle{}^{+}C^{r}=\prod_{k\geq 0}C^{r-2k}$
$\displaystyle{}^{-}C^{r}:=\prod_{k\geq 0}C^{r+2k}$ (1)
$\displaystyle\mathbb{P}C^{2r+1}=\prod_{k\geq 0}C^{2k+1}$ $\displaystyle\ \
\mathbb{P}C^{2r}=\prod_{k\geq 0}C^{2k}\ \text{for any}\ r$ $\displaystyle
PC^{2r+1}=\bigoplus_{k\geq 0}C^{2k+1}$ $\displaystyle\ \
PC^{2r}=\bigoplus_{k\geq 0}C^{2k}\text{for any}\ r.$

Since our vector spaces are $\mathbb{Z}_{\geq 0}$-graded the direct product
${}^{+}C^{r}$ involves only finitely many factors.

Next introduce

$\displaystyle{}^{+}D^{r}_{\beta}(w_{r},w_{r-2},\cdots):=$
$\displaystyle(\delta\omega_{r},(\delta\omega_{r-2}+\beta\omega_{r}),\cdots)$
(2) $\displaystyle{}^{+}D^{\delta}_{r}(w_{r},w_{r-2},\cdots):=$
$\displaystyle((\beta\omega_{r}+\delta\omega_{r-2}),(\beta\omega_{r-2}+\delta\omega_{r-4}),\cdots)$
$\displaystyle{}^{-}D^{r}_{\beta}(\cdots,w_{r+2},w_{r}):=$
$\displaystyle(\cdots,(\beta\omega_{r+4}+\delta\omega_{r+2}),(\beta\omega_{r+2}+\delta\omega_{r}))$
$\displaystyle{}^{-}D^{\delta}_{r}(\cdots,w_{r+2},w_{r}):=$
$\displaystyle(\cdots,(\delta\omega_{r}+\beta\omega_{r+2}),\beta\omega_{r})$
$\displaystyle D^{2r}(\cdots,\omega_{2r+2},\omega_{2r},\cdots\omega_{0})=$
$\displaystyle(\cdots,(\delta\omega_{2r}+\beta\omega_{2r+2}),\cdots)$
$\displaystyle D^{2r+1}(\cdots,\omega_{2r+3},\omega_{2r+1}\cdots\omega_{1})=$
$\displaystyle(\cdots,(\delta\omega_{2k+1}+\beta\omega_{2k+3}),\cdots).$

Finally consider the cochain complexes

$\mathcal{C}:=(C^{\ast},\delta^{\ast}),\ \
^{+}\mathcal{C}_{\beta}:=(^{+}C^{\ast},^{+}D^{\ast}_{\beta}),\ \
^{-}\mathcal{C}_{\beta}:=(^{-}C^{\ast},^{-}D^{\ast}_{\beta}),$

the chain complexes

$\mathcal{H}:=(C^{\ast},\beta_{\ast}),\ \
^{+}\mathcal{H}^{\delta}:=(^{+}C^{\ast},^{+}D^{\delta}_{\ast}),\ \
^{-}\mathcal{H}^{\delta}:=(^{-}C\ast,^{-}D^{\delta}_{\ast})$

and the $2-$periodic cochain complexes 222 here
$(\mathbb{P}C^{\ast},D^{\ast})$ is regarded as a cochain complex with
$D^{\ast}$ obtained from the degree $+1$ derivation $\delta$ perturbed by the
degree $-1$ derivation $\beta$ ; the same complex can be regarded chain
complex with $D^{\ast}$ obtained from the degree $-1$ derivation $\beta$
perturbed by the degree $+1$ derivation $\delta$; the cohomology for the first
is the same as homology for the second

$P\mathcal{C}:=(PC^{\ast},D^{\ast}),\ \
\mathbb{P}\mathcal{C}:=(\mathbb{P}C^{\ast},D^{\ast})$

whose cohomology, homology, $2-$periodic cohomology are denoted by

$\displaystyle H^{\ast}:=H^{\ast}(C,\delta),$
$\displaystyle{}^{+}H^{\ast}_{\beta}:=^{+}H^{\ast}_{\beta}(C,\delta,\beta),\ $
$\displaystyle{}^{-}H^{\ast}_{\beta}$
$\displaystyle:=^{-}H^{\ast}_{\beta}(C,\delta,\beta),$ $\displaystyle
H_{\ast}:=H_{\ast}(C,\beta),$
$\displaystyle{}^{+}H^{\delta}_{\ast}:=^{+}H^{\delta}_{\ast}(C,\delta,\beta),\
$ $\displaystyle{}^{-}H^{\delta}_{\ast}$
$\displaystyle:=^{-}H^{\delta}_{\ast}(C,\delta,\beta),$ $\displaystyle
PH^{\ast}:=PH^{\ast}(C,\delta,$ $\displaystyle\beta),\ $
$\displaystyle\mathbb{P}H^{\ast}$
$\displaystyle:=\mathbb{P}H^{\ast}(C,\delta,\beta).$

In this paper the chain complexes $\mathcal{H},^{\pm}{\mathcal{H}}^{\delta},$
will only be used to derive conclusions about the cochain complexes
$\mathcal{C},^{\pm}{\mathcal{C}}_{\beta},\ \mathbb{P}\mathcal{C}.$

The obvious inclusions and projections lead to the following commutative
diagrams of short exact sequences

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{}^{-}\mathcal{H}^{\delta}_{\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{P}\mathcal{C}^{\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{}^{+}\mathcal{H}^{\delta}_{\ast-2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{H}_{\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{}^{+}\mathcal{H}^{\delta}_{\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{}^{+}\mathcal{H}^{\delta}_{\ast-2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{}^{+}\mathcal{C}^{\ast-2}_{\beta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{\ast-2}}$$\scriptstyle{id}$$\textstyle{{}^{+}\mathcal{C}^{\ast}_{\beta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{I^{\ast}}$$\textstyle{\mathcal{C}^{\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{}^{+}\mathcal{C}^{\ast-2}_{\beta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{I^{\ast-2}}$$\textstyle{\mathbb{P}\mathcal{C}^{\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{}^{-}\mathcal{C}^{\ast}_{\beta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

They give rise to the following commutative diagram of long exact sequences

$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{}^{-}H_{r}^{\delta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{r}}$$\textstyle{\mathbb{P}H(r)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{}^{+}H_{r-2}^{\delta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id}$$\textstyle{{}^{-}H_{r-1}^{\delta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{r-1}}$$\textstyle{\cdots}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{J_{r}}$$\textstyle{{}^{+}H_{r}^{\delta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{S_{r}}$$\textstyle{{}^{+}H_{r-2}^{\delta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{B_{r-2}}$$\textstyle{H_{r-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$

Fig 1.

$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{}^{+}H^{r-2}_{\beta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{S^{r-2}}$$\scriptstyle{id}$$\textstyle{{}^{+}H^{r}_{\beta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{I}^{\ast}}$$\textstyle{H^{r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{B^{r}}$$\scriptstyle{i^{r}}$$\textstyle{{}^{+}H_{\beta}^{r-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id}$$\textstyle{\cdots}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{}^{+}H^{r-2}_{\beta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{I}^{r-2}}$$\textstyle{\mathbb{P}H^{r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{J}^{r}}$$\textstyle{{}^{-}H^{r}_{\beta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{B}^{r}}$$\textstyle{{}^{+}H^{r-1}_{\beta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$

Fig 2.

and

$\textstyle{{}^{+}H^{r-2}_{\beta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{I}^{r-2}}$$\scriptstyle{S^{r-2}}$$\textstyle{{}^{+}H^{r}_{\beta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{I}^{r}}$$\textstyle{\mathbb{P}H^{r}=\mathbb{P}H^{r+2}}$$\textstyle{{\underset{\underset{S}{\rightarrow}}{\lim}^{+}H^{r+2k}_{\beta}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

The diagram (Fig1) is the one familiar in the homological algebra of
Hochschild versus cyclic homologies, cf [Lo]. The diagram Fig 2 is the one we
will use in this paper.

Note that Hochschild, cyclic, periodic cyclic, negative cyclic homology of an
associative unital algebra $A$ as defined in [Lo], is
$H_{\ast},^{+}H^{\delta}_{\ast},$ $\mathbb{P}H^{\ast},^{-}H^{\beta}_{\ast}$ of
the Hochschild mixed complex with $C^{r}:=A^{\otimes(r+1)}$ , $\beta$ the
Hochschild boundary, and $\delta^{r}=(1-\tau_{r+1})\cdot
s_{r}\cdot(1+\tau_{r}+\cdots\tau^{r}_{r})$ where $\tau_{r}(a_{0}\otimes
a_{1}\otimes\cdots a_{r})=(a_{r}\otimes a_{0}\otimes\cdots a_{r-1})$ and
$s_{r}(a_{0}\otimes a_{1}\otimes\cdots a_{r})=(1\otimes a_{0}\otimes
a_{1}\otimes\cdots a_{r}).$

A morphism
$f:(C^{\ast}_{1},\delta^{\ast}_{1},\beta^{1}_{\ast})\to(C^{\ast}_{2},\delta^{\ast}_{2},\beta^{2}_{\ast})$
is a degree preserving linear map which intertwines $\delta^{\prime}$s and
$\beta^{\prime}$s. It induces degree preserving linear maps between any of the
homologies /cohomologies defined above. The following elementary observations
will be used below.

######  Proposition 1.

Let $(C,\delta,\beta)$ be a mixed cochain complex.

1.$PH^{r}=\underset{\underset{S}{\rightarrow}}{\lim}^{+}H^{r+2k}_{\beta},$
where $S^{k+2r}:^{+}H^{k+2r}_{\beta}\to^{+}H^{k+2r+2}_{\beta}$ is induced by
the inclusion
${}^{+}\mathcal{C}^{\ast}_{\beta}\to^{+}\mathcal{C}^{\ast+2}_{\beta}.$

2\. The following is an exact sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\underset{\underset{S}{\leftarrow}}{\lim^{\prime}}\
^{+}H^{\delta}_{r-1+2k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{P}H^{r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\underset{\underset{S}{\leftarrow}}{\lim}\
^{+}H^{\delta}_{r+2k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

with $S_{k+2r}:^{+}H_{k+2r}^{\delta}\to^{+}H_{k+2r-2}^{\delta}$ induced by the
projection
${}^{+}\mathcal{H}^{\delta}_{\ast}\to^{+}\mathcal{H}^{\delta}_{\ast-2}.$

Let
$f^{\ast}:(C^{\ast}_{1},\delta^{\ast}_{1},\beta^{1}_{\ast})\to(C^{\ast}_{2},\delta^{\ast}_{2},\beta^{2}_{\ast})$
be a morphism of mixed complexes.

3\. If $H^{\ast}(f)$ is an isomorphism then so is ${}^{+}H^{\ast}_{\beta}(f)$
and $PH^{\ast}(f).$

4\. If $H_{\ast}(f)$ is an isomorphism then so is ${}^{+}H^{\delta}_{\ast}(f)$
and $\mathbb{P}H^{\ast}(f).$

5\. If $H^{\ast}(f)$ and $H_{\ast}(f)$ are both isomorphisms then, in addition
to the conclusions in (3) and (4), ${}^{-}H^{\ast}_{\beta}(f)$ is an
isomorphism.

###### Proof.

(1): Recall that a direct sequence of cochain complexes

$\mathcal{C}^{\ast}_{0}\overset{i_{0}}{\rightarrow}\mathcal{C}^{\ast}_{1}\overset{i_{1}}{\rightarrow}\mathcal{C}^{\ast}_{2}\overset{i_{2}}{\rightarrow}\cdots$

induces, by passing to cohomology, the direct sequence

$H^{\ast}(\mathcal{C}^{\ast}_{0})\overset{H(i_{0})}{\rightarrow}H^{\ast}(\mathcal{C}^{\ast}_{1})\overset{H(i_{1})}{\rightarrow}H^{\ast}(\mathcal{C}^{\ast}_{2})\overset{i_{2}}{\rightarrow}\cdots$

and that
$H^{j}(\underset{\rightarrow}{\lim}\mathcal{C}^{\ast}_{i})=\underset{\rightarrow}{\lim}H^{j}(\mathcal{C}^{\ast}_{i})$
for any $j.$

(2): Recall that an inverse sequence of chain complexes

$\mathcal{H}_{\ast}^{0}\overset{p_{0}}{\leftarrow}\mathcal{H}_{\ast}^{1}\overset{p_{1}}{\leftarrow}\mathcal{H}_{\ast}^{2}\overset{p_{2}}{\leftarrow}\cdots$

induces, by passing to homology, the sequence

$H_{\ast}(\mathcal{H}^{\ast}_{0})\overset{H(p_{0})}{\leftarrow}H_{\ast}(\mathcal{H}^{\ast}_{1})\overset{H(p_{1})}{\leftarrow}H_{\ast}(\mathcal{H}^{\ast}_{2})\overset{p_{2}}{\leftarrow}\cdots$

and the following short exact sequence cf.[Lo] 5.1.9.

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\underset{\leftarrow}{\lim^{\prime}}H_{j-1}(\mathcal{H}^{i}_{\ast})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{j}(\underset{\leftarrow}{\lim}\mathcal{H}^{i}_{\ast})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\underset{\leftarrow}{\lim}H_{j}(\mathcal{H}^{i}_{\ast})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

for any $j.$

Item (3) follows by induction on degree from the naturality of the first exact
sequence in the diagram Fig 2 and (1).

Item (4) follows by induction from the naturality of the second exact sequence
of the diagram Fig 1 and from (2).

Item (5) follows from the naturality of the second exact sequence in diagram
Fig 2 and from (3) and (4). ∎

The mixed complex $(C^{\ast},\delta^{\ast},\beta_{\ast})$ is called
$\beta-$acyclic if $\beta_{1}$ is surjective and
$\ker(\beta_{r})=\text{im}(\beta_{r+1}).$ If so consider the diagram whose
rows are short exact sequences of cochain complexes

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\text{Im}(\beta)^{\ast},\delta^{\ast})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j}$$\scriptstyle{i}$$\textstyle{(C^{\ast},\delta^{\ast})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\scriptstyle{id}$$\textstyle{((\text{Im}(\beta))^{\ast-1},\delta^{\ast-1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(^{+}C^{\ast-2}_{\beta},^{+}D^{\ast-2}_{\beta})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{\ast-2}}$$\textstyle{(^{+}C^{\ast}_{\beta},^{+}D^{\ast}_{\beta})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(C^{\ast},\delta^{\ast})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

Each row induces the long exact sequence in the diagram below and a simple
inspection of boundary map in these long exact sequences permits to construct
linear maps $\theta^{r}$ and to verify that the diagram below is commutative.

$\textstyle{H^{r-2}(\text{Im}(\beta),\delta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta^{r-2}}$$\textstyle{H^{r}(\text{Im}(\beta),\delta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta^{r}}$$\textstyle{H^{r}(C,\delta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id}$$\textstyle{H^{r-1}(\text{Im}(\beta),\delta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta^{r-1}}$$\textstyle{{}^{+}H^{r-2}_{\beta}(C,\delta,\beta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{}^{+}H^{r}_{\beta}(C,\delta,\beta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{r}(C,\delta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{}^{+}H^{r-1}_{\beta}(C,\delta,\beta)}$

As a consequence one verifies by induction on degree that the inclusion
$j:(\text{Im}\beta,\delta)\to(^{+}C,^{+}D_{\beta})$ induces an isomorphism
$H^{\ast}(\text{Im}(\beta),\delta)\to^{+}H^{\ast}_{\beta}(C,\delta,\beta).$

Mixed complex with power maps and involution

A collection of degree zero (degree preserving) linear maps
$\Psi_{k},k=1,2.\cdots,\tau:=\Psi_{-1}$ which satisfy

  1. (i)

$\Psi_{k}\circ\delta=\delta\circ\Psi_{k},$

  2. (ii)

$\Psi_{k}\circ\beta=k\beta\circ\Psi_{k},$

  3. (iii)

$\Psi_{k}\circ\Psi_{r}=\Psi_{r}\circ\Psi_{k}=\Psi_{kr},\ \ \Psi_{1}=id$

will be referred to as “power maps and involution”, or simpler ”power maps”,
$\Psi_{k},k=-1,1,2,\cdots.$ 333We use the notation $\tau$ for $\Psi_{-1}$ to
emphasize that is an involution and to suggest consistency with other familiar
involutions in homological algebras and topology.. They provide the morphisms
of cochain complexes

$\displaystyle\Psi_{k}:\mathcal{C}\to$ $\displaystyle\mathcal{C},$
$\displaystyle{}^{\pm}\Psi_{k}:^{\pm}\mathcal{C}_{\beta}\to$
$\displaystyle{}^{\pm}\mathcal{C}_{\beta}$
$\displaystyle\mathbb{P}\Psi_{k}:\mathbb{P}\mathcal{C}\to$
$\displaystyle\mathbb{P}\mathcal{C}$

defined as follows

$\displaystyle{}^{+}\Psi_{k}^{r}(w_{r},w_{r-2},\cdots)=$
$\displaystyle(\Psi^{r}(\omega_{r}),\frac{1}{k}\Psi^{r-2}(\omega_{r-2}),\cdots)$
$\displaystyle{}^{-}\Psi_{k}^{r}(\cdots,w_{r+2},w_{r}):=$
$\displaystyle(\cdots
k\Psi_{k}^{2r+2}(\omega_{r+2}),\Psi_{k}^{r}(\omega_{r}))\ $

and

$\displaystyle\mathbb{P}\Psi_{k}^{2r}(\cdots,\omega_{2r+2},\omega_{2r},\omega_{r-2},\cdots,\omega_{0})=$
$\displaystyle=(\cdots,k\Psi_{k}^{2r+2}(\omega_{r+2}),\Psi_{k}^{2r}(\omega_{2r}),\frac{1}{k}\Psi_{k}^{2r-2}(\omega_{2r-2}),\cdots,\frac{1}{k^{r}}\Psi_{k}^{0}(\omega_{0}))$
$\displaystyle\mathbb{P}\Psi_{k}^{2r+1}(\cdots,\omega_{2r+3},\omega_{2r+1},\omega_{2r-1},\cdots,\omega_{1})=$
$\displaystyle=(\cdots,k\Psi_{k}^{2r+3}(\omega_{2r+3}),\Psi_{k}^{2r+1}(\omega_{2r+1}),\frac{1}{k}\Psi_{k}^{2r-1}(\omega_{2r-1}),\cdots\frac{1}{k^{r}}\Psi_{k}^{1}(\omega_{1}))$

Consequently they induce the endomorphisms,

$\underline{\Psi}_{k}^{\ast}:H^{\ast}\to H^{\ast}$
$\underline{{}^{\pm}\Psi}_{k}^{\ast}:^{\pm}H^{\ast}_{\beta}\to^{\pm}H^{\ast}_{\beta}$
$\underline{\mathbb{P}\Psi^{\ast}}_{k}:\mathbb{P}H^{\ast}\to\mathbb{P}H^{\ast}$

Note that in diagram (Fig2):

$\mathbb{J}^{\ast},$ $J^{\ast}$ and the vertical arrows intertwine the
endomorphisms induced by $\Psi_{k},$

$\mathbb{B}^{\ast}$ resp. $B^{\ast}$ intertwine $k(^{-}\underline{\Psi}_{k})$
resp. $k\underline{\Psi}_{k}$ with ${}^{+}\underline{\Psi}_{k},$

$\mathbb{I}^{\ast-2}$ resp. $S^{\ast-2}$ intertwine
${}^{+}\underline{\Psi}_{k}$ with $k\underline{\mathbb{P}\Psi}_{k}$ resp.
$k(^{+}\underline{\Psi}_{k}),$

The above elementary linear algebra will be applied to CDGA’s in the next
sections.

##  3 Mixed commutative differential graded algebras

Let $\kappa$ be a field of characteristic zero (for example
$\mathbb{Q},\mathbb{R},\mathbb{C}$).

######  Definition 1.

  1. (i)

A commutative graded algebra abbreviated CGA, is an associative unital
augmentable graded algebra $\mathcal{A}^{\ast},$ (the augmentation is not part
of the data) which is commutative in the graded sense, i.e.

$a_{1}\cdot a_{2}=(-1)^{r_{1}r_{2}}a_{2}\cdot a_{1},\ \
a_{i}\in\mathcal{A}^{r_{i}},i=1,2.$

  2. (ii)

An exterior differential
$d^{\ast}_{\mathcal{A}}:\mathcal{A}^{\ast}\to\mathcal{A}^{\ast+1},$ is a
degree +1-linear map which satisfies

$d(a_{1}\cdot a_{2})=d(a_{1})\cdot a_{2}+(-1)^{r_{1}}a_{1}\cdot
d(a_{2}),a_{1}\in\mathcal{A}^{r_{1}},\ \
d^{*+1}_{\mathcal{A}}d^{*}_{\mathcal{A}}=0.$

  3. (iii)

An interior differential
$\beta^{\mathcal{A}}_{\ast}:\mathcal{A}^{\ast}\to\mathcal{A}^{\ast-1}$ is a
degree -1 linear map which satisfies

$\beta(a_{1}\cdot a_{2})=\beta(a_{1})\cdot
a_{2}+(-1)^{r_{1}}a_{1}\cdot\beta(a_{2}),a_{1}\in\mathcal{A}^{r_{1}},\ \
\beta_{\ast-1}^{\mathcal{A}}\beta_{\ast}^{\mathcal{A}}=0.$

  4. (iv)

The exterior and interior differentials $d^{\ast}$ and $\beta_{\ast}$ are
compatible if

$d^{\ast-1}\cdot\beta_{\ast}+\beta_{\ast+1}\cdot d^{\ast}=0.$

  5. (v)

A pair $(\mathcal{A}^{\ast},d^{\ast}),\ \mathcal{A}^{\ast}$ a CGA and
$d^{\ast}$ exterior differential, is called CDGA and a triple
$(\mathcal{A}^{\ast},d^{\ast},\beta_{\ast}),\ \mathcal{A}^{\ast}$ a CGA,
$d^{\ast}$ exterior differential and $\beta_{\ast}$ interior differential,
with $d^{\ast}$ and $\beta_{\ast}$ compatible, is called a mixed CDGA.

A mixed CDGA is a mixed cochain complex .

A degree preserving linear map
$f^{\ast}:\mathcal{A}^{\ast}\to\mathcal{B}^{\ast}$ is a morphism of CGA’s,
resp. CDGA’s, resp. mixed CDGA’s if is a unit preserving graded algebra
homomorphism and intertwines $d^{\prime}$s and $\beta^{\prime}$s when the
case.

We will consider the categories of CGA’s, CDGA’s and mixed CDGA’s. In all
these three categories there is a canonical tensor product and in the category
of CDGA’s a well defined concept of homotopy between two morphisms (cf. [Lo],
[Ha]) 444 let $k(t,dt)$ be the free commutative graded algebra generated by
the symbol $t$ of degree zero and $dt$ of degree one, equipped with the
differential $d(t)=dt.$ A morphism
$F:(\mathcal{A},d_{\mathcal{A}})\to(\mathcal{B},d_{\mathcal{B}})\otimes_{k}(k(t,dt),d),$
is called elementary homotopy from $f$ to $g$ ,
$f,g:(\mathcal{A},d_{\mathcal{A}})\to(\mathcal{B},d_{\mathcal{B}}),$ if
$\rho_{0}\cdot F=f,$ and $\rho_{1}\cdot F=g$ where
$\displaystyle\rho_{0}(a\otimes p(t))=$ $\displaystyle p(0)a,\
\rho_{0}(a\otimes p(t)dt)=0,$ $\displaystyle\rho_{1}(a\otimes p(t))=$
$\displaystyle p(1)a,\ \rho_{1}(a\otimes p(t)dt)=0,$ The homotopy is the
equivalence relation generated by elementary homotopy.. The category of mixed
CDGA’s is a subcategory of mixed cochain complexes and all definitions and
considerations in section 2 can be applied.

For a (commutative) differential graded algebra
$(\mathcal{A}^{\ast},d^{\ast}_{\mathcal{A}}),$ the graded vector space
$H^{\ast}(\mathcal{A}^{\ast},d^{\ast})=Ker(d^{\ast})/{Im(d^{\ast-1})}$ is a
commutative graded algebra whose multiplication is induced by the
multiplication in $\mathcal{A}^{\ast}.$ A morphism
$f=f^{\ast}:(\mathcal{A}^{\ast},d^{\ast}_{\mathcal{A}})\to(\mathcal{B}^{\ast},d^{\ast}_{\mathcal{B}})$
induces a degree preserving linear map,
$H^{\ast}(f):H^{\ast}(\mathcal{A}^{\ast},d_{\mathcal{A}}^{\ast})\to
H^{\ast}(\mathcal{B}^{\ast},d_{\mathcal{B}}^{\ast}),$ which is an algebra
homomorphism.

######  Definition 2.

A morphism of CDGA’s $f,$ with $H^{k}(f)$ isomorphism for any $k,$ is called a
quasi isomorphism.

The CDGA $(\mathcal{A},d_{\mathcal{A}})$ is called homologically connected if
$H^{0}(\mathcal{A},d_{\mathcal{A}})$ $=\kappa$ and homologically 1-connected
if homologically connected and $H^{1}(\mathcal{A},d_{\mathcal{A}})=0.$

The full subcategory of homologically connected CDGA’s will be denoted by
c–CDGA. For all practical purposes (related to geometry and topology) it
suffices to consider only c-CDGA’ s.

######  Definition 3.

1\. The CDGA $(\mathcal{A},d)$ is called free if $\mathcal{A}=\Lambda[V],$
where $V=\sum_{i\geq 0}V^{i}$ is a graded vector space and $\Lambda[V]$
denotes the free commutative graded algebra generated by $V.$ If in addition
$V^{0}=0$ is called free connected commutative differential graded algebra,
abbreviated fc-CDGA.

2\. The CDGA $(\mathcal{A},d)$ is called minimal if it is a fc-CDGA and in
addition

i. $d(V)\subset\Lambda^{+}[V]\cdot\Lambda^{+}[V],$ with $\Lambda^{+}[V]$ the
ideal generated by $V$,

ii. $V^{1}=\oplus_{\alpha\in I}V_{\alpha}$ with $I$ a well ordered set and
$d(V_{\beta})\subset\Lambda[\oplus_{\alpha<\beta}V_{\alpha}]$ (the set $I$ and
its order are not part of the data)

######  Observation 1.

If $(\Lambda[V],d_{V})$ is minimal and 1-connected, then $V^{1}=0$ and, for
$v\in V^{i},$ $d_{V}(v)$is a linear combination of products of elements
$v_{j}\in V^{j}$ with $j<i$. In particular for $v\in V^{2}$ one has $dv=0.$

The interest of minimal algebras comes from the following result [L], [Ha].

######  Theorem 1.

1 (D. Sullivan)

1\. A quasi isomorphism between two minimal CDGA’s is an isomorphism.

2\. For any homologically connected CDGA, $(\mathcal{A},d_{\mathcal{A}}),$
there exists quasi isomorphisms
$\theta:(\Lambda[V],d_{V})\to(\mathcal{A},d_{\mathcal{A}})$ with
$(\Lambda[V],d_{V})$ minimal. Such $\theta$ will be called minimal model of
$(\mathcal{A},d_{\mathcal{A}}).$

3\. Given a morphism
$f:(\mathcal{A},d_{\mathcal{A}})\to(\mathcal{B},d_{\mathcal{B}})$ and the
minimal models $\theta_{A}:(\Lambda[V_{A}],d_{V_{A}})\to(A,d_{A})$ and
$\theta_{B}:(\Lambda[V_{B}],d_{V_{B}})\to(B,d_{B}),$ there exists morphisms
$f^{\prime}:(\Lambda[V_{A}],d_{V_{A}})\to(\Lambda[V_{B}],d_{V_{B}})$ such that
$f\cdot\theta_{A}$ and $\theta_{B}\cdot f^{\prime}$ are homotopic; moreover
any two such $(f^{\prime})$’s are homotopic.

We can therefore consider the homotopy category of c–CDGA’s, whose the
morphisms are homotopy classes of morphisms of CDGA’s. By the above theorem
the full subcategory of fc-CDGA is a skeleton, and therefore any homotopy
functor a priori defined on fc–CDGA’s admits extensions to homotopy functors
defined on the full homotopy category of c–CDGA’s and all these extensions are
isomorphic as functors. In particular any statement about a homotopy functor
on the category c-CDGA suffices to be verified for fc–CDGA.

Precisely for any c–GDGA, $(\mathcal{A},d_{\mathcal{A}}),$ choose a minimal
model
$\theta_{\mathcal{A}}:(\Lambda[V_{\mathcal{A}}],d_{V_{\mathcal{A}}})\to(\mathcal{A},d_{\mathcal{A}})$
and for any $f:(\mathcal{A},d_{\mathcal{A}})\to(\mathcal{B},d_{\mathcal{B}})$
choose a morphism
$f^{\prime}:(\Lambda[V_{\mathcal{A}}],d_{V_{\mathcal{A}}})\to(\Lambda[V_{\mathcal{B}}],d_{V_{\mathcal{B}}})$
so that $\theta_{B}\cdot f^{\prime}$ and $f\cdot\theta_{A}$ are homotopic.
Define the value of the functor on $(\mathcal{A},d_{\mathcal{A}})$ to be the
value on $(\Lambda[V_{\mathcal{A}}],d_{V_{\mathcal{A}}})$ and the value on a
morphism $f:(\mathcal{A},d_{\mathcal{A}})\to(\mathcal{B},d_{\mathcal{B}})$ to
be the value on the morphism
$f^{\prime}:(\Lambda[V_{\mathcal{A}}],d_{V_{\mathcal{A}}})\to(\Lambda[V_{\mathcal{B}}],d_{V_{\mathcal{B}}}).$

There are two natural examples of mixed CDGA’s; one is provided by a smooth
manifold equipped with a smooth vector field, the other by a construction
referred to as ”the free loop”, considered first by Sullivan-Vigué. The free
loop construction applies directly only to a fc-CDGA but in view of Theorem 1
can be indirectly used for any c–CDGA.

The first will lead to (the de–Rham version of) a new homotopy functor defined
on the category of possibly infinite dimensional manifolds (hence on the
homotopy category of all countable CW complexes) , the s-cohomology, and its
relationship with other familiar homotopy functors 555this functor was called
in [B2] and [B4] string cohomology for its unifying role explained below, cf.
Observation 2. The name ”string homology” was afterwards used by Sullivan and
his school to designate the homology and equivariant homology of the free loop
space of a closed manifold when endowed with additional structures induced by
intersection theory and the Pontrjagin product in the chains of based pointed
loops cf. [CS]., cf section 4 below. The second leads to simple definitions of
three homotopy functors defined on the full category of c–CDGA’s (via the
minimal model theorem) with values in the graded vector spaces endowed with
weight decomposition, cf section 5 below. Their properties lead to interesting
results about cohomology of the free loop space of 1-connected spaces.

##  4 De-Rham Theory in the presence of a smooth vector field

Let $M$ be a smooth manifold, possibly of infinite dimension. In the last case
the manifold is modeled on a good Frechet space 666this is a Frechet space
with countable base which admits a smooth partition of unity; Note that if a
Frechet space $V$ is good then the space of smooth maps $C^{\infty}(S^{1},V)$
equipped with the $C^{\infty}-$ topology is also good., for which the
differential calculus can be performed as expected.

Consider the CDGA of differential forms $\Omega^{\ast}(M)$ with exterior
differential $d^{\ast}:\Omega^{\ast}(M)\to\Omega^{\ast+1}(M)$ and interior
differential $i^{X}_{\ast}:\Omega^{\ast}(M)\to\Omega^{\ast-1}(M),$ the
contraction along the vector field $X.$ They are not compatible. However we
can consider the Lie derivative $L_{X}:=d\cdot i^{X}+i^{X}\cdot d$ and define
$\Omega_{X}(M):=\\{\omega\in\Omega(M)|L_{X}\omega=0\\};$ $\Omega_{X}(M)$
consists of the smooth forms invariant by the flow induced by $X.$ The graded
vector space $\Omega^{\ast}_{X}(M)$ is a commutative graded algebra, a sub
algebra of $\Omega^{\ast}(M),$ and the restriction of $d^{\ast}$ and of
$i^{X}_{\ast}$ leave invariant $\Omega^{\ast}_{X}(M)$ and are compatible.
Consequently $(\Omega^{\ast}_{X}(M),d^{\ast},i^{X}_{\ast})$ is a mixed CDGA.

Denote by

  1. (i)

$H^{\ast}_{X}(M):=H^{\ast}(\Omega^{\ast}_{X},d^{\ast}),$

  2. (ii)

${}^{\pm}H^{\ast}_{X}(M):=^{\pm}H^{\ast}(\Omega^{\ast}_{X},d^{\ast},i^{X}_{\ast}),$

  3. (iii)

$PH^{\ast}_{X}(M):=PH^{\ast}(\Omega^{\ast}_{X},d^{\ast},i^{X}_{\ast}),$

  4. (iv)

$\mathbb{P}H^{\ast}_{X}(M):=\mathbb{P}H^{\ast}(\Omega^{\ast}_{X},d^{\ast},i^{X}_{\ast}).$

The diagram Fig 2 becomes

$\textstyle{{}^{+}H^{r}_{X}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{S^{r}}$$\scriptstyle{id}$$\textstyle{{}^{+}H^{r+2}_{X}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{J^{r+2}}$$\scriptstyle{\mathbb{I}^{r+2}}$$\textstyle{H^{r+2}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{B^{r+2}}$$\textstyle{{}^{+}H_{X}^{r+1}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$$\textstyle{{}^{+}H^{r}_{X}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{I}^{r}}$$\textstyle{\mathbb{P}H^{r+2}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{J}^{r+2}}$$\textstyle{{}^{-}H^{r+2}_{X}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{B}^{r+2}}$$\textstyle{{}^{+}H^{r+1}_{X}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$

Fig 3

The above diagram becomes more interesting if the vector field $X$ is induced
from an $S^{1}$ action $\mu:S^{1}\times M\to M$ (i.e. if $x\in M$ then $X(x)$
is the tangent to the orbits through $x$). In this section we will explore
particular cases of this diagram.

Observe that since $\mu$ is a smooth action, the subset $F$ of fixed points is
a smooth sub manifold. For any $x\in F$ denote by $\rho_{x}:S^{1}\times
T_{x}(M)\to T_{x}(M)$ the linearization of the action at $x$ which is a linear
representation. The inclusion $F\subset M$ induces the morphism
$r^{\ast}:(\Omega^{\ast}_{X}(M),d^{\ast},i^{X}_{\ast})\to(\Omega^{\ast}(F),d^{\ast},0).$

For a linear representation $\rho:S^{1}\times V\to V$ on a good Frechet space
denote by $V^{f}$ the fixed point set and by $X$ the vector field associated
to $\rho$ when regarded as a smooth action.

######  Definition 4.

A linear representation $\rho:S^{1}\times V\to V$ on the good Frechet space is
good if the following conditions hold:

a. $V^{f},$ the fixed point set, is a good Frechet space,

b. The map $r^{\ast}:\Omega^{\ast}(V)\to\Omega^{\ast}(V^{f})$ induced by the
inclusion is surjective,

c. $(\Omega^{\ast}_{X}(V,V^{f}),i^{X}_{\ast})$ with
$(\Omega^{\ast}_{X}(V,V^{f})=\ker r^{\ast}$ is acyclic.

We have:

######  Proposition 2.

1\. Any representation on a finite dimensional vector space is good.

2\. If $V$ is a good Frechet space then the regular representation
$\rho:S^{1}\times C^{\infty}(S^{1},V)\to C^{\infty}(S^{1},V)$, with
$C^{\infty}(S^{1},V),$ the Frechet space of smooth functions, is good.

For a proof consult Appendix [B1]. The proof is based on an explicit formula
for $i^{X}$ in the case of irreducible $S^{1}-$ representation and on the
writing of the elements of $C^{\infty}(S^{1},V)$ as Fourier series.

######  Definition 5.

A smooth action $\mu:S^{1}\times M\to M$ is good if its linearization at any
fixed point is a good representation.

Then a smooth action on any finite dimensional manifold is good and so is the
canonical smooth action of $S^{1}$ on $P^{S^{1}},$ the smooth manifold of
smooth maps from $S^{1}$ to $P$ where $P$ is any smooth Frechet manifold (in
particular a finite dimensional manifold). In view of the definitions above
observe the following.

######  Proposition 3.

If $\tilde{M}=(M,\mu)$ is a smooth $S^{1}-$manifold and $X$ is the associated
vector field, then:

1\. $H^{\ast}_{X}(M)=H^{\ast}(M),$

2\. ${}^{+}H^{\ast}_{X}(M)=H^{\ast}_{S^{1}}(\tilde{M})$ and $S:H^{\ast}(M)\to
H^{\ast+2}(M)$ identifies to the multiplication with $u\in H^{2}_{S^{1}}(pt),$
the generator of the equivariant cohomology of one point space,

3\.
$PH^{\ast}_{X}(M)=\underset{\underset{S}{\rightarrow}}{\lim}H^{\ast+2k}_{S^{1}}(\tilde{M}).$

If the action is good then:

4\. $\mathbb{P}H^{\ast}_{X}(M)=K^{\ast}(F)\ $ where

$\displaystyle K^{r}(F)=$ $\displaystyle\prod_{k}H^{2k}(F)\ \text{if}\ \ r\ \
even$ $\displaystyle K^{r}(F)=$ $\displaystyle\prod_{k}H^{2k+1}(F)\ \text{if}\
\ r\ \ odd.$

If $M$ is a closed of $n-$dimension manifold then :

5\. ${}^{-}H^{\ast}_{X}(M)=H^{S^{1}}_{n-1-\ast}(\tilde{M},\mathcal{O}_{M})$
with $H^{S^{1}}_{\ast}(\tilde{M},\mathcal{O}_{M})$ the equivariant homology
with coefficients in the orientation bundle 777 Recall that
$H^{S^{1}}_{\ast}(M,\mathcal{O}_{M})=H_{\ast}(M//S^{1},\mathbb{O}_{M})$ where
$M//S^{1}$ is the homotopy quotient of this action. This equivariant homology
can be derived from invariant currents in the same way as equivariant
cohomology from invariant forms, cf. [AB86]. The complex of invariant currents
(with coefficients in orientation bundle) contains the complex
$(\Omega^{n-\ast}_{X}(M,\mathcal{O}_{M}),\partial_{n-\ast})$ as a quasi
isomorphic sub complex. of $M.$

###### Proof.

1\. The verification is standard since $S^{1}$ is compact and connected; one
construct
$av^{\ast}:(\Omega^{\ast}(M),d^{\ast})\to(\Omega_{X}^{\ast},d^{\ast})$ by
$S^{1}-$ averaging using the compacity of $S^{1}.$ The homomorphism induced in
cohomology by $av^{\ast}$ is obviously surjective. To check it is injective
one has to show that any closed $k$ differential form $\omega$ which becomes
exact after applying $av$ is already exact, precisely $\int_{c}\omega=0$ for
any smooth $k-$ cycle $c.$ Indeed, since the connectivity of $S^{1}$ implies
$\int_{c}\omega=\int_{\mu(-\theta,c)}\omega,\ \theta\in S^{1}$ one has:

$\displaystyle\int_{c}\omega=1/2\pi\int_{S^{1}}(\int_{c}\omega)d\theta=$
$\displaystyle
1/2\pi\int_{S^{1}}(\int_{\mu(-\theta,c)}\omega)d\theta=1/2\pi\int_{S^{1}}(\int_{c}\mu_{\theta}^{\ast}(\omega))d\theta=$
$\displaystyle\int_{c}(1/2\pi\int_{S^{1}}\mu_{\theta}^{\ast}(\omega)d\theta)=\int_{c}av^{\ast}(\omega)=0.$

Here $\mu_{\theta}$ denotes the diffeomorphism $\mu(\theta,\cdot):M\to M.$

2\. Looking at the definition in section 2 one recognizes one of the most
familiar definition of equivariant cohomology using invariant differential
forms cf. [AB86].

3\. The proof is a straightforward consequence of Proposition 1 in section 2
and (2) above.

4\. Let $F$ be the smooth sub manifold of the fixed points of $\mu.$ Clearly
$r^{\ast}:(\Omega^{\ast}_{X}(M),d^{\ast},i^{X}_{\ast})\to(\Omega^{\ast}(F),d^{\ast}_{F},0)$
is a morphism of mixed CDGA, hence of mixed complexes. If the smooth action is
good then the above morphism induces an isomorphism in homology
$H_{\ast}(\Omega^{\ast}_{X},i^{X}_{\ast})\to H_{\ast}(\Omega^{\ast}(F),0).$ To
check this we have to show that $(\ker r^{\ast},i^{X}_{\ast})$, with $\ker
r^{\ast}:=\\{\omega\in\Omega_{X}^{\ast}(M)|\omega|_{F}=0\\},$ is acyclic. This
follows (by $S^{1}-$average) from the acyclicity of the chain complex
$(\Omega^{\ast}(M,F),i^{X}_{\ast})$ which in turn can be derived using the
linearity w.r. to functions of of $i^{X}.$ Indeed, using a ”partition of
unity” argument it suffices to verify this acyclicity locally. For points
outside $F$ the acyclicity follows from the acyclicity of the complex

$\cdots\Lambda^{\ast-1}(V)\overset{i^{e}}{\to}\Lambda^{\ast}(V)\overset{i^{e}}{\to}\Lambda^{\ast-1}(V)\to\cdots,$
where $V$ is a Frechet space , $\Lambda^{k}(F)$ the space of skew symmetric
$k-$linear maps from $V$ to $\kappa=\mathbb{R},\mathbb{C}$ and $e\in
V\setminus 0.$ For points $x\in F$ this follows from the fact that the
linearization of the action at $x$ is a good representation, as stated in
Proposition 2.

5\. If $\tilde{M}$ is a finite dimensional smooth $S^{1}-$manifold we can
equip $M$ with an invariant Riemannian metric $g$ and consider
$\star:\Omega^{\ast}(M)\to\Omega^{n-\ast}(M;\mathcal{O}_{M})$ the Hodge star
operator. Denote by $\omega\in\Omega^{1}(M)$ the $1-$form corresponding to $X$
w.r. to the metric $g$, by
$e_{\omega}:\Omega^{\ast}(M;\mathcal{O}_{M})\to\Omega^{\ast+1}(M;\mathcal{O}_{M})$
the exterior multiplication with $\omega$ and by
$\partial_{\ast}:\Omega^{\ast}(M;\mathcal{O}_{M})\to\Omega^{\ast-1}(M;\mathcal{O}_{M})$
the formal adjoint of $d^{\ast-1}$ w.r. to $g$ i.e.
$\partial_{\ast}=\pm\star\cdot\ d^{n-\ast}\cdot\ \star^{-1}.$ Note that
$e_{\omega}=\pm\star\cdot\ i^{X}\cdot\star^{-1}.$ All these operators leave
$\Omega_{X}$ invariant since $g$ is invariant. Clearly
$(\Omega^{\ast}_{X}(M;\mathcal{O}_{M}),e^{\ast}_{\omega},\partial_{\ast})$ is
a mixed cochain complex and we have

${}^{-}H^{\ast}_{i^{X}}(\Omega_{X}(M),d,i^{X})=^{+}H^{\partial}_{n-\ast}(\Omega_{X}(M;\mathcal{O}_{M}),e^{\omega},\partial).$

The equivariant homology of $\tilde{M}$ with coefficients in the orientation
bundle can be calculated from the complex of invariant currents which, if $M$
closed, contains the complex
$(\Omega^{n-\ast}_{X}(M,\mathcal{O}_{M}),\partial_{n-\ast})$ as a quasi
isomorphic sub complex. As a consequence we have

$\displaystyle H_{n-\ast}(\Omega_{X}(M;\mathcal{O}_{M}),\partial)=$
$\displaystyle H_{\ast}(M;\mathcal{O}_{M})$
$\displaystyle{}^{+}H^{e_{\omega}}_{n-\ast}(\Omega_{X}(M;\mathcal{O}_{M}),e_{\omega},\partial)=$
$\displaystyle H_{\ast}^{S^{1}}(M;\mathcal{O}_{M})$

(cf. section 2 for notations). ∎

As a consequence of Proposition 3 (1)-(4) for any smooth $S^{1}$ manifold with
good $S^{1}-$ action the second long exact sequence in diagram Fig 3 becomes

|  |
---|---|---
$\textstyle{\cdots\to
H^{r-2}_{S^{1}}(\tilde{M})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{I}^{r-2}}$$\textstyle{K^{r}(F)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{J}^{r}}$$\textstyle{{{}^{-}H}^{r}_{S^{1}}(\tilde{M})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{B}^{r}}$$\textstyle{H^{r-1}_{S^{1}}(\tilde{M})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$$\textstyle{\underset{\underset{S}{\rightarrow}}{\lim}H^{r+2k}_{S^{1}}(\tilde{M})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

Fig 4 The sequence above is obviously natural in the sense that
$f:\tilde{M}\to\tilde{N},$ an $S^{1}-$ equivariant smooth map, induces a
commutative diagram whose rows are the above exact sequence Fig 4 for
$\tilde{M}$ and $\tilde{N}.$ Then if $f$ and its restriction to the fixed
point set induce isomorphisms in cohomology it induces isomorphisms in
$H^{\ast}_{S^{1}}$ and $K^{\ast}$ and then all other types of equivariant
cohomologies ${}^{-}H^{\ast}_{S^{1}},\ \ PH^{\ast}_{S^{1}},\ \
\mathbb{P}H^{\ast}_{S^{1}}.$

If $\tilde{M}$ is a compact smooth $S^{1}-$ manifold in view of Proposition 3
(5) one identifies ${}^{-}H^{r}_{S^{1}}(M)$ to
$H^{S^{1}}_{n-r}(M;\mathcal{O}_{M})$ and in view of this identification write
$Pd_{n-r}.$ instead of $\mathbb{B}^{r}.$ The long exact sequence becomes

$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K^{r}(F)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{H}_{n-r}^{S^{1}}(\tilde{M};\mathcal{O}_{M})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Pd_{n-r}}$$\textstyle{H^{r-1}_{S^{1}}(\tilde{M})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K^{r+1}(F)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots.}$

Fig 5

In case that the fixed point set is empty we conclude that

$Pd_{n-r}:{H}_{n-r}^{S^{1}}(\tilde{M},\mathcal{O}_{M})\to
H^{r-1}_{S^{1}}(\tilde{M})$

is an isomorphism. In this case the orbit space $M/S^{1}$ is a
$\mathbb{Q}-$homological manifold of dimension $(n-1)$, hence

$\displaystyle H_{n-r}^{S^{1}}(\tilde{M};\mathcal{O}_{M})=$ $\displaystyle
H_{n-r}(M/S^{1};\mathcal{O}_{M/S^{1}})$ $\displaystyle
H_{S^{1}}^{r-1}(\tilde{M};\mathcal{O}_{M})=$ $\displaystyle
H^{r-1}(M/S^{1};\mathcal{O}_{M/S^{1}})$

and $Pd_{\ast}$ is nothing but the Poincaré duality isomorphism for
$\mathbb{Q}-$ homology manifolds. In general the long exact sequence Fig 5
measures the failure of the Poincaré duality map, $Pd_{\ast},$ to be an
isomorphism.

##  5 The free loop space and s-cohomology

A more interesting example is provided by the $S^{-}$manifold
$\tilde{P^{S^{1}}}:=(P^{S^{1}},\mu).$ Here $P^{S^{1}}$ denotes the smooth
manifold of smooth maps from $S^{1}$ to $P$ modeled by the Frechet space
$C^{\infty}(S^{1},V)$ where $V$ is the model for $P$ (finite or infinite
dimensional Frechet space) cf [B1]. This smooth manifold is equipped with the
canonical smooth $S^{1}-$action $\mu:S^{1}\times P^{S^{1}}\to P^{S^{1}}$
defined by

$\mu(\theta,\alpha)(\theta^{\prime})=\alpha(\theta+\theta^{\prime}),\ \
\alpha:S^{1}\to P,\ \ \theta,\ \theta^{\prime}\in S^{1}=\mathbb{R}/2\pi.$

The fixed points set of the action $\mu$ consists of the constant maps hence
identifies with $P.$ This action is the restriction of the canonical action of
$O(2),$ the group of isometries of $S^{1},$ to the subgroup of orientation
preserving isometries identified to $S^{1}$ itself. For any $x\in P$ viewed as
a fixed point of $\mu$ the linearization representation is the regular
representation of $S^{1}$ on $V=T_{x}(P).$ In view of Proposition 2 the action
$\mu$ is good. The space $P^{S^{1}}$ is also equipped with the natural maps
$\psi_{k},\ k=1,2,\cdots,$ the geometric power maps and with the involution
$\tau,$ defined by

$\displaystyle\psi_{k}(\alpha)(\theta)=\alpha(k\theta)$
$\displaystyle\tau(\alpha)(\theta)=\alpha(-\theta)$

with $\alpha\in P^{S^{1}}$, and $\theta\in S^{1}.$

The involution $\tau$ is the restriction of the action of $O(2)$ to the
reflexion $\theta\to-\theta$ in $S^{1}.$ Then
$(\Omega^{\ast}_{X}(P^{S^{1}}),d^{\ast},i^{X}_{\ast})$ is a mixed CDGA, hence
a mixed complex with power maps $\Psi_{k},\tau$ and involution $\tau$ induced
from $\psi_{k}$ and $\tau.$

Suppose $f:P_{1}\to P_{2}$ is a smooth map. It induces a smooth equivariant
map $f^{S^{1}}:{P_{1}^{S^{1}}}\to{P_{2}^{S^{1}}}$ whose restriction to the
fixed points set is exactly $f.$ If $f$ is a homotopy equivalence then so is
$f^{S^{1}}.$

Introduce the notation

$\displaystyle hH^{\ast}(P):=H^{\ast}(P^{S^{1}}),$ $\displaystyle
cH^{\ast}(P):=H^{\ast}_{S^{1}}(\tilde{P^{S^{1}}}),$ $\displaystyle
sH^{\ast}(P):=^{-}H^{\ast}_{S^{1}}(\tilde{P^{S^{1}}}).$

The assignments $P\rightsquigarrow hH^{\ast}(P),$ $P\rightsquigarrow
cH^{\ast}(P),$ $P\rightsquigarrow sH^{\ast}(P)$ are functors 888the notations
$hH^{\ast},cH^{\ast}$ are motivated by the Hochschild resp. cyclic homology
interpretation of these functors, while $sH^{\ast}$ is abbreviation from
string cohomology. with the property that
$hH^{\ast}(f),cH^{\ast}(f),sH^{\ast}(f)$ are isomorphism if $f$ is a homotopy
equivalence, hence they are all homotopy functors . They are related by the
commutative diagram below. This diagram is the same as diagram (Fig 3) applied
to $\tilde{M}=\tilde{P^{S^{1}}}$ with the specifications provided by
Proposition 3.

|  |
---|---|---
$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{cH^{r-2}(P)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{S^{r-2}}$$\scriptstyle{id}$$\textstyle{cH^{r}(P)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{J^{r}}$$\scriptstyle{\mathbb{I}^{r}}$$\textstyle{hH^{r}(P)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{B^{r}}$$\textstyle{cH^{r-1}(P)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id}$$\textstyle{\cdots}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{cH^{r-2}(P)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{I}^{r-2}}$$\scriptstyle{S}$$\textstyle{K^{r}(P)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{J}^{r}}$$\textstyle{sH^{r}(P)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{B}^{r}}$$\textstyle{cH^{r-1}(P)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$$\textstyle{cH^{r}(P)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{I}^{r}}$$\textstyle{\underset{\rightarrow}{\lim}\
cH^{r+2k}(P)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

Fig 6

where $\underset{\rightarrow}{\lim}\ \
cH^{r+2k}(P)=\underset{\rightarrow}{\lim}\\{\cdots\to
cH^{r+2k}(P)\overset{S}{\rightarrow}cH^{r+2k+2}(P)\to\cdots\\}.$

The linear map

$cH^{\ast}(P)=H^{\ast}_{S^{1}}(\tilde{P^{S^{1}})}\overset{\mathbb{I}^{r}}{\rightarrow}K^{\ast}$

factors through $\underset{\rightarrow}{\lim}\ cH^{r+2k}(P)$ which depends
only on the fundamental group of $P.$ Indeed, it is shown in [B3] that if
$P(1)$ 999the notation for the first stage Postnikov term of $P.$ is a smooth
manifold (possibly of infinite dimension) which has the homotopy type of
$K(\pi,1)$ and $p(1):P\to P(1)$ is smooth map inducing an isomorphism for the
fundamental group then
$\underset{\rightarrow}{\lim}H^{r+2k}_{S^{1}}(\tilde{P^{S^{1}}})\to\underset{\rightarrow}{\lim}H^{r+2k}_{S^{1}}(\tilde{P(1)^{S^{1}}})$
is an isomorphism. cf [B3]. Then if one denotes by
$\overline{cH^{\ast}}(M):=\text{coker}(cH^{\ast}(M)\to cH^{\ast}(pt))$ and
$\overline{K^{\ast}}(M):=\text{coker}(K^{\ast}(M)\to K^{\ast}(pt))$101010
clearly $K^{r}(pt)=H^{r}_{S^{1}}(pt)=\kappa\ \ \text{resp.}\ \ 0$ if $r$ is
even resp. odd one obtains

######  Theorem 2.

If $P$ is a 1-connected smooth manifold then we have the following short exact
sequence:

$0\to\overline{K}^{r}(P)\otimes\kappa\overset{\mathbb{J}^{r}}{\rightarrow}{sH}^{r}(P)\overset{\mathbb{B}^{r}}{\rightarrow}\overline{c}H^{r-1}(P)\to
0$

where $\kappa=\mathbb{R}$ or $\mathbb{C}.$

######  Observation 2.

The vector space $K^{\ast}(P)$ can be identified via the Chern character to
the Atiyah–Hirtzebruch (complex) $K-$theory tensored with the field
$\kappa=\mathbb{R}\text{or}\ \mathbb{C},$ depending of what sort of
differential forms one consider (real or complex valued). When $P$ is
1-connected $\overline{c}H^{\ast}(P)$ identifies to
$\text{Hom}(\tilde{A}_{\ast}(P),k)$ where $\tilde{A}_{\ast}(P)$ denotes the
reduced Waldhaussen algebraic $K-$theory111111often referred to as $A-$
theory., cf [B2]. From this perspective $sH^{\ast}$ unifies topological
(Atiyah–Hirtzebruch) $K-$theory and Waldhaussen algebraic $K-$theory.

######  Observation 3.

In view of the definition of
${}^{-}H^{\ast}_{\beta}(C^{\ast},\delta^{\ast},\beta_{\ast}),$ cf. section 2,
observe that $sH^{\ast}(P)$ is represented by infinite sequences
121212$sH^{\ast}(P)$ is the cohomology of the cochain complex
$(^{-}C^{\ast},^{-}D^{\ast})$ with ${}^{-}C^{r}=\prod_{k\geq
0}\Omega^{r+2k}_{inv}(P^{S^{1}})$ and
${}^{-}D^{r}(\cdots,\omega_{r+2},\omega_{r})=(\cdots,(i^{X}\omega_{r+2}+d\omega_{r})),$
cf. section 2., rather than eventually finite sequences of invariant
differential forms on $P^{S^{1}}.$ If instead of “infinite sequences” we would
have considered “eventually finite sequences” the outcome would have been
different for infinite dimensional manifolds. The difference between “infinite
sequences” and ”eventually finite sequences” exists only for infinite
dimensional manifolds which $P^{S^{1}}$ always is.

The power maps $\psi_{k}$ induce the endomorphisms $h\Psi_{k},$ $c\Psi_{k}$
$s\Psi_{k}$ and $K\Psi_{k}$ on $hH^{\ast},cH^{\ast},sH^{\ast},$ and
$K^{\ast}.$

In general only $K\Psi_{k}$ are easy to describe. Precisely if $r$ is even
then $K^{r}=\prod_{i\geq 0}H^{2i}(P)$ and if $r$ is odd then
$K^{r}=\prod_{i\geq 0}H^{2i+1}(P),$ and in both cases $K\Psi_{k}=\prod_{i\geq
0}k^{i-r}Id.$

The symmetric part with respect to the involution $c\Psi_{-1},$ i.e. the
eigenspaces corresponding to the eigenvalues $+1$ identifies with
$H^{\ast}_{O(2)}(P^{S^{1}}),$ the equivariant cohomology for the canonical
$O(2)-$action.

However, if $P$ is 1-connected, in view of the section 6, one can describe
both the eigenvalues and the eigenspaces of the power maps $h\Psi_{k}$ and
$c\Psi_{k}$ and then of $s\Psi_{k}.$ We have:

######  Theorem 3.

Let $P$ be a 1-connected manifold.

1\. All eigenvalues of the endomorphisms $h\Psi_{k}$ and $c\Psi_{k}$ are
$k^{r},r=0,1,2\cdots,$ and the eigenspaces corresponding to $k^{r}$ are
independent of $k$ provided $k\geq 2.$

2\. Denotes these eigenspaces by $hH^{\ast}(M)(r)$ and $cH^{\ast}(M)(r).$ Then

$hH^{\ast}(0)=H^{\ast}(X;\kappa),cH^{\ast}(0)=H^{\ast+1}(X;\kappa),$ and

$hH^{r}(p)=cH^{r}(p)=0,p\geq r+1.$

3\. If $\sum_{i}\dim\pi_{i}(P)\otimes\kappa<\infty,$ $\kappa$ the field of
real or complex numbers and $\sum_{i}\dim(H^{i}(P))<\infty$ then for any
$r\geq 0$ one has

$\sum_{i}\dim hH^{i}(P)(r)<\infty,\ \ \sum_{i}\dim cH^{i}(P)(r)<\infty.$

If $P$ is ”formal” in the sense of rational homotopy theory 131313 i.e. for
each connected component of $P$ De-Rham algebra and the cohomology algebra
equipped with the differential $0$ are homotopy equivalent, cf section 6., (a
projective complex algebraic variety, or more general a closed Kaehler
manifold is formal, cf [DGMS] ) then the Euler Poincaré characteristic

$\chi^{h}(\lambda):=\sum_{i,r}\dim hH^{i}(r)\lambda^{r}$

and

$\chi^{c}(\lambda):=\sum_{i,r}\dim cH^{i}(r)\lambda^{r}$

can be explicitly calculated in terms of the numbers $\dim H^{i}(P),$ cf [B].
The explicit formulae are quite complicated. They require the results of
P.Hanlon [H] about the eigenspaces of Adams operations in Hochschild and
cyclic homology as well as the identification of $hH^{\ast}(P)$ resp.
$cH^{\ast}(P)$ with the Hochschild resp. cyclic homology of the graded algebra
$H^{\ast}(P).$ These are not discussed in this paper but the reader can
consult [BFG] and [B4] for precise statements.

The functor $\overline{sH}^{r}(P)$ is of particular interest in geometric
topology. In the case $P$ is 1-connected it calculates in some ranges the
homotopy groups of the (homotopy) quotient space of homotopy equivalences by
the group of diffeomorphisms [B1], [B4].

##  6 The free loop construction for CDGA

The ”free loop ” construction associates to a free connected CDGA,
$(\Lambda[V],d_{V})$ a mixed CDGA,
$(\Lambda[V\oplus\overline{V}],\delta_{V},i^{V}),$ endowed with power maps
$\Psi_{k}$ and involution $\tau$ defined as follows.

  1. (i)

Let $\overline{V}=\oplus_{i\geq 0}\overline{V}^{i}$ with
$\overline{V}^{i}:=V^{i+1}$ and let $\Lambda[V\oplus\overline{V}]$ be the
commutative graded algebra generated by $V\oplus\overline{V}.$

  2. (ii)

Let $i^{V}:\Lambda[V\oplus\overline{V}]\to\Lambda[V\oplus\overline{V}]$be the
unique internal differential (of degree $-1$) which extends
$i^{V}(v)=\overline{v}$ and $i^{V}(\overline{v})=0.$

  3. (iii)

Let $\delta_{V}:\Lambda[V\oplus\overline{V}]\to\Lambda[V\oplus\overline{V}]$
be the unique external differential (of degree $+1$) which extends
$\delta_{V}(v)=d(v)$ and $\delta(\overline{v})=-i^{V}(d(v)).$

  4. (iv)

Let
$\Psi_{k}:(\Lambda[V\oplus\overline{V}],\delta_{V})\to(\Lambda[V\oplus\overline{V}],\delta_{V}),k=-1,1,2,\cdots$
be the unique morphisms of CDGA which extends
$\Psi_{k}(v)=v,\Psi_{k}(\overline{v})=k\overline{v}.$ We put
$\tau:=\Psi_{-1}.$ The maps $\Psi_{k}$ $k\geq 1$ are called the power maps and
$\tau$ the canonical involution. One has

$\displaystyle\Psi_{k}\cdot\Psi_{r}=$ $\displaystyle\Psi_{kr}$
$\displaystyle\Psi_{k}\cdot i_{V}=$ $\displaystyle ki_{V}\cdot\Psi_{k}$

  5. (v)

Let $\Lambda^{+}[V\oplus\overline{V}]$ be the ideal of
$\Lambda[V\oplus\overline{V}]$ generated by $V\oplus\overline{V}$ or the
kernel of the augmentation which vanishes on $V\oplus\overline{V}.$

Note that :

######  Observation 4.

1\. $\text{(}{Im}(i^{V}),\delta_{V},0)$ is a mixed sub complex of
($\Lambda^{+}[V\oplus\overline{V}],\delta_{V},i^{V})\subset(\Lambda[V\oplus\overline{V}],\delta_{V},i^{V})$

2\. $\Psi_{k}$, $k=-1,1,2,\cdots$ leave
$(\Lambda^{+}[V\oplus\overline{V}],\delta_{V},i^{V})$ and
$(\text{Im}(i^{V}),\delta)$ invariant and have $k^{r}$ $r=0,1,2,\cdots$ as
eigenvalues. These are all eigenvalues.

For $k\geq 2$ the eigenspace of
$\Psi_{r}:\Lambda[V\oplus\overline{V}]\to\Lambda[V\oplus\overline{V}]$
corresponding to the eigenvalue $k^{r}$ is exactly
$\Lambda[V]\otimes\overline{V}^{\otimes r},$ resp.
$\Lambda^{+}[V\oplus\overline{V}]\cap\Lambda^{[}V]\otimes\overline{V}^{\otimes
r},$ resp.
$\text{Im}(i^{V})(r)=\text{Im}(i^{V})\cap\Lambda[V]\otimes\overline{V}^{\otimes
r}$, hence independent of $k.$ Each such eigenspace is $\delta_{V}-$invariant.

3\. The mixed complex $(\Lambda^{+}[V\oplus\overline{V}],\delta_{V},i^{V})$ is
$i^{V}-$acyclic,

4\. We have the decomposition

$(\Lambda[V\oplus\overline{V}],\delta_{V})=\bigoplus_{r\geq
1}(\Lambda[V]\otimes\overline{V}^{\otimes r},\delta_{V})$

and the analogous decomposition for
$(\Lambda^{+}[V\oplus\overline{V}],\delta_{V})$ and
$(\text{Im}(i^{V}),\delta_{V})$ referred from now on as the weight
decompositions.

Consider the complex $(\Lambda[V]\otimes\overline{V}^{\otimes r},\delta_{V})$
and the filtration provided by $\Lambda[V]\otimes F_{p}(\overline{V}^{\otimes
r})$ with $F_{p}(\overline{V}^{\otimes r})$ the span of elements in
$\overline{V}^{\otimes r}$ of total degree $\leq p.$

For a graded vector space $W=\oplus_{i}W^{i}$ denote by $\dim W=\sum\dim
W^{i}.$

######  Observation 5.

1\. $(\Lambda[V]\otimes F_{p}(\overline{V}^{\otimes r}),\delta_{V})$ is a sub
complex of $(\Lambda[V]\otimes\overline{V}^{\otimes r},\delta_{V}).$

2.If $(\Lambda[V],d_{V})$ is minimal and one connected then, by Observation 1,

$\delta(F_{p}(\overline{V}^{\otimes r}))\subset\Lambda[V]\otimes
F_{p-1}(\overline{V}^{\otimes r})$ and then

$({\Lambda[V]\otimes F_{p}(\overline{V}^{\otimes r})}/{\Lambda[V]\otimes
F_{p-1}(\overline{V}^{\otimes r})}\
,\delta_{V})=(\Lambda[V],d_{V})\otimes{F_{p}(\overline{V}^{\otimes
r})}/{F_{p-1}(\overline{V}^{\otimes r})}.$

2\. $\sum_{p}\dim(F_{p}(\overline{V}^{\otimes
r})/F_{p-1}(\overline{V}^{\otimes r}))=\dim(\overline{V}^{\otimes r})=(\dim
V)^{r}.$

If $f:(\Lambda[V],d_{V})\to\Lambda(W,d_{W})$ is a morphism of CDGAs then it
induces
$\tilde{f}:(\Lambda[V\oplus\overline{V}],\delta_{V},i^{V})\to(\Lambda[W\oplus\overline{W}],\delta_{W},i^{W})$
which intertwines $\Psi_{k}^{\prime}$s and then preserves the weight
decompositions.

We introduce the the notation $HH^{\ast},CH^{\ast}PH^{\ast}$

$\displaystyle HH^{\ast}(\Lambda[V],d_{V}):=$ $\displaystyle
H^{\ast}(\Lambda[V\oplus\overline{V}],\delta_{V})$ $\displaystyle
CH^{\ast}(\Lambda[V],d_{V}):=$
$\displaystyle{}^{+}H^{\ast}_{i^{V}}(\Lambda[V\oplus\overline{V}],\delta_{V},i^{V})$
$\displaystyle PH^{\ast}(\Lambda[V],d_{V}):=$ $\displaystyle
PH^{\ast}_{i^{V}}(\Lambda[V\oplus\overline{V}],\delta_{V},i^{V})$

and for a morphism $f$ denote by $HH(f),CH(f),PH(f)$ the linear maps induced
by $\tilde{f}.$ The assignments $HH^{\ast},CH^{\ast},PH^{\ast}$ provide
functors from the category of fc–CDGA’s to graded vector spaces. They come
equipped with the operations $H\Psi_{k},C\Psi_{k}$ etc. induced from
$\Psi_{k}.$ Since for $f$ quasi isomorphisms $HH^{\ast}(f),CH^{\ast}(f),$
$PH^{\ast}(f)$ are isomorphisms these functors, as shown in section 3 extend
to the category of c–CDGA’s. We have the following result.

######  Theorem 4.

Let $(\mathcal{A},d_{\mathcal{A}})$ be a connected CDGA.

1\. All eigenvalues of the endomorphisms $H\Psi_{k}$ and $C\Psi_{k}$ are
$k^{r},r=0,1,2\cdots,$ and their eigenspaces are independent of $k$ provided
$k\geq 2.$ One denotes them by $HH(\mathcal{A},d_{\mathcal{A}})(r),$ and
$CH(\mathcal{A},d_{\mathcal{A}})(r).$

2.

$\displaystyle HH^{\ast}(\mathcal{A},d_{\mathcal{A}})(0)=$ $\displaystyle
H^{\ast}(\mathcal{A},d_{\mathcal{A}}),$ $\displaystyle
CH^{\ast}(\mathcal{A},d_{\mathcal{A}})(0)=$ $\displaystyle
H^{\ast+1}(\mathcal{A},d_{\mathcal{A}})$ $\displaystyle
HH^{r}(\mathcal{A},d_{\mathcal{A}})(p)=$ $\displaystyle
CH^{r}(\mathcal{A},d_{\mathcal{A}})(p)=0,\ \ p\geq r+1$

3.Suppose $(\mathcal{A},d_{\mathcal{A}})$ is 1-connected with minimal model
$(\Lambda[V],d_{V}).$ If $\sum_{i}\dim V^{i}<\infty$ and $\sum_{i}\dim
H^{i}(\mathcal{A},d_{\mathcal{A}})<\infty$ then for any $r\geq 0$ one has

$\sum_{i}\dim HH^{i}(\mathcal{A},d_{\mathcal{A}})(r)<\infty,\ \ \sum_{i}\dim
CH^{i}(\mathcal{A},d_{\mathcal{A}})(r)<\infty.$

###### Proof.

It suffices to check the statements for
$(\mathcal{A},d_{\mathcal{A}})=(\Lambda[V],d_{V})$ minimal. Items 1) and 2)
are immediate consequences of Observation 4.

Item 3) follows from Observation 5. Indeed for a fixed $r$ one has

$\displaystyle\sum_{i}\dim H^{i}(\Lambda[V]\otimes\overline{V}^{\otimes
r},\delta_{V}))\leq$ $\displaystyle\sum_{i,p}\dim H^{i}({\Lambda[V]\otimes
F_{p}(\overline{V}^{\otimes r}}/{\Lambda[V]\otimes
F_{p-1}(\overline{V}^{\otimes r})},\ \delta_{V})=$ $\displaystyle(\dim
V)^{r}\cdot\sum_{i}\dim H^{i}(\Lambda[V],d_{V})$

∎

In addition to $\chi(\mathcal{A},d_{\mathcal{A}}):=\sum(-1)^{i}\dim
H^{i}(\mathcal{A},d_{\mathcal{A}})$ one can consider

$\chi^{H}(\mathcal{A},d_{\mathcal{A}})(r):=\sum(-1)^{i}\dim
HH^{i}(\mathcal{A},d\mathcal{A})(r)$ and

$\chi^{C}(\mathcal{A},d_{\mathcal{A}})(r):=\sum(-1)^{i}\dim
CH^{i}(\mathcal{A},d_{\mathcal{A}})(r),$

and then the power series in $\lambda$,

$\chi^{H}(\mathcal{A},d_{\mathcal{A}})(\lambda):=\sum\chi^{H}(\mathcal{A},d_{\mathcal{A}})(r)\lambda^{r},\
\
\chi^{C}(\mathcal{A},d\mathcal{A})(\lambda):=\sum\chi^{C}(\mathcal{A},d\mathcal{A})(r)\lambda^{r}.$

Theorem 4 (3) implies that for $(\mathcal{A},d_{\mathcal{A}})$ 1-connected
with $\sum_{i}\dim V^{i}<\infty$ and $\sum_{i}\dim
H^{i}(\mathcal{A},d_{\mathcal{A}})<\infty$ the partial Euler–Poincaré
characteristics $\chi^{H}(\mathcal{A},d_{\mathcal{A}})(r)$ and
$\chi^{C}(\mathcal{A},d_{\mathcal{A}})(r)$ and therefore the power series
$\chi^{H}(\mathcal{A},d_{\mathcal{A}})(\lambda)$ and
$\chi^{C}(\mathcal{A},d_{\mathcal{A}})(\lambda)$ are well defined. The results
of Hanlon [H] permit to calculate explicitly $\chi^{H}(\lambda)$ and
$\chi^{C}(\lambda)$ in terms of $\dim H^{i}(\mathcal{A},d_{A})$ if
$(\mathcal{A},d_{\mathcal{A}})$ is 1-connected and formal, i.e. there exists a
quasi isomorphims $(\Lambda[V],d)\to(H^{\ast}(\Lambda[V],d),0),$
$(\Lambda[V],d)$ a minimal model of $(\mathcal{A},d_{\mathcal{A}}).$

We want to define an algebraic analogue of the functor $sH^{\ast}$ on the
category of cCDGA’s. Recall that for a morphism
$f^{\ast}:(C^{\ast}_{1},d^{\ast}_{1})\to(C^{\ast}_{2},d^{\ast}_{2})$ the
“mapping cone” $\textit{Cone}(f^{\ast})$ is the cochain complex with
components $C^{\ast}_{f}=C^{\ast}_{2}\oplus C^{\ast+1}_{1}$ and with

$d^{\ast}_{f}=\begin{pmatrix}d^{\ast}_{2}&f^{\ast+1}\\\
0&-d^{\ast+1}_{1}\end{pmatrix}.$

Notice that, when $f^{\ast}$ is injective, the morphism
$\textit{Cone}(f^{\ast})\to C^{\ast}_{2}/f^{\ast}(C^{\ast}_{1})$ defined by
the composition $C^{\ast}_{2}\oplus C^{\ast+1}_{1}\rightarrow
C^{\ast}_{2}\rightarrow C^{\ast}_{2}/f^{\ast}(C^{\ast}_{1})$ is a quasi
isomorphism.

We will consider the composition

$\underline{I}^{\ast-2}:^{+}\mathcal{C}^{\ast-2}_{i^{V}}(\Lambda[V\oplus\overline{V}],\delta_{V},i^{V})\overset{I^{\ast-2}}{\rightarrow}\mathbb{P}\mathcal{C}^{\ast}(\Lambda[V\oplus\overline{V}],\delta_{V},i^{V})\overset{\mathbb{P}^{\ast}(p)}{\rightarrow}\mathbb{P}\mathcal{C}^{r}(\Lambda[V],d_{V},0)$

with the fist arrow provided by the natural transformation
$I^{\ast-2}:^{+}C^{\ast-2}_{\beta}\to\mathbb{P}C^{\ast}$ described in section
2 applied to the mixed complex
$(\Lambda[V\oplus\overline{V}],\delta_{V},i^{V})$ and the second induced by
the projection on the zero weight component of
$(\Lambda[V\oplus\overline{V}],\delta_{V},i^{V}).$

The mapping cone $\textit{Cone}(\underline{I}^{\ast-2}),$ is functorial when
regarded on the category of fc–CDGA’s. Define

$SH^{\ast}(\Lambda[V],d_{V}):=H^{\ast}(\textit{Cone}(\underline{I}^{\ast})).$

The assignment $(\Lambda[V],d_{V})\rightsquigarrow
SH^{\ast}(\Lambda[V],d_{V})$ is an homotopy functor.

Consider the commutative diagrams

$\textstyle{{}^{+}\mathcal{C}^{\ast-2}_{i^{V}}(\Lambda[V\oplus\overline{V}],\delta_{V},i^{V})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{\ast-2}}$$\scriptstyle{id}$$\textstyle{{}^{+}\mathcal{C}^{\ast}_{i^{V}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces(\Lambda[V\oplus\overline{V}],\delta_{V},i^{V})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\underline{I}^{\ast}}$$\textstyle{\textit{Cone}(i^{\ast-2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{}^{+}\mathcal{C}^{\ast-2}_{i^{V}}(\Lambda[V\oplus\overline{V}],\delta_{V},i^{V})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\underline{I}^{\ast-2}}$$\textstyle{\mathbb{P}\mathcal{C}^{\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces(\Lambda[V],d_{V},0)}$$\textstyle{\textit{Cone}(\underline{I}^{\ast-2}).}$

and

$\textstyle{{}^{+}\mathcal{C}^{\ast-2}_{i^{V}}(\Lambda[V\oplus\overline{V}],\delta_{V},i^{V})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{\ast-2}}$$\scriptstyle{id}$$\textstyle{{}^{+}\mathcal{C}^{\ast}_{i^{V}}(\Lambda[V\oplus\overline{V}],\delta,i^{V})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id}$$\textstyle{\textit{Cone}(i^{\ast-2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{}^{+}\mathcal{C}^{\ast-2}_{i^{V}}(\Lambda[V\oplus\overline{V}],\delta_{V},i^{V})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{\ast-2}}$$\textstyle{{}^{+}\mathcal{C}^{\ast}_{i^{V}}(\Lambda[V\oplus\overline{V}],\delta_{V},i^{V})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C^{\ast}(\Lambda[V\oplus\overline{V}],\delta_{V})}$

with the last vertical arrow in the second diagram a quasi isomorphism as
noticed above.

The long exact sequence induced by passing to cohomology in the first diagram
combined with the identifications implied by the second diagram lead to

|  |
---|---|---
$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{CH^{r-2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{S^{r-2}}$$\scriptstyle{id}$$\textstyle{CH^{r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{J^{r}}$$\scriptstyle{T^{r}}$$\textstyle{HH^{r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{B^{r}}$$\textstyle{CH^{r-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id}$$\textstyle{\cdots}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{CH^{r-2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{I}^{r-2}}$$\scriptstyle{S^{r-2}}$$\textstyle{K^{r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{J}^{r}}$$\textstyle{SH^{r+2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{B}^{r}}$$\textstyle{CH^{r-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$$\textstyle{CH^{r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{I}^{r}}$$\textstyle{PH^{r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

Fig 7 with $PH^{r}=\underset{\rightarrow}{\lim}CH^{r+2k}$ and
$K^{r}:=K^{r}(\Lambda[V],d_{V})$ given by $=\prod_{k}H^{2k}(\Lambda[V],d_{V})$
resp. $\prod_{k}H^{2k+1}(\Lambda[V],d_{V})$if $r$ is even resp. odd. It is
immediate that Theorem 2 remains true for $sH^{\ast},cH^{\ast}$ replaced by
$SH^{\ast},CH^{\ast}$ as follows easily from diagram Fig 7. The diagram Fig 7
should be compared with diagram Fig 6. This explains why
$SH^{\ast}(\Lambda[V],d^{V})$ will be regarded as the algebraic analogue of
$sH^{\ast}(P).$

It is natural to ask if the functors $HH^{\ast},CH^{\ast},SH^{\ast}$ applied
to $(\Omega^{\ast}(P),d^{\ast})$ calculate $hH^{\ast},cH^{\ast},sH^{\ast}$
applied to $P$ and the diagram Fig 7 identifies to the diagram Fig 6. The
answer is in general no, but is yes if $P$ 1-connected .

The minimal model theory, discussed in the next section, permits to identify,
$HH^{\ast}(\Omega^{\ast}(P),d^{\ast}),CH^{\ast}(\Omega^{\ast}(P),d^{\ast})$ to
$hH^{\ast}(P),CH^{\ast}(P)$ and then $SH^{\ast}(\Omega^{\ast}(P),d^{\ast})$ to
$SH^{\ast}(P)$ and actually diagram Fig 6 to diagram Fig 7 when $P$ is
1-connected.

##  7 Minimal models and the proof of Theorem 3

Observe that if $(\mathcal{A}^{\ast},d^{\ast},\beta_{\ast})$ is a mixed CDGA
equipped with the power maps and involution $\Psi_{k},k=-1,1,2,\cdots,$ then
the diagram

$\textstyle{{}^{+}H^{\ast}_{\beta}(\mathcal{A}^{\ast},d^{\ast},\beta_{\ast})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{}^{+}\underline{\Psi}_{k}}$$\textstyle{H^{\ast}(\mathcal{A}^{\ast},d^{\ast})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{}^{+}\underline{\Psi}_{k}}$$\textstyle{{}^{+}H^{\ast}_{\beta}(\mathcal{A}^{\ast},d^{\ast},\beta_{\ast})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{\ast}(\mathcal{A}^{\ast},d^{\ast})}$

can be derived by passing to cohomology in the commutative diagram of CDGA’s.

$\textstyle{(\mathcal{A}^{\ast}\otimes\Lambda[u],\mathcal{D}[u])\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi_{k}[u]}$$\textstyle{(\mathcal{A}^{\ast},d^{\ast})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi_{k}}$$\textstyle{(\mathcal{A}^{\ast}\otimes\Lambda[u],\mathcal{D}[u])\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\mathcal{A}^{\ast},d^{\ast})}$

where $\Lambda[u]$ is the free commutative graded algebra generated by the
symbol $u$ of degree 2, $\mathcal{D}[u](a\otimes u^{r})=d(a)\otimes
u^{r}+\beta(a)\otimes u^{r+1}$ and $\Psi[u](a\otimes
u^{r})=1/k^{r}\Psi(a)\otimes u^{r}.$

For $P$ 1-connected and $(\Lambda([V],d_{V})$ a minimal model of
$(\Omega^{\ast}(P),d^{\ast})$ we want to establish the existence of the
homotopy commutative diagram
---
$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi^{P}_{k}[u]}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi^{P}_{k}}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\theta}}$$\scriptstyle{\Psi_{k}[u]}$$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta}$$\scriptstyle{\Psi_{k}}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\theta}}$$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta}$

where

$\displaystyle A=$
$\displaystyle(\Omega_{X}(P^{S^{1}})\otimes\Lambda[u],\mathcal{D}[u])$
$\displaystyle B=$ $\displaystyle(\Omega_{X}(P^{S^{1}}),d)$ $\displaystyle C=$
$\displaystyle(\Lambda[V\oplus\overline{V}]\otimes\Lambda[u],\delta[u])$
$\displaystyle D=$ $\displaystyle(\Lambda[V\oplus\overline{V}],\delta)$

with

$\displaystyle\mathcal{D}[u](\omega\otimes u^{r})=$ $\displaystyle
d(\omega)\otimes u^{r}+i^{X}(\omega)\otimes u^{r+1}$
$\displaystyle\delta[u](a\otimes u^{r})=$ $\displaystyle\delta(a)\otimes
u^{r}+i^{V}(\omega)\otimes u^{r+1}.$

The existence of the quasi isomorphism $\theta$ was established in in [SV].
The existence of the quasi isomorphism $\tilde{\theta}$ and the homotopy
commutativity of the top square was established in [BV] and the homotopy
commutativity of the side squares was verified in [BFG]. The right side square
resp. left side square in this diagram provide identifications of
$HH^{\ast}(\Lambda[V],d_{V})$ with $hH^{\ast}(P)$ resp. of
$CH^{\ast}(\Lambda[V],d_{V})$ with $cH^{\ast}(P).$ These identifications are
compatible with all natural transformations defined above and with the
endomorphisms induced by the algebraic resp. geometric power maps. In
particular one drive Theorem 3 from Theorem 4. It is tedious but
straightforward to derive, under the hypothesis of 1– connectivity for $P,$
the identification of the diagram Fig 6 for $P$ and the diagram Fig 7 for
$(\Omega^{\ast}(P),d^{\ast}).$

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##  Appendix A Comparative results on DWIE

Model | IgnF1 | F1
---|---|---
CNN $\spadesuit$ | 34.65 | 46.14
LSTM $\spadesuit$ | 40.81 | 52.60
BLSTM $\spadesuit$ | 42.03 | 54.47
Context-Aware $\spadesuit$ | 45.37 | 56.58
CorefBERT $\clubsuit$ | 61.71 | 66.59
GAIN $\spadesuit$ | 62.37 | 67.57
SSAN $\clubsuit$ | 62.58 | 69.39
ATLOP $\spadesuit$ | 62.09 | 69.94
CorefBERT+RSMAN (Yu et al., 2022) | 62.01 | 67.52
SSAN+RSMAN (Yu et al., 2022) | 63.42 | 70.95
BERTbase (2022) (Han et al., 2022) | 62.92 | 69.12
BERT+Correlation (Han et al., 2022) | 65.64 | 71.56
Table 7. Comparative Results on the DWIE dataset (Zaporojets et al., 2021).
Results with $\spadesuit$ and $\clubsuit$ come from Ru et al. (2021) and Yu et
al. (2022), respectively.