# On the String Topology Coproduct for Lie Groups

Maximilian Stegemeyer  Max Planck Institute for Mathematics in the Sciences,
Inselstraße 22, 04103 Leipzig, Germany  Mathematisches Institut, Universität
Leipzig, Augustusplatz 10, 04109 Leipzig, Germany
<EMAIL_ADDRESS>
(Date: August 28, 2024)

###### Abstract.

The free loop space of a Lie group is homeomorphic to the product of the Lie
group itself and its based loop space. We show that the coproduct on the
homology of the free loop space that was introduced by Goresky and Hingston
splits into the diagonal map on the group and a based coproduct on the
homology of the based loop space. This result implies that the coproduct is
trivial for even-dimensional Lie groups. Using results by Bott and Samelson,
we show that the coproduct is trivial as well for a large family of simply
connected Lie groups.

###### Contents

  1. 1 Introduction
  2. 2 Free and Based Coproduct
  3. 3 Splitting of the Coproduct
  4. 4 Triviality of the Coproduct for Even-Dimensional Lie Groups
  5. 5 Triviality of the Coproduct for certain Simply Connected Lie Groups

##  1\. Introduction

In the last two decades the study of string topology structures on the
homology or cohomology of the free loop space has received a great deal of
attention. The operation that has been studied the most is certainly the Chas-
Sullivan product which was introduced by Chas and Sullivan in [CS99]. For an
oriented closed manifold $M$ this is a product on the singular homology of the
free loop space $\Lambda M$ of the form

$\wedge:\mathrm{H}_{i}(\Lambda M)\otimes\mathrm{H}_{j}(\Lambda
M)\to\mathrm{H}_{i+j-n}(\Lambda M)$

where $n$ is the dimension of $M$. In [CKS08], Cohen, Klein and Sullivan show
that the Chas-Sullivan product is a homotopy invariant.

Going back to ideas by Sullivan [Sul04], Goresky and Hingston [GH09] define a
coproduct on the homology of the free loop space of an oriented closed
$n$-dimensional manifold $M$ relative to the constant loops which takes the
form

$\vee:\mathrm{H}_{i}(\Lambda M,M)\to\mathrm{H}_{i+1-n}(\Lambda M\times\Lambda
M,\Lambda M\times M\cup M\times\Lambda M).$

As a dual operation, a product on the cohomology of the free loop space can be
defined (see [GH09, Section 9]). This homology coproduct and cohomology
product have been further studied by Hingston and Wahl in [HW17]. The homology
coproduct is not a homotopy invariant as an example of Naef [Nae21] shows.
However, Hingston and Wahl show in [HW19] that the homology coproduct is
invariant under homotopy equivalences that satisfy an additional assumption.

It is not surprising that explicit computations with these string topology
operations can become very complicated. There are some results, where the
Chas-Sullivan product was computed explicitly. Using integer coefficients the
Chas-Sullivan product has been computed for spheres and complex projective
spaces (see [CJY03]) and complex Stiefel manifolds (see [Tam06]). With
rational coefficients, the Chas-Sullivan product has furthermore been computed
for complex and quaternionic projective spaces (see [Yan13]). Hepworth has
computed the Chas-Sullivan product with rational coefficients for the special
orthogonal groups (see [Hep10]).

The Goresky-Hingston product in cohomology and the corresponding coproduct in
homology have only been computed explicitly in a few instances. In [GH09] the
authors compute the Goresky-Hingston product for spheres of dimension greater
than or equal to $3$. In [HW17] the homology coproduct is computed for odd-
dimensional spheres.

The goal of this article is to study the homology coproduct for Lie groups.
The free loop space of a compact Lie group $G$ splits into the product

$\Lambda G\cong G\times\Omega_{e}G$

where $\Omega_{e}G$ is the space of loops based at the neutral element $e$.
Since the homology of $\Omega_{e}G$ with coefficients in a commutative ring
$R$ is free (see [Bot56]), the homology of $\Lambda G$ is isomorphic to the
tensor product
$\mathrm{H}_{\bullet}(G)\otimes\mathrm{H}_{\bullet}(\Omega_{e}G)$. The
following main result of this article shows that the homology coproduct
behaves well under this isomorphism. This result should be thought of as an
analogue to [Hep10, Theorem 1.1]. There, Hepworth shows that for a Lie group
$G$, the Chas-Sullivan ring is isomorphic to the tensor product of the
intersection ring on the Lie group $G$ and the Pontryagin ring on the based
loop space $\Omega_{e}G$.

######  Theorem (Theorem 3.1).

Let $G$ be a compact Lie group of dimension $n$ and consider homology with
coefficients in a commutative ring $R$. Under the isomorphism
$\mathrm{H}_{\bullet}(\Lambda
G,G)\cong\mathrm{H}_{\bullet}(G)\otimes\mathrm{H}_{\bullet}(\Omega_{e}G,e)$
the coproduct $\vee$ can be expressed by the tensor product of the map $d_{*}$
that is induced by the diagonal map $d:G\to G\times G$ and the based coproduct
$\vee_{\Omega}$ up to a sign-correction. More precisely, the following diagram
commutes

${\bigoplus_{k+j=i}\mathrm{H}_{k}(G)\otimes\mathrm{H}_{j}(\Omega,e)}$${\mathrm{H}_{i}(\Lambda,G)}$${\bigoplus_{k+j=i}\mathrm{H}_{k}(G^{2})\otimes\mathrm{H}_{j+1-n}(\Omega^{2},\Omega\times
e\cup e\times\Omega)}$${\mathrm{H}_{i+1-n}(\Lambda^{2},\Lambda\times G\cup
G\times\Lambda)}$$\scriptstyle{(-1)^{ni}d_{*}\otimes\vee_{\Omega}}$$\scriptstyle{\cong}$$\scriptstyle{\vee}$$\scriptstyle{\cong}$

where $\Omega=\Omega_{e}G$ and $\Lambda=\Lambda G$.

With this result, one can prove directly that the coproduct is trivial for
even-dimensional Lie groups. Combining this with explicit cycles in the based
loop space of a compact, simply connected Lie group of rank $r\geq 2$ we are
able to show the following.

######  Theorem (Theorem 4.1 and Theorem 5.6).

Let $G$ be a compact Lie group. If $G$ is even-dimensional or if $G$ is simply
connected and of rank $r\geq 2$, then the homology coproduct is trivial.

This article is organized as follows. In Section 2, we review some facts about
loop spaces and define the based homology coproduct as well as the free
homology coproduct. We also show a compatibility statement between these two
coproducts. The goal of Section 3 is to show that the splitting of the free
loop space of a compact Lie group is respected by the homology coproduct. We
also show that the dual cohomology product behaves nicely under this
splitting. In the brief Section 4, we conclude that the coproduct is trivial
for even-dimensional Lie groups. Finally, in Section 5 we define explicit
cycles of the based loop space of a Lie group that were first introduced by
Bott and Samelson [BS58] to prove that the free homology coproduct vanishes
for simply connected compact Lie groups of rank $r\geq 2$.

_In this article all manifolds, are assumed to be smooth and connected and all
Riemannian metrics are assumed to be smooth. In particular, all Lie groups are
assumed to be connected._

##  2\. Free and Based Coproduct

In this section, we introduce the based coproduct and the free coproduct. We
will then show that these two coproducts are compatible. Furthermore, the dual
cohomology products are introduced.

In the following, let $M$ be an oriented closed $n$-dimensional Riemannian
manifold. We consider absolutely continuous curves in $M$ (see [Kli95,
Definition 2.3.1]). Let

$PM=\big{\\{}\gamma:I\to M\,|\,\gamma\,\,\text{absolutely
continuous},\,\,\int_{0}^{1}|\dot{\gamma}(t)|^{2}\,\mathrm{d}t<\infty\big{\\}}$

be the set of absolutely continuous curves in $M$ with square integrable
derivative, where $I=[0,1]$ is the unit interval. This set can be given a
topology and a differentiable structure that make it a Hilbert manifold (see
[Kli95, Theorem 2.3.12]). Note that $PM$ with this topology is homotopy
equivalent to the space

$C^{0}(I,M)=\\{\gamma:I\to M\,|\,\gamma\,\,\,\text{continuous}\\}$

of continuous paths in $M$ with the compact-open topology (see [Kli78, Theorem
1.2.10]).

We consider the following submanifolds of $PM$: The free loop space of $M$ is
defined to be

$\Lambda M=\\{\gamma\in PM\,|\,\gamma(0)=\gamma(1)\\}$

and for a fixed point $p_{0}\in M$, the based loop space of $M$ in $p_{0}$ is

$\Omega_{p_{0}}M=\\{\gamma\in PM\,|\,\gamma(0)=\gamma(1)=p_{0}\\}.$

If it is clear what the basepoint of $\Omega_{p_{0}}M$ is, we may suppress the
index $p_{0}$ from the notation. If the manifold in question is clear from the
context, we will also write $\Omega$ and $\Lambda$ for $\Omega_{p_{0}}M$ and
$\Lambda M$, respectively. Note that the trivial loops in $M$ form a
submanifold of $\Lambda M$ which is diffeomorphic to $M$ (see [Kli78,
Proposition 1.4.6]).

On the path space $PM$ we consider the function

$\mathcal{L}:PM\to[0,\infty),\qquad\mathcal{L}(\gamma)=\sqrt{\int_{0}^{1}|\dot{\gamma}(t)|^{2}\mathrm{d}t}$

which is the square root of the energy functional and which is well-defined by
definition of $PM$. The energy functional is a continuous function on $PM$
(see [Kli95, Theorem 2.3.20]), hence the function $\mathcal{L}$ is continuous
as well.

Assume that we have fixed a basepoint $p_{0}\in M$ and choose an $\epsilon>0$
smaller than the injectivity radius of $M$. In order to define the based
coproduct and the free coproduct, we need to make the following preparations.
For the definition of the free coproduct, we closely follow [HW17, Section
1.5]. Fix a commutative ring $R$ and consider homology and cohomology with
coefficients in $R$.

First, let $\Delta M$ be the diagonal in $M\times M$. The diagonal has a
tubular neighborhood in $M\times M$ which can be chosen as

$U_{M}=\\{(p,q)\in M\times M\,|\,\mathrm{d}(p,q)<\epsilon\\}$

where $\mathrm{d}$ is the distance function on $M\times M$ induced by the
Riemannian metric on $M$. This choice of $U_{M}$ is made as in [HW17, Section
1.3]. Choose $\epsilon_{0}>0$ such that $\epsilon_{0}<\epsilon$ and define

$U_{M,\geq\epsilon_{0}}=\\{(p,q)\in
U_{M}\,|\,\mathrm{d}(p,q)\geq\epsilon_{0}\\}.$

A tubular neighborhood of the diagonal in $M\times M$ is homeomorphic to the
normal bundle of $\Delta M\hookrightarrow M\times M$ which itself is
isomorphic to the tangent bundle of $M$. Consequently, the pair
$(U_{M},U_{M,\geq\epsilon_{0}})$ is homeomorphic to the pair
$(TM^{<\epsilon},TM^{<\epsilon}_{\geq\epsilon_{0}})$ where $TM^{<\epsilon}$ is
the open disk bundle

$TM^{<\epsilon}=\\{v\in TM\,|\,|v|<\epsilon\\}$

and $TM^{<\epsilon}_{\geq\epsilon_{0}}$ is the fiber bundle

$TM^{<\epsilon}_{\geq\epsilon_{0}}=\\{v\in
TM^{<\epsilon}\,|\,|v|\geq\epsilon_{0}\\}.$

The Thom class in $\mathrm{H}^{n}(TM,TM\smallsetminus M)$ that is defined by
the orientation of $M$ induces a class in
$\mathrm{H}^{n}(TM^{<\epsilon},TM^{<\epsilon}_{\geq\epsilon_{0}})$ (see [HW17,
Section 1.3]) and therefore we obtain a class

$\tau_{M}\in\mathrm{H}^{n}(U_{M},U_{M,\geq\epsilon_{0}}).$

Consider the open ball

$B_{p_{0}}=\\{q\in M\,|\,\mathrm{d}(p_{0},q)<\epsilon\\}\subseteq M$

and the inclusion $\iota:B_{p_{0}}\hookrightarrow U_{M}$ given by
$\iota(q)=(p_{0},q).$ If we define

$B_{p_{0},\geq\epsilon_{0}}=\\{q\in
B_{p_{0}}\,|\,\mathrm{d}(p_{0},q)\geq\epsilon_{0}\\}$

then the inclusion $\iota$ can be understood as a map of pairs
$\iota:(B_{p_{0}},B_{p_{0},\geq\epsilon_{0}})\hookrightarrow(U_{M},U_{M,\geq\epsilon_{0}})$.

Clearly, we have

$\mathrm{H}^{n}(B_{p_{0}},B_{p_{0},\geq\epsilon_{0}})\cong\mathrm{H}^{n}(\mathbb{D}^{n},\mathbb{S}^{n-1})\cong
R.$

Furthermore, under the identification $U_{M}\cong TM$, one observes that the
inclusion $\iota:B_{p_{0}}\hookrightarrow U_{M}$ corresponds to the inclusion
of the fiber $T_{p_{0}}M\hookrightarrow TM$.

Recall that the Thom class $\tau$ of an orientable vector bundle $E\to B$ of
rank $k$ has the following property. Let $b\in B$ be a point and denote by
$E_{b}$ the fiber over $b$. If we consider the inclusion
$i:(E_{b},E_{b}\smallsetminus\\{0\\})\hookrightarrow(E,E\smallsetminus B)$
then the pull back
$i^{*}\tau\in\mathrm{H}^{k}(E_{b},E_{b}\smallsetminus\\{0\\})$ is the
generator induced by the orientation of $E_{b}$. Hence, in our particular
situation one obtains the following.

######  Lemma 2.1.

The generator $\tau_{0}$ of
$\mathrm{H}^{n}(B_{p_{0}},B_{p_{0},\geq\epsilon_{0}})$ which is induced by the
orientation of $M$ and the class $\tau_{M}$ satisfy

$\iota^{*}\tau_{M}=\tau_{0}.$

Furthermore, define the spaces

$\displaystyle U_{\Lambda}$ $\displaystyle=$
$\displaystyle\\{(\gamma,s)\in\Lambda M\times
I\,|\,\mathrm{d}(\gamma(0),\gamma(s))<\epsilon\\}\qquad\text{and}$
$\displaystyle U_{\Lambda,\geq\epsilon_{0}}$ $\displaystyle=$
$\displaystyle\\{(\gamma,s)\in
U_{\Lambda}\,|\,\mathrm{d}(\gamma(0),\gamma(s))\geq\epsilon_{0}\\}$

and their based counterparts

$\displaystyle U_{\Omega}$ $\displaystyle=$
$\displaystyle\\{(\gamma,s)\in\Omega M\times
I\,|\,\mathrm{d}(p_{0},\gamma(s))<\epsilon\\}\qquad\text{and}$ $\displaystyle
U_{\Omega,\geq\epsilon_{0}}$ $\displaystyle=$ $\displaystyle\\{(\gamma,s)\in
U_{\Omega}\,|\,\mathrm{d}(p_{0},\gamma(s))\geq\epsilon_{0}\\}.$

Consider the evaluation maps $\mathrm{ev}_{\Lambda}:\Lambda\times I\to M\times
M$ and $\mathrm{ev}_{\Omega}:\Omega\times I\to M$ given by

$\mathrm{ev}_{\Lambda}(\gamma,s)=(\gamma(0),\gamma(s))\qquad\text{and}\qquad\mathrm{ev}_{\Omega}(\sigma,t)=\sigma(t)$

for $\gamma\in\Lambda M$, $\sigma\in\Omega_{p_{0}}M$ and $s,t\in I$. These
maps restrict to maps of pairs

$\displaystyle\mathrm{ev}_{\Lambda}:(U_{\Lambda},U_{\Lambda,\geq\epsilon_{0}})\to(U_{M},U_{M,\geq\epsilon_{0}})\qquad\text{and}$
$\displaystyle\mathrm{ev}_{\Omega}:(U_{\Omega},U_{\Omega,\geq\epsilon_{0}})\to(B_{p_{0}},B_{p_{0},\geq\epsilon_{0}}).$

Therefore we can define the classes

$\tau_{\Lambda}=\mathrm{ev}_{\Lambda}^{*}\tau_{M}\in\mathrm{H}^{n}(U_{\Lambda},U_{\Lambda,\geq\epsilon_{0}})\qquad\text{and}\qquad\tau_{\Omega}=\mathrm{ev}_{\Omega}^{*}\tau_{0}\in\mathrm{H}^{n}(U_{\Omega},U_{\Omega,\geq\epsilon_{0}}).$

######  Lemma 2.2.

The classes $\tau_{\Lambda}$ and $\tau_{\Omega}$ satisfy
$j^{*}\tau_{\Lambda}=\tau_{\Omega}$ where
$j:(U_{\Omega},U_{\Omega,\geq\epsilon_{0}})\hookrightarrow(U_{\Lambda},U_{\Lambda,\geq\epsilon_{0}})$
is the inclusion.

###### Proof.

We have

$j^{*}\tau_{\Lambda}=j^{*}\mathrm{ev}_{\Lambda}^{*}\tau_{M}=\mathrm{ev}_{\Omega}^{*}\iota^{*}\tau_{M}=\mathrm{ev}_{\Omega}^{*}\tau_{0}=\tau_{\Omega}$

where we used Lemma 2.1 and the fact that $\mathrm{ev}_{\Lambda}\circ
j=\iota\circ\mathrm{ev}_{\Omega}$ with $\iota$ as above. ∎

Consider the spaces

$\displaystyle F_{\Lambda}=\\{(\gamma,s)\in\Lambda M\times
I\,|\,\gamma(s)=\gamma(0)\\}\qquad\text{and}$ $\displaystyle
F_{\Omega}=\\{(\gamma,s)\in\Omega M\times I\,|\,\gamma(s)=p_{0}\\}.$

We now want to define a retraction map $\mathrm{R}_{GH}:U_{\Lambda}\to
F_{\Lambda}$. Let $\gamma\in PM$ be a path and $s\in I$. In order to make the
restriction $\gamma|_{[0,s]}$ a well-defined element of $PM$, we introduce the
map $\rho_{s}:PM\to PM$ given by

$\rho_{s}(\gamma)(t)=\gamma(st)\qquad\text{for}\,\,\,\gamma\in PM.$

Similarly, define $\mu_{s}:PM\to PM$ by

$\mu_{s}(\gamma)(t)=\gamma(s+(1-s)t)\qquad\text{for}\,\,\,\gamma\in PM$

The path $\mu_{s}(\gamma)$ is the restriction of $\gamma$ to the interval
$[s,1]$ combined with a reparametrization to make it an element of $PM$.
Furthermore, for a fixed $s\in I$ and two paths $\gamma,\delta\in PM$ with
$\gamma(1)=\delta(0)$, we denote by $\mathrm{concat}_{s}(\gamma,\delta)$ the
concatenation of these two paths such that the path $\gamma$ is run through
during the interval $[0,s]$ and the path $\delta$ is run through during the
interval $[s,1]$.

Given two paths $\gamma,\delta\in PM$ such that $\gamma(1)=\delta(0)$, we
define their optimal concatenation to be the path

$\gamma*\delta=\mathrm{concat}_{t_{*}}(\gamma,\delta),\qquad\text{where}\,\,\,t_{*}=\frac{\mathcal{L}(\gamma)}{\mathcal{L}(\gamma)+\mathcal{L}(\delta)}$

if one of $\gamma$ or $\delta$ has positive length. If
$\mathcal{L}(\gamma)=\mathcal{L}(\delta)=0$, then $\gamma=\delta=p_{0}$ is a
trivial loop at some point $p_{0}$ in $M$ and we set $\gamma*\delta=p_{0}$.
For two points $p,q\in M$ with $\mathrm{d}(p,q)<\epsilon$, the path
$\overline{pq}\in PM$ is chosen to be the unique distance-minimizing geodesic
connecting these two points parametrized on the unit interval $I$.

Using these definitions, we define $\mathrm{R}_{GH}:U_{\Lambda}\to
F_{\Lambda}$ by

$\mathrm{R}_{GH}(\gamma,s)=\big{(}\mathrm{concat}_{s}\big{(}\rho_{s}(\gamma)*\overline{\gamma(s)\gamma(0)}\,,\,\,\overline{\gamma(0)\gamma(s)}*\mu_{s}(\gamma)\big{)}\,,\,s\big{)}$

for $(\gamma,s)\in U_{\Lambda}$. One checks that this indeed a continuous map.

If we restrict $\mathrm{R}_{GH}$ to $U_{\Omega}\subseteq U_{\Lambda}$, we
obtain a map $\mathrm{R}_{GH}:U_{\Omega}\to F_{\Omega}$. Finally, let
$\mathrm{cut}:F_{\Lambda}\to\Lambda M\times\Lambda M$ be the map

$\mathrm{cut}(\gamma,s)=(\rho_{s}(\gamma),\mu_{s}(\gamma))\qquad\text{for}\,\,\,(\gamma,s)\in
F_{\Lambda}.$

This restricts to a map
$\mathrm{cut}:F_{\Omega}\to\Omega_{p_{0}}M\times\Omega_{p_{0}}M$.

Let $[I]$ be a generator of $\mathrm{H}_{1}(I,\partial I)$. If $p_{0}\in M$ is
the base point of $M$, then by a slight abuse of notation, we shall write
$p_{0}$ for the set $\\{p_{0}\\}\subseteq\Omega M$ that consists of the
trivial loop at $p_{0}$.

######  Definition 2.3.

Let $M$ be an oriented closed $n$-dimensional Riemannian manifold with base
point $p_{0}\in M$.

  1. (1)

The based homology coproduct is defined as the composition

$\displaystyle\vee_{\Omega}:\mathrm{H}_{\bullet}(\Omega,p_{0})$
$\displaystyle\xrightarrow{\times[I]}$
$\displaystyle\mathrm{H}_{\bullet+1}(\Omega\times I,\Omega\times\partial I\cup
p_{0}\times I)$ $\displaystyle\xrightarrow{\tau_{\Omega}\cap}$
$\displaystyle\mathrm{H}_{\bullet+1-n}(U_{\Omega},\Omega\times\partial I\cup
p_{0}\times I)$ $\displaystyle\xrightarrow{(\mathrm{R}_{GH})_{*}}$
$\displaystyle\mathrm{H}_{\bullet+1-n}(F_{\Omega},\Omega\times\partial I\cup
p_{0}\times I)$ $\displaystyle\xrightarrow{(\mathrm{cut})_{*}}$
$\displaystyle\mathrm{H}_{\bullet+1-n}(\Omega\times\Omega,\Omega\times
p_{0}\cup p_{0}\times\Omega).$

  2. (2)

The free homology coproduct is defined as the composition

$\displaystyle\vee:\mathrm{H}_{\bullet}(\Lambda,M)$
$\displaystyle\xrightarrow{\times[I]}$
$\displaystyle\mathrm{H}_{\bullet+1}(\Lambda\times I,\Lambda\times\partial
I\cup M\times I)$ $\displaystyle\xrightarrow{\tau_{\Lambda}\cap}$
$\displaystyle\mathrm{H}_{\bullet+1-n}(U_{\Lambda},\Lambda\times\partial I\cup
M\times I)$ $\displaystyle\xrightarrow{(\mathrm{R}_{GH})_{*}}$
$\displaystyle\mathrm{H}_{\bullet+1-n}(F_{\Lambda},\Lambda\times\partial I\cup
M\times I)$ $\displaystyle\xrightarrow{(\mathrm{cut})_{*}}$
$\displaystyle\mathrm{H}_{\bullet+1-n}(\Lambda\times\Lambda,\Lambda\times
M\cup M\times\Lambda).$

Note that the relative cap product that we use, requires some care. For
example, in the case of the based coproduct, it is understood as a map

$\mathrm{H}^{n}(U_{\Omega},U_{\Omega,\geq\epsilon_{0}})\otimes\mathrm{H}_{i}(\Omega\times
I,\Omega\times\partial I\cup p_{0}\times
I)\to\mathrm{H}_{i-n}(U_{\Omega},\Omega\times\partial I\cup p_{0}\times I)$

and analogously for the free coproduct. See [HW17, Appendix A] for details of
this construction and a naturality statement.

We now want to show that the based coproduct and the free coproduct are
compatible.

######  Proposition 2.4.

Let $M$ be an oriented closed Riemannian manifold with base point $p_{0}\in M$
and consider singular homology with coefficients in a commutative ring $R$.
The based coproduct and the free coproduct are compatible in the sense that
the diagram

${\mathrm{H}_{k}(\Omega,p_{0})}$${\mathrm{H}_{k}(\Lambda,M)}$${\mathrm{H}_{k+1-n}(\Omega^{2},\Omega\times
p_{0}\cup p_{0}\times\Omega)}$${\mathrm{H}_{k+1-n}(\Lambda^{2},\Lambda\times
M\cup
M\times\Lambda)}$$\scriptstyle{i_{*}}$$\scriptstyle{\vee_{\Omega}}$$\scriptstyle{\vee}$$\scriptstyle{(i,i)_{*}}$

commutes. Here $i:\Omega\to\Lambda$ is the inclusion.

###### Proof.

The claimed commutativity of the diagram in the statement of the proposition
follows if we verify that all subdiagrams of the following diagram commute

${\mathrm{H}_{k}(\Omega,p_{0})}$${\mathrm{H}_{k}(\Lambda,M)}$${\mathrm{H}_{k+1}(\Omega\times
I,\Omega\times\partial I\cup p_{0}\times I)}$${\mathrm{H}_{k+1}(\Lambda\times
I,\Lambda\times\partial I\cup M\times
I)}$${\mathrm{H}_{k+1-n}(U_{\Omega},\Omega\times\partial I\cup p_{0}\times
I)}$${\mathrm{H}_{k+1-n}(U_{\Lambda},\Lambda\times\partial I\cup M\times
I)}$${\mathrm{H}_{k+1-n}(\Omega\times\Omega,\Omega\times p_{0}\cup
p_{0}\times\Omega)}$${\mathrm{H}_{k+1-n}(\Lambda\times\Lambda,\Lambda\times
M\cup
M\times\Lambda).}$$\scriptstyle{i_{*}}$$\scriptstyle{\times[I]}$$\scriptstyle{\times[I]}$$\scriptstyle{(i,\mathrm{id}_{I})_{*}}$$\scriptstyle{\tau_{\Omega}\cap}$$\scriptstyle{\tau_{\Lambda}\cap}$$\scriptstyle{(i,\mathrm{id}_{I})_{*}}$$\scriptstyle{(\mathrm{cut}\circ\mathrm{R}_{GH})_{*}}$$\scriptstyle{(\mathrm{cut}\circ\mathrm{R}_{GH})_{*}}$$\scriptstyle{(i,i)_{*}\,\,}$

It is clear that the first and the last square of the above diagram commute.
For the middle square, let $X\in\mathrm{H}_{\bullet}(\Omega\times
I,\Omega\times\partial I\cup p_{0}\times I)$. Then if we write $j$ for
$(i,\mathrm{id}_{I})$, we see that

$j_{*}(\tau_{\Omega}\cap X)=j_{*}(j^{*}\tau_{\Lambda}\cap
X)=\tau_{\Lambda}\cap(j_{*}X)$

by Lemma 2.2 and the naturality of the cap product. Hence, the above diagram
commutes and therefore the coproducts are compatible. ∎

We will now focus on field coefficients for the rest of this section. If
$\mathbb{F}$ is a field and if the homology of $\Omega$, resp. $\Lambda$ is of
finite type, then the based and free coproduct induce products in cohomology.
Note that the following product on the cohomology of the free loop space was
first introduced by Goresky and Hingston in [GH09].

######  Definition 2.5.

Let $M$ be an oriented, closed manifold with base point $p_{0}\in M$ and let
$\mathbb{F}$ be a field. Assume that the homology of $\Omega M$, resp.
$\Lambda M$ is of finite type.

  1. (1)

If $\alpha\in\mathrm{H}^{i}(\Omega,p_{0};\mathbb{F})$,
$\beta\in\mathrm{H}^{j}(\Omega,p_{0};\mathbb{F})$, then their based cohomology
product
$\alpha\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\ast$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\ast$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\ast$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\ast$\cr}}}_{\Omega}\beta$
is defined to be the unique class in
$\mathrm{H}^{i+j+n-1}(\Omega,p_{0};\mathbb{F})$ that satisfies

$\langle\alpha\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\ast$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\ast$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\ast$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\ast$\cr}}}_{\Omega}\beta,X\rangle=\langle\alpha\times\beta,\vee_{\Omega}X\rangle\qquad\text{for
all}\,\,\,X\in\mathrm{H}_{\bullet}(\Omega,p_{0};\mathbb{F}).$

  2. (2)

If $\alpha\in\mathrm{H}^{i}(\Lambda,M;\mathbb{F})$,
$\beta\in\mathrm{H}^{j}(\Lambda,M;\mathbb{F})$, then their free cohomology
product
$\alpha\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\ast$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\ast$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\ast$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\ast$\cr}}}\beta$
is defined to be the unique class in
$\mathrm{H}^{i+j+n-1}(\Lambda,M;\mathbb{F})$ that satisfies

$\langle\alpha\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\ast$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\ast$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\ast$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\ast$\cr}}}\beta,X\rangle=\langle\alpha\times\beta,\vee
X\rangle\qquad\text{for
all}\,\,\,X\in\mathrm{H}_{\bullet}(\Lambda,M;\mathbb{F}).$

######  Remark 2.6.

Let $M$ be an oriented closed Riemannian manifold.

  1. (1)

The free homology coproduct was first introduced in [GH09, Section 8]. In the
above exposition we followed the definition of the coproduct in [HW17], where
the authors give a chain-level definition. They then show that the induced
coproduct in homology is equivalent to the definition in [GH09]. The free
cohomology product is more often referred to as the Goresky-Hingston product,
since it was introduced by Goresky and Hingston in [GH09, Section 9]. Using
the chain level definition of the free coproduct, Hingston and Wahl obtain an
equivalent definition in [HW17]. In order to make a better distinction to the
based cohomology product, we will always refer to the Goresky-Hingston product
as the free cohomology product.

  2. (2)

Note that we used the names product and coproduct. However, the operations
only have the algebraic properties of what is usually understood as product
and coproduct if we introduce additional sign conventions (see [HW17]). Since
this paper mostly considers situations in which the products and coproducts
are trivial, we will not deal with the signs and stick to the above
definitions.

  3. (3)

One can also extend all the coproducts and products to absolute homology and
cohomology, respectively. As an example, consider the free homology coproduct

$\vee:\mathrm{H}_{i}(\Lambda,M)\to\mathrm{H}_{i+1-n}(\Lambda\times\Lambda,\Lambda\times
M\cup M\times\Lambda).$

The homology of $\Lambda$ with coefficients in a commutative ring $R$ is
isomorphic to the direct sum

$\mathrm{H}_{i}(\Lambda)\cong\mathrm{H}_{i}(\Lambda,M)\oplus\mathrm{H}_{i}(M).$

The idea of extending the free homology coproduct to a map

$\hat{\vee}:\mathrm{H}_{i}(\Lambda)\to\mathrm{H}_{i+1-n}(\Lambda\times\Lambda)$

is therefore to extend the relative homology coproduct $\vee$ on
$\mathrm{H}_{i}(\Lambda,M)$ by the trivial map on $\mathrm{H}_{i}(M)$ in order
to get a well-defined map on $\mathrm{H}_{i}(\Lambda)$. Then, one has to lift
the resulting class in $\mathrm{H}_{i+1-n}(\Lambda^{2},\Lambda\times M\cup
M\times\Lambda)$ to a class in $\mathrm{H}_{i+1-n}(\Lambda^{2})$. See [HW17]
for details. While it might indeed be more convenient to work on the absolute
homology of the free loop space, these extensions do not bring any new
topological information into play. Therefore, for the rest of this article, we
will stick to the relative definitions.

##  3\. Splitting of the Coproduct

In this section, we will discuss how the based coproduct and the free
coproduct are related for Lie groups. While for arbitrary manifolds, we only
have the compatibility statement of Proposition 2.4, we will now show that in
the case of a Lie group the free coproduct is completely determined by the
based coproduct and the push-forward of the diagonal map of the group itself.
After that, we will examine how this result transfers to the cohomology
products.

Let $G$ be a compact Lie group. As base point we will always choose the unit
element $e\in G$. There is a homeomorphism

$\Phi:G\times\Omega G\to\Lambda G,\qquad\Phi(g,\gamma)=(t\mapsto g\gamma(t)).$

Observe that its inverse is the map

$\Psi:\Lambda G\to G\times\Omega
G,\qquad\Psi(\gamma)=(\gamma(0),t\mapsto(\gamma(0)^{-1}\gamma(t))).$ (3.1)

Bott [Bot56] has shown that the integer homology of $\Omega G$ is free and
that it is non-trivial only in even degrees. By the universal coefficient
theorem, this property then holds for homology with coefficients in an
arbitrary commutative ring $R$ as well. The same also holds for the relative
homology $\mathrm{H}_{\bullet}(\Omega G,e)$. Consequently, there is a Künneth
isomorphism

$\mathrm{H}_{i}(G\times\Omega G,G\times
e)\cong\bigoplus_{k+j=i}\mathrm{H}_{k}(G)\otimes\mathrm{H}_{j}(\Omega G,e).$

Combining this with the homeomorphism $\Phi$ we get an isomorphism

$\Theta_{*}:(\mathrm{H}_{\bullet}(G)\otimes\mathrm{H}_{\bullet}(\Omega
G,e))_{i}\xrightarrow[]{\times}\mathrm{H}_{i}(G\times\Omega G,G\times
e)\xrightarrow[]{\Phi_{*}}\mathrm{H}_{i}(\Lambda G,G).$

Hepworth [Hep10] has shown that the isomorphism of groups $\Theta_{*}$ becomes
a ring isomorphism if we equip $\mathrm{H}_{\bullet}(G)$ with the intersection
product, $\mathrm{H}_{\bullet}(\Omega G)$ with the Pontryagin product and
$\mathrm{H}_{\bullet}(\Lambda G)$ with the Chas-Sullivan product. The
following result can be seen as an analogous result for the coproduct.

For the statement of the theorem, observe that there is a homeomorphism

$\widetilde{\Phi}:G\times G\times\Omega G\times\Omega G\to\Lambda
G\times\Lambda G$

given by

$\widetilde{\Phi}(g_{1},g_{2},\gamma_{1},\gamma_{2})=((t\mapsto
g_{1}\gamma_{1}(t)),(s\mapsto g_{2}\gamma_{2}(s))).$ (3.2)

Since the relative homology $\mathrm{H}_{\bullet}(\Omega G,e)$ is free and
concentrated in even degrees, there is a Künneth isomorphism

$\mathrm{H}_{i}(\Omega G\times\Omega G,\Omega G\times e\cup e\times\Omega
G)\cong\bigoplus_{k+j=i}\mathrm{H}_{k}(\Omega G,e)\otimes\mathrm{H}_{j}(\Omega
G,e)\,\,,$

see [Spa95, Theorem 5.3.10] for a relative version of the Künneth theorem
which applies to this situation. Hence, the relative homology
$\mathrm{H}_{\bullet}(\Omega G\times\Omega G,\Omega G\times e\cup
e\times\Omega G)$ is also free and concentrated in even degrees.

Consequently, there is again a Künneth isomorphism

$\mathrm{H}_{i}(G^{2}\times\Omega^{2},G^{2}\times(\Omega\times e\cup
e\times\Omega))\cong\bigoplus_{k+j=i}\mathrm{H}_{k}(G^{2})\otimes\mathrm{H}_{j}(\Omega^{2},\Omega\times
e\cup e\times\Omega).$

This yields an isomorphism

$\displaystyle\widetilde{\Theta}_{*}:\big{(}\mathrm{H}_{\bullet}(G^{2})\otimes\mathrm{H}_{\bullet}(\Omega^{2},\Omega\times
e\cup e\times\Omega)\big{)}_{i}$ $\displaystyle\xrightarrow[]{\times}$
$\displaystyle\mathrm{H}_{i}(G^{2}\times\Omega^{2},G^{2}\times(\Omega\times
e\cup e\times\Omega))$ $\displaystyle\xrightarrow[]{\widetilde{\Phi}_{*}}$
$\displaystyle\mathrm{H}_{i}(\Lambda^{2},\Lambda\times G\cup G\times\Lambda).$

######  Theorem 3.1.

Let $G$ be a compact Lie group of dimension $n$. Consider homology with
coefficients in a commutative ring $R$. Under the isomorphisms $\Theta_{*}$
and $\widetilde{\Theta}_{*}$ the free coproduct $\vee$ can be expressed by the
tensor product of the map $d_{*}$ that is induced by the diagonal map $d:G\to
G\times G$ and the based coproduct $\vee_{\Omega}$ up to a sign-correction.
More precisely, for all $i\in\mathbb{N}_{0}$ the following diagram commutes

${\bigoplus_{k+j=i}\mathrm{H}_{k}(G)\otimes\mathrm{H}_{j}(\Omega,e)}$${\mathrm{H}_{i}(\Lambda,G)}$${\bigoplus_{k+j=i}\mathrm{H}_{k}(G^{2})\otimes\mathrm{H}_{j+1-n}(\Omega^{2},\Omega\times
e\cup e\times\Omega)}$${\mathrm{H}_{i+1-n}(\Lambda^{2},\Lambda\times G\cup
G\times\Lambda)}$$\scriptstyle{(-1)^{ni}d_{*}\otimes\vee_{\Omega}}$$\scriptstyle{\Theta_{*}}$$\scriptstyle{\vee}$$\scriptstyle{\widetilde{\Theta}_{*}}$

###### Proof.

For the proof, we fix a left-invariant metric on $G$. Let $k,j\in\mathbb{N}$
and put $l=k+j+1-n$. Let $\mathrm{pr}:G\times\Omega\times I\to\Omega\times I$
be the projection onto the last two factors.

We begin by proving that the following diagram commutes

${\mathrm{H}_{k}(G)\otimes\mathrm{H}_{j}(\Omega,e)}$${\mathrm{H}_{k+j}(\Lambda,G)}$${\mathrm{H}_{k+j}(G\times\Omega,G\times
e)}$${\mathrm{H}_{k+j}(\Lambda,G)}$${\mathrm{H}_{k+j+1}(G\times\Omega\times
I,G\times(e\times I\cup\Omega\times\partial
I))}$${\mathrm{H}_{k+j+1}(\Lambda\times I,G\times I\cup\Lambda\times\partial
I)}$${\mathrm{H}_{l}(G\times U_{\Omega},G\times(e\times
I\cup\Omega\times\partial I))}$${\mathrm{H}_{l}(U_{\Lambda},G\times
I\cup\Lambda\times\partial I)}$${\mathrm{H}_{l}(G\times
F_{\Omega},G\times(e\times I\cup\Omega\times\partial
I))}$${\mathrm{H}_{l}(F_{\Lambda},G\times I\cup\Lambda\times\partial
I)}$${\mathrm{H}_{l}(G^{2}\times\Omega^{2},G^{2}\times(\Omega\times e\cup
e\times\Omega))}$${\mathrm{H}_{l}(\Lambda^{2},\Lambda\times G\cup
G\times\Lambda)}$$\scriptstyle{\times}$$\scriptstyle{\Theta_{*}}$$\scriptstyle{=}$$\scriptstyle{\times[I]}$$\scriptstyle{\Phi_{*}}$$\scriptstyle{\times[I]}$$\scriptstyle{(\Phi,\operatorname{id}_{I})_{*}}$$\scriptstyle{\mathrm{pr}^{*}\tau_{\Omega}\cap}$$\scriptstyle{\tau_{\Lambda}\cap}$$\scriptstyle{(\Phi,\operatorname{id}_{I})_{*}}$$\scriptstyle{(\operatorname{id}_{G},\mathrm{R}_{GH})_{*}}$$\scriptstyle{(\mathrm{R}_{GH})_{*}}$$\scriptstyle{(\Phi,\operatorname{id}_{I})_{*}}$$\scriptstyle{(d,\mathrm{cut})_{*}}$$\scriptstyle{\mathrm{cut}_{*}}$$\scriptstyle{\widetilde{\Phi}_{*}}$

The commutativity of the first two squares is clear. Consider the third square

${\mathrm{H}_{k+j+1}(G\times\Omega\times I,G\times(e\times
I\cup\Omega\times\partial I))}$${\mathrm{H}_{k+j+1}(\Lambda\times I,G\times
I\cup\Lambda\times\partial I)}$${\mathrm{H}_{l}(G\times
U_{\Omega},G\times(e\times I\cup\Omega\times\partial
I))}$${\mathrm{H}_{l}(U_{\Lambda},G\times I\cup\Lambda\times\partial
I)}$$\scriptstyle{(\Phi,\operatorname{id}_{I})_{*}}$$\scriptstyle{\mathrm{pr}^{*}\tau_{\Omega}\cap}$$\scriptstyle{\tau_{\Lambda}\cap}$$\scriptstyle{(\Phi,\operatorname{id}_{I})_{*}}$

To prove that this diagram commutes, let
$X\in\mathrm{H}_{\bullet}(G\times\Omega\times I,G\times(e\times
I\cup\Omega\times\partial I))$. We need to show that

$\tau_{\Lambda}\cap((\Phi,\operatorname{id}_{I})_{*}X)=(\Phi,\operatorname{id}_{I})_{*}(\mathrm{pr}^{*}\tau_{\Omega}\cap
X).$ (3.3)

By naturality, the left hand side of this equation is

$\tau_{\Lambda}\cap((\Phi,\operatorname{id}_{I})_{*}X)=(\Phi,\operatorname{id}_{I})_{*}((\Phi,\operatorname{id}_{I})^{*}\tau_{\Lambda}\cap
X)$ (3.4)

so it suffices to show that
$\mathrm{pr}^{*}\tau_{\Omega}=(\Phi,\operatorname{id}_{I})^{*}\tau_{\Lambda}$.

Consider the tubular neighborhood

$U_{G}=\\{(p,q)\in G\times G\,|\,\mathrm{d}(p,q)<\epsilon\\}$

where $\mathrm{d}$ is the distance-function induced by the chosen Riemannian
metric. The tubular neighborhood $U_{G}$ is globally trivial in the sense that
the map $\chi\colon U_{G}\to G\times B_{e}$ given by

$\chi(g,h)=(g,g^{-1}h)\qquad\text{for}\qquad(g,h)\in U_{G}$

is a homeomorphism. Here we use that the metric on $G$ is left-invariant,
therefore also the distance function is $G$-invariant.

It follows that the Thom class
$\tau_{G}\in\mathrm{H}^{n}(U_{G},U_{G,\geq\epsilon_{0}})$ can be written as

$\tau_{G}=\chi^{*}(1_{G}\times\tau_{e})$

where $\tau_{e}\in\mathrm{H}^{n}(B_{e},B_{e,\geq\epsilon_{0}})$ is the class
defined for the based coproduct as in Lemma 2.1. Using this identity, we see
that

$(\Phi,\operatorname{id}_{I})^{*}\tau_{\Lambda}=[\chi\circ\mathrm{ev}_{\Lambda}\circ(\Phi,\operatorname{id}_{I})]^{*}(1_{G}\times\tau_{e}).$

It furthermore holds that

$(\chi\circ\mathrm{ev}_{\Lambda}\circ(\Phi,\operatorname{id}_{I}))(g,\gamma,s)=(g,\gamma(s))=(\operatorname{id}_{G},\mathrm{ev}_{\Omega})(g,\gamma,s)$

for $(g,\gamma,s)\in G\times\Omega\times I$ . Consequently,

$(\Phi,\operatorname{id}_{I})^{*}\tau_{\Lambda}=(\operatorname{id}_{G},\mathrm{ev}_{\Omega})^{*}(1_{G}\times\tau_{e})=1_{G}\times\mathrm{ev}_{\Omega}^{*}\tau_{e}=\mathrm{pr}^{*}\tau_{\Omega}.$

Consequently, we see from equations (3.3) and (3.4) that the third square
commutes.

For the fourth square

${\mathrm{H}_{l}(G\times U_{\Omega},G\times(e\times I\cup\Omega\times\partial
I))}$${\mathrm{H}_{l}(U_{\Lambda},M\times I\cup\Lambda\times\partial
I)}$${\mathrm{H}_{l}(G\times F_{\Omega},G\times(e\times
I\cup\Omega\times\partial I))}$${\mathrm{H}_{l}(F_{\Lambda},M\times
I\cup\Lambda\times\partial
I)}$$\scriptstyle{(\Phi,\operatorname{id}_{I})_{*}}$$\scriptstyle{(\operatorname{id}_{G},\mathrm{R}_{GH})_{*}}$$\scriptstyle{(\mathrm{R}_{GH})_{*}}$$\scriptstyle{(\Phi,\operatorname{id}_{I})_{*}}$

we prove its commutativity by showing that the diagram of maps

${G\times U_{\Omega}}$${U_{\Lambda}}$${G\times
F_{\Omega}}$${F_{\Lambda}}$$\scriptstyle{(\Phi,\operatorname{id}_{I})}$$\scriptstyle{(\operatorname{id}_{G},\mathrm{R}_{GH})}$$\scriptstyle{\mathrm{R}_{GH}}$$\scriptstyle{(\Phi,\operatorname{id}_{I})}$

commutes. Let $(g,\gamma,s)\in G\times U_{\Omega}$, i.e.
$\mathrm{d}(e,\gamma(s))<\epsilon$. Using the shorthand notation
$\gamma_{t}=\gamma(t)$ for $t\in[0,1]$, we have

$\displaystyle((\Phi,\mathrm{id}_{I})\circ(\mathrm{id}_{G},\mathrm{R}_{GH}))(g,\gamma,s)$
$\displaystyle=$
$\displaystyle\big{(}g\cdot\mathrm{concat}_{s}\big{[}(\rho_{s}(\gamma)*\overline{\gamma_{s}\gamma_{0}}),(\overline{\gamma_{0}\gamma_{s}}*g\cdot\mu_{s}(\gamma))\big{]},\,s\big{)}$
$\displaystyle=$
$\displaystyle\big{(}\mathrm{concat}_{s}\big{[}g\cdot\rho_{s}(\gamma)*\overline{(g\gamma_{s})(g\gamma_{0})},\overline{(g\gamma_{0})(g\gamma_{s})}*g\cdot\mu_{s}(\gamma)\big{]},s\big{)}$
$\displaystyle=$
$\displaystyle(\mathrm{R}_{GH}\circ(\Phi,\mathrm{id}_{I}))(g,\gamma,s).$

The second equality holds by our use of a left-invariant metric on $G$ which
implies that the geodesic segment from $g\gamma_{0}$ to $g\gamma_{s}$ can be
expressed as

$\overline{(g\gamma_{0})(g\gamma_{s})}=g\cdot\overline{\gamma_{0}\gamma_{s}}$

since left-multiplication by $g\in G$ is an isometry. This proves that the
fourth square commutes.

The commutativity of the last square can be seen in an analogous manner by
checking that the underlying maps commute. Thus we have shown that the large
diagram commutes.

Call the map that is defined by the left vertical side of the large diagram

$\Xi:\mathrm{H}_{k}(G)\otimes\mathrm{H}_{j}(\Omega,e)\to\mathrm{H}_{l}(G^{2}\times\Omega^{2},G^{2}\times(\Omega\times
e\cup e\times\Omega)).$

In order to complete the proof of the theorem, we need to show that the
following diagram commutes

${\mathrm{H}_{k}(G)\otimes\mathrm{H}_{j}(\Omega,e)}$${\mathrm{H}_{k}(G^{2})\otimes\mathrm{H}_{j+1-n}(\Omega^{2},\Omega\times
e\cup
e\times\Omega)}$${\mathrm{H}_{l}(G^{2}\times\Omega^{2},G^{2}\times(\Omega\times
e\cup
e\times\Omega))}$$\scriptstyle{(-1)^{n(k+j)}d_{*}\otimes\vee_{\Omega}}$$\scriptstyle{\Xi}$$\scriptstyle{\times}$

This is easily seen by unwinding the definitions. Let $X\in\mathrm{H}_{k}(G)$
and $Y\in\mathrm{H}_{j}(\Omega,e)$, then

$\displaystyle\Xi(X\otimes Y)$ $\displaystyle=$
$\displaystyle(d,\mathrm{cut})_{*}(\operatorname{id}_{G},\mathrm{R}_{GH})_{*}(\mathrm{pr}^{*}\tau_{\Omega}\cap(X\times
Y\times[I]))$ $\displaystyle=$
$\displaystyle(d,\mathrm{cut}\circ\mathrm{R}_{GH})_{*}((1_{G}\times\tau_{\Omega})\cap(X\times(Y\times[I])))$
$\displaystyle=$
$\displaystyle(-1)^{nk}(d,\mathrm{cut}\circ\mathrm{R}_{GH})_{*}(X\times(\tau_{\Omega}\cap(Y\times[I])))$
$\displaystyle=$ $\displaystyle(-1)^{nk}(d_{*}X\times\vee_{\Omega}Y).$

Now, if $Y$ is non-trivial, then $j$ is even, hence we see that

$\Xi(X\otimes Y)=(-1)^{n(k+j)}(d_{*}X\times\vee_{\Omega}Y).$

The commutativity of this last diagram combined with the commutativity of the
large diagram at the beginning of the proof show the claim. ∎

We now prove a dual result about the cohomology products introduced in Section
2. We will only deal with cohomology with coefficients in a field
$\mathbb{F}$. Note that since the cohomology of the Lie group is of finite
type, the cross product in cohomology induces an isomorphism

$(\mathrm{H}^{\bullet}(G;\mathbb{F})\otimes\mathrm{H}^{\bullet}(\Omega
G,e;\mathbb{F}))^{i}\cong\mathrm{H}^{i}(G\times\Omega G,G\times e;\mathbb{F})$

see [Spa95, Theorem 5.6.1]. Combining this with the pull-back of the map
$\Psi$ defined in equation (3.1), we get an isomorphism

$\Theta^{*}:(\mathrm{H}^{\bullet}(G;\mathbb{F})\otimes\mathrm{H}^{\bullet}(\Omega,e;\mathbb{F}))^{i}\xrightarrow[]{\times}\mathrm{H}^{i}(G\times\Omega,G\times
e;\mathbb{F})\xrightarrow[]{\Psi^{*}}\mathrm{H}^{i}(\Lambda,G;\mathbb{F}).$

######  Theorem 3.2.

Let $G$ be a compact Lie group. Assume that the homology
$\mathrm{H}_{\bullet}(\Omega G;\mathbb{F})$ is of finite type. The free
cohomology product on $\Lambda G$ can be expressed by the tensor product of
the cup product on the cohomology of $G$ and the based cohomology product on
$\Omega G$. More precisely, if $\alpha\in\mathrm{H}^{i}(\Lambda,G;\mathbb{F})$
and $\beta\in\mathrm{H}^{j}(\Lambda,G;\mathbb{F})$ are of the form

$\alpha=\Psi^{*}(a\times A)\qquad\text{and}\qquad\beta=\Psi^{*}(b\times B)$

where $a,b\in\mathrm{H}^{\bullet}(G;\mathbb{F})$ and
$A,B\in\mathrm{H}^{\bullet}(\Omega,e;\mathbb{F})$, then

$\alpha\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\ast$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\ast$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\ast$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\ast$\cr}}}\beta=(-1)^{n(i+j)}\Psi^{*}[(a\cup
b)\times(A\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\ast$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\ast$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\ast$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\ast$\cr}}}_{\Omega}B)].$

###### Proof.

Since the homology $\mathrm{H}_{\bullet}(\Omega G;\mathbb{F})$ is of finite
type, so is $\mathrm{H}_{\bullet}(\Lambda G;\mathbb{F})$ and thus the
cohomology products are defined. The free cohomology product of $\alpha$ and
$\beta$ is determined by the natural pairing of homology and cohomology, i.e.
we have

$\langle\alpha\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\ast$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\ast$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\ast$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\ast$\cr}}}\beta,X\rangle=\langle\alpha\times\beta,\vee
X\rangle\qquad\text{for
all}\,\,\,X\in\mathrm{H}_{\bullet}(\Lambda,G;\mathbb{F}).$ (3.5)

Since the natural pairing is trivial if the homology class and the cohomology
class have different degrees, it is sufficient to consider homology classes
$X$ of degree $k=i+j+n-1$.

Let $X\in\mathrm{H}_{k}(\Lambda,G;\mathbb{F})$ and assume that the class $X$
is of the form

$X=\Phi_{*}(x\times\xi)\qquad\text{with}\,\,\,\,x\in\mathrm{H}_{\bullet}(G;\mathbb{F}),\,\xi\in\mathrm{H}_{\bullet}(\Omega,e;\mathbb{F}).$
(3.6)

Equation (3.5) then becomes

$\displaystyle\langle\alpha\times\beta,\vee X\rangle$ $\displaystyle=$
$\displaystyle\langle\Psi^{*}(a\times A)\times\Psi^{*}(b\times
B),(-1)^{nk}\widetilde{\Phi}_{*}(d_{*}x\times\vee_{\Omega}\xi)\,\rangle$ (3.7)
$\displaystyle=$
$\displaystyle(-1)^{nk}\langle\widetilde{\Phi}^{*}(\Psi,\Psi)^{*}(a\times
A\times b\times B),d_{*}x\times\vee_{\Omega}\xi\rangle$ $\displaystyle=$
$\displaystyle(-1)^{nk}\langle((\Psi,\Psi)\circ\widetilde{\Phi})^{*}(a\times
A\times b\times B),d_{*}x\times\vee_{\Omega}\xi\rangle$

where $\widetilde{\Phi}$ was defined in equation (3.2) and where we used
Theorem 3.1 in the first equality.

A direct computation shows that

$((\Psi,\Psi)\circ\widetilde{\Phi})(g_{1},g_{2},\gamma_{1},\gamma_{2})=(g_{1},\gamma_{1},g_{2},\gamma_{2})\qquad\text{for}\,\,\,g_{1},g_{2}\in
G,\,\gamma_{1},\gamma_{2}\in\Omega.$

Therefore, $(\Psi,\Psi)\circ\widetilde{\Phi}$ is just the swapping map

$G\times G\times\Omega\times\Omega\to G\times\Omega\times G\times\Omega$

interchanging the second and the third factor. By the standard properties of
the cross product we obtain

$((\Psi,\Psi)\circ\widetilde{\Phi})^{*}(a\times A\times b\times
B)=(-1)^{|A||b|}a\times b\times A\times B$

where $|\cdot|$ is the degree of a cohomology class. As for homology, the
cohomology of the based loop space is only non-trivial in even degrees, so if
the class $A$ is non-trivial, the sign $(-1)^{|A||b|}$ is always equal to $1$.
Going back to equation (3.7), we see that

$\displaystyle\langle\alpha\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\ast$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\ast$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\ast$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\ast$\cr}}}\beta,X\rangle$
$\displaystyle=$ $\displaystyle(-1)^{nk}\langle a\times b\times A\times
B,d_{*}x\times\vee_{\Omega}\xi\rangle$ $\displaystyle=$
$\displaystyle(-1)^{nk}\langle a\times b,d_{*}x\rangle\langle A\times
B,\vee_{\Omega}\xi\rangle$ $\displaystyle=$
$\displaystyle(-1)^{nk}\langle(a\cup
b)\times(A\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\ast$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\ast$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\ast$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\ast$\cr}}}_{\Omega}B),x\times\xi\rangle$
$\displaystyle=$ $\displaystyle(-1)^{nk}\langle\Psi^{*}((a\cup
b)\times(A\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\ast$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\ast$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\ast$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\ast$\cr}}}_{\Omega}B)),X\rangle$

where we used the definitions of the cup product and of the based cohomology
product, respectively. For the sign, we observe that

$(-1)^{nk}=(-1)^{n(i+j)}(-1)^{n(n-1)}=(-1)^{n(i+j)}.$

This shows that

$\langle\alpha\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\ast$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\ast$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\ast$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\ast$\cr}}}\beta,X\rangle=(-1)^{n(i+j)}\langle\Psi^{*}((a\cup
b)\times(A\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\ast$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\ast$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\ast$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\ast$\cr}}}_{\Omega}B)),X\rangle$
(3.11)

for all classes $X$ that can be written as in equation (3.6). But since these
classes span all of $\mathrm{H}_{k}(\Lambda,G;\mathbb{F})$, it follows that
equation (3.11) holds for all $X\in\mathrm{H}_{k}(\Lambda,G;\mathbb{F})$. By
the non-degeneracy of the natural pairing this completes the proof of the
theorem. ∎

##  4\. Triviality of the Coproduct for Even-Dimensional Lie Groups

We are now going to show that the coproduct is trivial for large classes of
Lie groups. In this section, we will consider compact, even-dimensional Lie
groups, where the triviality follows directly from Theorem 3.1.

######  Theorem 4.1.

Let $G$ be an even-dimensional compact Lie group. Consider homology with
coefficients in a commutative ring $R$. Then the based coproduct
$\vee_{\Omega}$ and the free coproduct $\vee$ are trivial.

###### Proof.

As we have mentioned earlier, the relative homology of the based loop space
$\mathrm{H}_{\bullet}(\Omega G,e)$ is free and concentrated in even degrees
(see [Bot56]). Consequently, by the Künneth isomorphism the same properties
hold for the relative homology $\mathrm{H}_{\bullet}(\Omega G\times\Omega
G,\Omega G\times e\cup e\times\Omega G)$.

The based coproduct is a map

$\vee_{\Omega}:\mathrm{H}_{i}(\Omega G,e)\to\mathrm{H}_{i+1-n}(\Omega
G\times\Omega G,\Omega G\times e\cup e\times\Omega G)$

with degree shift $1-n$. Since $n$ is even, this degree shift is odd. Consider
now the case where $i$ is even. Then the degree $i+1-n$ is odd, so
$\mathrm{H}_{i+1-n}(\Omega G^{2},\Omega G\times e\cup e\times\Omega G)$ is
trivial. Hence, for even $i$, the based coproduct must be trivial. In the
other case, one can see similarly that the based coproduct must be trivial as
well. This shows that the based coproduct must be trivial.

By Theorem 3.1, the free coproduct $\vee$ on $\mathrm{H}_{\bullet}(\Lambda,G)$
can be expressed as

$d_{*}\otimes\vee_{\Omega}:(\mathrm{H}_{\bullet}(G)\otimes\mathrm{H}_{\bullet}(\Omega
G,e))_{i}\to(\mathrm{H}_{\bullet}(G^{2})\otimes\mathrm{H}_{\bullet}(\Omega
G^{2},\Omega G\times e\cup e\times\Omega G))_{i+1-n}$ (4.1)

under the isomorphisms described in Theorem 3.1. We can drop the sign that
appears in Theorem 3.1, since the dimension $n$ of $G$ is even. As we have
already shown, the based coproduct $\vee_{\Omega}$ is trivial in this
situation. This implies that the map in equation (4.1) is trivial and by
Theorem 3.1 this implies the triviality of the free coproduct. ∎

######  Corollary 4.2.

Let $G$ be a compact even-dimensional Lie group and let $\mathbb{F}$ be a
field. Assume that $\mathrm{H}_{\bullet}(\Omega G;\mathbb{F})$ is of finite
type. Then the based cohomology product and the free cohomology product of $G$
are trivial.

###### Proof.

Since $\mathrm{H}_{\bullet}(\Omega G;\mathbb{F})$ is of finite type, so is
$\mathrm{H}_{\bullet}(\Lambda G;\mathbb{F})$. Hence, we can use Theorem 4.1 to
see that the cohomology products must vanish. ∎

##  5\. Triviality of the Coproduct for certain Simply Connected Lie Groups

We now turn to compact, simply connected Lie groups of rank $r\geq 2$. There
are lots of classical results about the topology of the based loop space of a
compact Lie group by Bott (see [Bot56] and [Bot58]) and by Bott and Samelson
(see [BS58]). We will use an explicit description of cycles in the based loop
space. Throughout this section, we consider homology with integer
coefficients.

Before, we investigate these cycles, let us define the intersection
multiplicity of a homology class. The following definition is given as in
[HW17, Definition 5.1].

######  Definition 5.1.

Let $M$ be a Riemannian manifold. Let $[X]\in\mathrm{H}_{\bullet}(\Lambda
M,M)$ be a non-trivial homology class with representing cycle
$X\in\mathrm{C}_{\bullet}(\Lambda M,M)$. Assume that the relative cycle $X$ is
represented by a cycle $x\in\mathrm{C}_{\bullet}(\Lambda M)$. The basepoint
intersection multiplicity $\mathrm{int}([X])$ of the class $[X]$ is the number

$\mathrm{int}([X])=\inf_{A\sim
x}\Big{(}\sup\big{[}\\#(\gamma^{-1}(\\{\gamma(0)\\}))\,|\,\gamma\in\mathrm{Im}(A),\,\,\,\mathcal{L}(\gamma)>0\big{]}\Big{)}-1$

where the infimum is taken over all cycles $A\in\mathrm{C}_{\bullet}(\Lambda
M)$ homologous to $x$.

Note that this definition does not depend on the choice of the cycle $x$,
since if $x^{\prime}\in\mathrm{C}_{\bullet}(\Lambda M)$ is another cycle
representing the relative cycle $X$, then $x$ and $x^{\prime}$ differ only by
elements of $\mathrm{C}_{\bullet}(M)$. However, in the above definition we do
not consider trivial loops, so the basepoint intersection multiplicity is
well-defined.

Also note that the correction of $-1$ comes from the fact that we consider
loops as maps $\gamma:[0,1]\to X$ with $\gamma(0)=\gamma(1)$. Hence, for any
loop, we would count the basepoint twice if we did not perform this
correction. By using finite-dimensional approximations of $\Lambda M$, one can
furthermore see that the number $\mathrm{int}([X])$ is always finite (see
[HW17, p. 44]).

Hingston and Wahl show the following result (see [HW17, Theorem 3.10 (C)]).

######  Proposition 5.2.

Let $M$ be an oriented, closed manifold. If $X\in\mathrm{H}_{i}(\Lambda M,M)$
is a non-trivial homology class with basepoint intersection multiplicity
$\mathrm{int}(X)=1$, then the free coproduct $\vee X$ vanishes.

From now on, let $G$ be a compact, simply connected Lie group of rank $r\geq
2$. In particular, $G$ is then semisimple (see [BtD95, Section V.7]). We shall
now describe some explicit cycles in $\Omega G$, closely following the
exposition in [Bot58, Section 3].

Denote the Lie algebra of $G$ by $\mathfrak{g}$. Let $T\subseteq G$ be a
maximal torus, which implies $\mathrm{dim}(T)=r$. Furthermore, let
$\mathfrak{g}_{\mathbb{C}}$ be the complexification of the Lie algebra
$\mathfrak{g}$. Then the Lie algebra $\mathfrak{t}$ of the maximal torus $T$
is a maximal abelian subalgebra of $\mathfrak{g}$ and its complexification
$\mathfrak{h}=\mathfrak{t}\oplus i\mathfrak{t}$ is a Cartan subalgebra of
$\mathfrak{g}_{\mathbb{C}}$.

A root of the complex semisimple Lie algebra $\mathfrak{g}_{\mathbb{C}}$ is an
element $\alpha$ of the dual space $\mathfrak{h}^{*}$ such that there is a
non-trivial subspace $\mathfrak{g}_{\alpha}\subseteq\mathfrak{g}_{\mathbb{C}}$
with

$[H,X]=\alpha(H)X\qquad\text{for
all}\,\,\,H\in\mathfrak{h},\,\,X\in\mathfrak{g}_{\alpha}.$

Denote the set of non-zero roots by $\Delta$, then one obtains a decomposition

$\mathfrak{g}_{\mathbb{C}}=\mathfrak{h}\oplus\bigoplus_{\alpha\in\Delta}\mathfrak{g}_{\alpha}.$

If a non-zero root $\alpha\in\Delta$ is restricted to
$\mathfrak{t}\subseteq\mathfrak{h}$, then its values are purely imaginary.

######  Definition 5.3.

Let $\alpha\in\Delta$ be a non-zero root and $n\in\mathbb{Z}$. The affine
plane

$\\{H\in\mathfrak{t}\,|\,\alpha(H)=2\pi in\\}\subseteq\mathfrak{t}$

is called the singular plane $(\alpha,n)$.

If $p=(\alpha,n)$ is a singular plane, we write $\overline{p}$ for its image
$\exp(p)\subseteq G$ under the Lie group exponential of $G$. Let $G(p)$ be the
centralizer of $\overline{p}$, i.e. we have

$G(p)=\\{g\in G\,|\,gh=hg\,\,\text{for all}\,\,h\in\overline{p}\\}.$

The group $G(p)$ is a closed subgroup of $G$.

Clearly, the torus $T$ is itself always a closed subgroup of $G(p)$ and one
can show that for all singular planes $p$ the dimension of $G(p)$ is strictly
larger than $r$ (see e.g. [Hel01, Lemma VII.4.5]).

Let $P=(p_{1},p_{2},\ldots,p_{m})$ be an ordered family of singular planes,
where $m\in\mathbb{N}$. Set

$W(P)=G(p_{1})\times G(p_{2})\times\ldots\times G(p_{m}).$

Following [Bot58], there is a right-action $\chi$ of the $m$-fold product of
the maximal torus $T$ on $W(P)$ given by

$\displaystyle\chi:W(P)\times T^{m}$ $\displaystyle\to$ $\displaystyle W(P)$
$\displaystyle((k_{1},\ldots,k_{m}),(t_{1},\ldots,t_{m}))$
$\displaystyle\mapsto$
$\displaystyle(k_{1}t_{1},t_{1}^{-1}k_{2}t_{2},\ldots,t_{m-1}^{-1}k_{m}t_{m}).$

Note that $T^{m}$ is a closed subgroup of $W(P)$ but the action above is not
the one induced by the group multiplication in $W(P)$. One can check that the
action $\chi$ is proper and free and consequently the quotient

$\Gamma_{P}=W(P)/T^{m}$

is a manifold. Bott and Samelson show in [BS58] that all $\Gamma_{P}$ are
connected and oriented.

We now construct a continuous map $f_{P}:\Gamma_{P}\to\Omega G$. Let
$c=(c_{0},c_{1},\ldots,c_{m})$ be an ordered sequence of polygons in
$\mathfrak{t}$ with the following property:

  * •

The polygon $c_{0}$ starts at the origin $0\in\mathfrak{t}$.

  * •

For $i=1,\ldots,m$ the endpoint of the polygon $c_{i-1}$ is the start point of
the polygon $c_{i}$ and lies on the singular plane $p_{i}$.

  * •

The polygon $c_{m}$ ends at the origin $0\in\mathfrak{t}$.

For all $i=0,1,\ldots,m$, we can parametrize the polygon $c_{i}$ on the
interval $[\frac{i}{m+1},\frac{i+1}{m+1}]$. Then we define a map
$\widetilde{f}_{P}:W(P)\to\Omega G$ by setting

$\widetilde{f}_{P}(g_{1},g_{2},\ldots,g_{m})=\mathrm{concat}\Big{[}\exp(c_{0}),g_{1}\exp(c_{1})g_{1}^{-1},\ldots,g_{1}\ldots
g_{m}\exp(c_{m})g_{m}^{-1}\ldots g_{1}^{-1}\Big{]}.$

By the choice of the polygons, this is a well-defined continuous map. One
checks that $\widetilde{f}_{P}$ is invariant under the right-action of $T^{m}$
by $\chi$, so it descends to a map

$f_{P}:\Gamma_{P}\to\Omega G.$

If we choose an orientation class
$[\Gamma_{P}]\in\mathrm{H}_{\bullet}(\Gamma_{P})$ of the manifold
$\Gamma_{P}$, we obtain a homology class

$P_{*}=(f_{P})_{*}[\Gamma_{P}]\in\mathrm{H}_{\bullet}(\Omega G).$

Bott and Samelson [BS58] have shown that the set

$\\{P_{*}\in\mathrm{H}_{\bullet}(\Omega
G)\,|\,P=(p_{1},\ldots,p_{m})\,\,\,\text{is an ordered family of singular
planes},\,\,\,m\geq 1\\}$ (5.1)

together with a generator of $\mathrm{H}_{0}(\Omega G)$ generates the homology
of $\Omega G$ if $G$ is simply connected. Hence, the image in
$\mathrm{H}_{\bullet}(\Omega G,e)$ of the set in equation (5.1) generates the
homology $\mathrm{H}_{\bullet}(\Omega G,e)$.

Note that the map $f_{P}$ depends on the choice of the polygons
$(c_{0},\ldots,c_{m})$. However, assume that we make a different choice, i.e.
we choose an ordered sequence of polygons $(d_{0},\ldots,d_{m})$ that satisfy
the same conditions with respect to the family of singular planes $P$. Then
the map $\widehat{f}_{P}$ which is induced by the polygons
$(d_{0},\ldots,d_{m})$ is homotopic to the map $f_{P}$, since we can
continuously deform each $d_{i}$ into $c_{i}$ and thus obtain the desired
homotopy from $\widehat{f}_{P}$ to $f_{P}$.

Let $\mathcal{F}$ be the lattice

$\mathcal{F}=\\{H\in\mathfrak{t}\,|\,\exp(H)=e\\}$

where $\exp$ is the Lie group exponential of $G$ and $e\in G$ is the unit
element. Recall that the dimension of $\mathfrak{t}$ is equal to the rank $r$.

######  Definition 5.4.

Let $P=(p_{1},\ldots,p_{m})$ be an ordered family of singular planes. We say
that an ordered family of polygons $(c_{0},c_{1},\ldots,c_{m})$ is lattice-
nonintersecting if the following holds: No polygon $c_{i}$ intersects the
lattice $\mathcal{F}$ apart from $c_{0}$ at its start point and $c_{m}$ at its
endpoint.

In case the rank $r$ of the Lie group $g$ satisfies $r\geq 2$, it is clear
that given an ordered family of singular planes $P=(p_{1},\ldots,p_{m})$ we
can choose the polygons $(c_{0},c_{1},\ldots,c_{m})$ to satisfy the conditions
with respect to the endpoints of the polygons and to be lattice-
nonintersecting at the same time. This choice is not possible however if the
rank is $r=1$.

######  Proposition 5.5.

Let $G$ be a compact, simply connected Lie group of rank $r\geq 2$. Then every
homology class in $\mathrm{H}_{\bullet}(\Lambda G,G)$ has basepoint
intersection multiplicity $1$.

###### Proof.

By the arguments from Section 3 the homology of the free loop space of $G$
relative to the constant loops is isomorphic to the tensor product of the
homology of $G$ and of the homology of the based loop space of $G$ relative to
the basepoint via the maps

$(\mathrm{H}_{\bullet}(G)\otimes\mathrm{H}_{\bullet}(\Omega
G,e))_{i}\xrightarrow[]{\times}\mathrm{H}_{i}(G\times\Omega G,G\times
e)\xrightarrow[]{\Phi_{*}}\mathrm{H}_{i}(\Lambda G,G).$

Choose classes $[x_{1}],\ldots,[x_{l}]\in\mathrm{H}_{\bullet}(G)$ that
generate the homology of $G$. Then by the considerations before this
proposition and the above isomorphism, it is clear that the set

$\\{\Phi_{*}([x_{i}]\times
P_{*})\,|\,i\in\\{1,\ldots,l\\},\,\,\,P\,\,\text{non-trivial ordered family of
singular planes}\\}$

generates the homology of $\Lambda G$ relative to the trivial loops $G$.

Let $i\in\\{1,2,\ldots,l\\}$ and let $P$ be a non-trivial ordered family of
singular planes. We want to determine the intersection multiplicity of the
homology class $\Phi_{*}([x_{i}]\times P_{*})$. Choose an arbitrary
representative $x_{i}$ of $[x_{i}]$ and a representative $\mathcal{P}$ of
$P_{*}$ that is obtained via a lattice-nonintersecting polygon. This is
possible, since the rank $r$ is greater than or equal to $2$.

A loop $\gamma\in\mathrm{Im}(\Phi_{*}(x_{i}\times\mathcal{P}))$ is of the form

$\gamma=k\cdot\mathrm{concat}\Big{[}\exp(c_{0}),g_{1}\exp(c_{1})g_{1}^{-1},\ldots,g_{1}\ldots
g_{m}\exp(c_{m})g_{m}^{-1}\ldots g_{1}^{-1}\Big{]}.$

Clearly, we have $\gamma(0)=\gamma(1)=k$. Assume that there is an $s\in(0,1)$
with $\gamma(s)=k$. Then there is an $i\in\\{0,\ldots,m\\}$ with
$\exp(c_{i}(s))=e$ which implies that $c_{i}(s)\in\mathcal{F}$. But by the
choice of a lattice-nonintersecting polygon, this is a contradiction to
$s\in(0,1)$ which proves that $\\#(\gamma^{-1}(k))=2$. We conclude that

$\mathrm{int}(\Phi_{*}([x_{i}]\times P_{*}))=1.$

Since the homology of the free loop space $\Lambda G$ relative to the constant
loops $G$ is generated by homology classes of the form that we considered,
this shows that every class in $\mathrm{H}_{\bullet}(\Lambda G,G)$ has
basepoint intersection multiplicity $1$. ∎

######  Theorem 5.6.

Let $G$ be a simply connected, compact Lie group of rank $r\geq 2$. Then the
free coproduct $\vee$ is trivial.

###### Proof.

This follows from Propositions 5.2 and 5.5. ∎

######  Remark 5.7.

  1. (1)

The condition on the rank in Theorem 5.6 cannot be given up. The $3$-sphere
$\mathbb{S}^{3}$ can be seen as the Lie group $\mathrm{SU}(2)$ which is a
compact, simply connected Lie group of rank $1$. The free coproduct on
$\mathbb{S}^{3}$ is non-trivial (see [HW17, Proposition 3.17] ).

  2. (2)

Classical examples of Lie groups to which Theorem 5.6 applies are the special
unitary groups $\mathrm{SU}(n)$ for $n\geq 3$, the Spin groups
$\mathrm{Spin}(n)$ for $n\geq 4$ and the compact symplectic groups
$\mathrm{Sp}(n)$ for $n\geq 2$.

### Acknowldegements

The author wants to thank Philippe Kupper and Stephan Mescher for many helpful
comments on this manuscript.

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