# Topological invariants of some chemical reaction networks

Jack Morava  Department of Mathematics, The Johns Hopkins University,
Baltimore, Maryland 21218<EMAIL_ADDRESS>
(Date: June 2020)

###### Abstract.

Certain toric dynamical systems [15, 21, 22, 25] studied in physical chemistry
have associated toric varieties which, when smooth, represent elements in the
homotopy groups $M\xi_{*}B{\mathbb{T}}$ of a symplectic variant of the
$A_{\infty}$ Baker-Richter spectrum $M\xi$ [9, 32]. The noncommutative Hopf
algebroid $(M\xi_{*},M\xi_{*}M\xi)$ has interesting connections to the group
of formal diffeomorphism of the noncommutative line, conjecturally defining a
noncommmutative analog of the Helmholtz free energy for systems of complex
symplectic (completely integrable) Hamiltonian toric manifolds.

Introduction and organization

In 2006 Baker and Richter [8, 9] defined an $A_{\infty}$ Thom spectrum $M\xi$
associated to the loop space $\Omega\Sigma{\mathbb{C}}P^{\infty}$, with close
relations to quasi- and noncommutative symmetric functions, and the remarkable
property that it splits $p$-locally as a wedge of copies of $BP$, even though
$M\xi$, being noncommutative, is not an $M{\rm U}$-algebra. Building on this
work a few years later, N Kitchloo and the author [32] sketched a theory of
$H_{*}M\xi$-valued characteristic numbers for the complex quasi-toric
manifolds defined by Davis and Januszkiewicz [16].

This paper applies these ideas to work of Guillemin, Ginzburg, and Karshon [1]
on cobordism of Hamiltonian group actions; in particular, to certain complex
symplectic toric manifolds related to classical completely integrable systems.
Smooth Hamiltonian toric manifolds associated to chemical reaction networks
[15, 21, 22, 25] provide a significant class of examples.

Part I of this paper begins with sketches (§1.1, 1.2) of background material
from geometry and mechanics, using the language of algebraic topology; the
recent monographs of Guillemin et al, and of Buchstaber and Panov [2] on toric
geometry, make this practical. A summary of some modern work on chemical
reaction networks follows, and the final section reviews characteristic
classes and numbers from non-equivariant cobordism theory. A brief account of
the motivation behind this paper is postponed to an afterword.

Part II is concerned more directly with the spectrum $M\xi$ and its associated
noncommutative Hopf algebroid $(M\xi_{*},M\xi_{*}M\xi)\otimes{\mathbb{Q}}$ of
rational cooperations, conjecturally closely related to a noncommutative Hopf
algebra of formal diffeomorphisms of the noncommutative line [10]. Its
connections with statistical mechanics [30, 36, 37] and free probability
theory [19, 20, 26, 29, 41] suggest a candidate for the Helmholtz free energy
for systems of symplectic Hamiltonian toric manifolds.

Apology to the reader: This paper takes its target audience to be generally
familiar with modern algebraic topology (such as cobordism theory and
equivariant language), with an interest in applications. Some knowledge of
toric geometry, classical mechanics, and combinatorics would also be useful
but §1.1-2 try to provide enough of an Ariadne’s thread through current
literature to make the rest of the paper accessible. The results of the paper
involve the algebraic theory of chemical reaction networks and quasi-symmetric
functions, neither of which is assumed to be known; §1.3 and 2.1 try to sketch
enough of these topics for the reader to see why the proposed applications may
be interesting.

Acknowledgements The author has profited from conversations and correspondence
with many helpful colleagues over the years, and would especially like to
thank Andrew Baker, Birgit Richter, Nitu Kitchloo, Michiel Hazewinkel, John
McKay, Roland Friedrich, Masoud Khalkali, and Jeremy Gunawardena for their
interest and help; but of course the excesses and misunderstandings in this
paper are his own responsibiity. He moreover thanks the editors and referees
for their insightful comments, recommendations, and patience.

§I Background

1.1 Some toric geometry

1.1.1 This paper is concerned with applications of the theory of Hamiltonian
$G$-cobordism developed in [1], specialized to the context of Delzant-like
half-dimensional toric manifolds. The general theory of Guillemin, Ginzburg,
and Karshon is in some sense naturally bigraded, by symmetry group as well as
dimension: the product of an $n$-dimensional $G$-manifold with an
$m$-dimensional $H$-manifold is most naturally an $n+m$-dimensional $G\times
H$ manifold. Here our focus will be a neighborhood of the diagonal of this
bigraded category.

We will be interested in compact smooth ‘${\sf T}$-manifolds’ $X$ endowed with
an effective111i.e. $(\forall x\in X)\;{\rm Lie\;}T\to{\rm End}(T_{x}X)$ is
injective; we will also fix an isomorphism of Lie $T$ with
$\pi_{1}(T)\otimes{\mathbb{R}}.$ We write
${\mathbb{T}}={\mathbb{R}}/{\mathbb{Z}}$, $T$ for a generic torus, and
$T^{k}\cong{\mathbb{T}}^{k}$ for a specific torus. smooth action of a compact
torus $T$, such that $\dim T=\lfloor{\textstyle{\frac{1}{2}}}\dim X\rfloor$;
thus the toruses acting on ${\sf T}$-manifolds $(X_{0},T_{0}),\;(X_{1},T_{1})$
will be distinct. The boundary $\partial X$ of an odd-dimensional ${\sf
T}$-manifold is again a ${\sf T}$-manifold, and if one or both of a pair of
${\sf T}$-manifolds as above is even-dimensional, then $M_{0}\times M_{1}$ is
a ${\sf T}$-manifold with an action of $T_{0}\times T_{1}$ and boundary
$\partial M_{0}\times M_{1}\coprod M_{0}\times\partial M_{1}$. The class of
${\sf T}$-manifolds is thus a cobordism category with products, graded by
dimension: an object is a smooth manifold with an action by a torus of half
its dimension, much like the spaces classified by Delzant [1](§1.3 p 17, §1.5
p 29), while odd-dimensional ${\sf T}$-manifolds will be the morphisms of the
category. This can be elaborated by specifying various orientations, but
terminology for structures (symplectic, complex-oriented …) on manifolds with
group actions can become confusing. Our convention is that a
$\mathcal{Z}$-structure on a ${\sf T}$-manifold will always mean a
$T$-equivariant $\mathcal{Z}$-structure.

To define the orientations relevant below, we work with $2n$-dimensional ${\sf
T}$-manifolds with an underlying Davis-Januszkiewicz quasi-toric structure,
i.e. such that the quotient

$M\to M/T=P$

is homeomorphic to a simple (i.e. with $n$ facets meeting at each vertex
[16](§1.13),[2](1.1 p 4, 1.7 p 46) polyhedron, with boundary dual to a
simplicial polytope $K$ with a set $[m]$ of vertices. The map which assigns to
a facet of $P$, its isotropy subgroup, lifts (nonuniquely) to an
omniorientation: a surjective homomorphism

$\Lambda:{\mathbb{Z}}^{m}\to{\mathbb{Z}}^{n}\cong\pi_{1}(T)$

[2](§7.3.9 p 247) such that the minor $\Lambda_{\\{i_{v}\\}}$ associated to
the columns $\\{i_{v}\\}$ indexed by the facets meeting at the vertex $v$ has
determinant $\pm 1$. Such an omniorientation defines, and is defined uniquely
by, an equivariant complex orientation [43], i.e. a stable complex structure
[2](B.6.1 p 434] with an associated equivariant decomposition

$T_{M}\oplus{\mathbb{C}}^{m-n}\cong\oplus_{1\leq i\leq m}L_{i}$

of the tangent bundle of $M$ as a sum of complex line bundles; see [7], and
§1.2.3 below.

1.1.2 Definition An (integral) Hamiltonian (complex symplectic) orientation on
$M$ is defined by

1) an equivariant complex line bundle ${\mathcal{L}}$ on $M$, with
characteristic class $c^{T}_{1}({\mathcal{L}})\in H^{2}_{T}(M;{\mathbb{Z}})$
[1](C.47 p 214), together with a lift of $c^{T}_{1}({\mathcal{L}})$ to

2) an equivariant symplectic form $\omega\in\Omega^{2}_{T}(M)$, i.e.
$d\omega=0$ with $\omega^{n}$ everywhere nonzero, together with an
equivariantly closed 2-form ${\bar{\omega}}=\omega-\Phi$ such that
$(2\pi)^{-1}[{\bar{\omega}}]$ is the image in $H^{2}_{T}(M;{\mathbb{R}})$ of
$c^{T}_{1}({\mathcal{L}})$. Then

3) $\Phi:M\to({\rm Lie\;}T)^{*}$ is a moment map for the action of $T$ on $M$
[1](§2.15 p 25, E.37 p 266) identifying $P$ with a convex polytope in $({\rm
Lie\;}T)^{*}$; moreover, there is a homotopically unique almost complex
structure on $M$ compatible with $\omega$, defining a canonical complex
omniorientation on $M$.

This is a generalization of the stable complex Hamiltonian $T$-structure on
$M$ defined by a smooth projective toric variety with an explicit
omniorientation222It is tempting to think of Hamiltonian toric manifolds as
something like compactifications of classical harmonic oscillators, much as
toric varieties are compactifications of algebraic torus actions. A complex-
oriented $2n+1$-manifold with $T^{n}$ action, quasi-toric on its boundary,
together with an equivariant complex line bundle and compatible two-forms
$({\bar{\omega}},\omega)$ as above, nondegenerate [1](Thm H.10 p 309) in the
sense that $\omega^{n}$ has one-dimensional kernel (trivial on the boundary),
defines a symplectic cobordism with a complex-oriented version of a formal
Hamiltonian structure as in [1](E.47 p 273, H1.1 p 301). Relaxing the
integrality condition on $[{\bar{\omega}}]$ in effect tensors the underlying
homotopy-theoretic constructions with ${\mathbb{R}}$.

There is however an interesting alternative notion of a complex-oriented
cobordism between such manifolds, defined not by a generalized moment map
[1](3.21 p 35) but by a stable decomposition of its tangent bundle into
complex line bundles, as above. This leads to the quite different theory of
characteristic numbers associated to the ring-spectrum $M\xi$ pursued in this
paper.

1.1.3 Examples

i) A projective toric manifold $M$ (defined by a smooth complex toric variety)
inherits a Kähler class $\omega$ from its projective embedding [2](§5.5.4 p
197, §7.3 p 243), as well as a moment map $\Phi$ defining a polyhedron in
$({\rm Lie\;}T)^{*}$ bounded by $m$ half-planes $\\{{\bf v}\>|\>{\bf v}\cdot
e_{i}\geq\lambda_{i}\\}$, such that [24]

$(2\pi)^{-1}[\omega]=-\sum\lambda_{i}[v_{i}]\in H^{2}(M,{\mathbb{R}})$

for certain classes $[v_{i}]\in H^{2}(M,{\mathbb{Z}})$ defined in §1.2.2
below; this is generalized in [11]. Similarly, toric orbifolds have underlying
toric varieties. Note that $(M,-\omega)$ is not projective.

ii) The orbits of the coadjoint action of a compact Lie group $G$ on the dual
$({\rm Lie\;}\>G)^{*}$ of its Lie algebra are canonically symplectic
manifolds, with the inclusion

$G/{\rm iso}(\lambda)\cong G\cdot\lambda\subset({\rm Lie\;}G)^{*}$

defining a moment map333See [49] for algebraic groups. They play a role in the
representation theory of general Lie groups, in Kirillov’s program, analogous
to that of conjugacy classes in the theory of finite groups. Such orbits are
foliated by tori, and are a rich source of classical (Liouville - Arnol’d)
completely integrable Hamiltonian dynamical systems, e.g. the Eulerian
mechanics of rigid bodies. This is pursued further in §III.

1.2 Some toric topology

Relations between the toric manifolds of Davis and Januszkiewicz [16] and the
toric varieties of algebraic geometry [14, 48] are a source of possible
ambiguity in the discussion below; [2] is helpful, especially for connections
with combinatorics. From the point of view of homotopy theory, the relevant
fact is that toric objects are abundantly supplied with interesting line
bundles, conveniently indexed in various ways by polyopes. The characteristic
classes of these line bundles provide algebraic invariants defined over the
integers, and thus over ${\mathbb{Q}}$; but in the latter context distinctions
between manifolds, orbifolds, and varieties [23, 34] can sometimes be usefully
backgrounded.

1.2.1 Let $K$ be a finite simplicial complex with vertices $[m]$, and let
$(X_{i},A_{i})$ be a family of reasonable pairs of spaces, also indexed by
$[m]$; then Browder’s polyhedral product

$K\mapsto(X_{*},A_{*})^{K}$

([2](§4.2.3 p 137), functorial under maps of families $(X_{*},A_{*})$ and
inclusions of complexes) associates to $K$, the moment-angle complex
[2](§4.1.1 p 131)

${\mathcal{Z}}_{K}:=(D^{2},S^{1})^{K}\in({\rm
Spaces\;with}\;{\mathbb{T}}^{m}-{\rm action})$

[2](§4.1.1 p 131), where now $(X_{i},A_{i})=(D^{2},S^{1})$ is a pair with an
action of the circle.

Remarks

1) $K\mapsto{\mathcal{Z}}_{K}$ is monoidal, taking joins of simplicial
complexes to Cartesian products of spaces [2](§4.5.5), and

2) If $|K|\cong S^{n-1}$ then ${\mathcal{Z}}_{K}$ is a closed topological
manifold of dimension $n+m$ [2](§4.1.4).

1.2.2 Following [2](§3.1.1 p 93), let ${\mathbb{Z}}[K]$ denote the Stanley-
Reisner face ring of the simplicial complex $K$: this is the quotient of the
augmented polynomial algebra ${\mathbb{Z}}_{1\leq i\leq
m}[v_{i}]={\mathbb{Z}}[v_{*}]$ generated by the vertices of $K$, modulo the
ideal generated by relations $v_{I}=\prod_{k\in I}v_{k}$, where the subset
$I\subset K$ is not a simplex of $K$. Evidently ${\mathbb{Z}}[K]$ is a
${\mathbb{Z}}[v_{*}]$-algebra.

There has been a great deal of interesting recent work relating combinatorics
(formulated for example in terms of face rings of simplicial complexes) and
the equivariant topology of their associated moment-angle complexes. Some of
this work will be summarized below; [2] contains further references to the
original literature. Thus [2](Prop 4.5.5 p 147):

${\rm Tor}^{*}_{{\mathbb{Z}}[v_{*}]}({\mathbb{Z}}[K],{\mathbb{Z}})\cong
H^{*}({\mathcal{Z}}_{K};{\mathbb{Z}})\;,\;H^{*}_{{\mathbb{T}}^{m}}({\mathcal{Z}}_{K};{\mathbb{Z}})\cong{\mathbb{Z}}[K]$

(as graded rings). Moreover, if $P$ is an associated omnioriented simple
polytope as in §1.1.1, and we let $T_{\Lambda}$ denote the kernel of the
surjection

$\Lambda\otimes{\mathbb{R}}/{\mathbb{Z}}:{\mathbb{T}}^{m}\to{\mathbb{T}}^{n}\;,$

then $T^{n}\cong{\mathbb{T}}^{m}/{\mathbb{T}}_{\Lambda}$ acts on
${\mathcal{Z}}_{K}/T_{\Lambda}:=M$, which carries the structure of a complex-
oriented quasi-toric manifold [2](§7.3.8 - 16 p 250); in particular, it is a
${\sf T}$-space. Moreover

$H^{*}_{T^{n}}(M;{\mathbb{Z}})\cong{\mathbb{Z}}[K]$

while $H^{*}(M;{\mathbb{Z}})$ is isomorphic to the quotient of
${\mathbb{Z}}[K]$ by the ideal generated by the image of $\Lambda$ [2](Th
5.3.1 p 186). These isomorphisms have natural interpretations in terms of
Eilenberg-Moore spectral sequences.

Remarks

1) The composition $T_{\Lambda}\subset{\mathbb{T}}^{m}=\prod_{1\leq i\leq
m}{\mathbb{T}}_{i}\to T_{i}$ defines a representation ${\mathbb{C}}(i)$ of
$T_{\Lambda}$ on the $i$th circle in the product ${\mathbb{T}}^{m}$, and thus
a complex line bundle

$L_{i}:={\mathcal{Z}}_{K}\times_{T_{\Lambda}}{\mathbb{C}}(i)\to M$

whose Chern classes can naturally be identified with the images of the classes
$v_{i}$ in $H^{*}_{{\mathbb{T}}^{m}}({\mathcal{Z}}_{P};{\mathbb{Z}})$. The
images $[v_{i}]$ of these classes in $H^{*}(M;{\mathbb{Z}})$ are those cited
above in Guillemin’s formula for the Kähler class of $M$ [2](§7.3.15 p 249).

2) When $K$ is a Gorenstein complex [2](Th 3.4.2 p 112), $M$ is a manifold
with fundamental class

---
$\textstyle{{H^{2n}_{T_{\Lambda}}({\mathcal{Z}}_{K};{\mathbb{Z}})\cong
H^{2n}(M;{\mathbb{Z}})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{Z}}$$\textstyle{{H^{2n}_{{\mathbb{T}}^{m}}({\mathcal{Z}}_{K};{\mathbb{Z}})\cong{\mathbb{Z}}[K]}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\;.}$

1.2.3 [2](Th 7.3.15 p 249) We have a $T^{n}$-equivariant splitting

$T_{M}\oplus{\mathbb{C}}^{m-n}\cong\bigoplus_{1\leq i\leq m}L_{i}$

of the tangent bundle of $M$ [7],[2](Th 7.3.15 p 249). This suggests the
interest of the class of complex-oriented manifolds, perhaps odd-dimensional,
possessing stable splittings of their tangent or normal bundles as sums of
complex line bundles.

1.3 Toric dynamical systems in theoretical chemistry

Work on the formal theory of chemical reaction networks dates back at least to
the 1970s [28] but work of Gatermann [21, 22, 25] and others around 2000
brought new ideas from toric geometry into the subject. Following
[15],[47](§2) closely, we summarize fundamental constructions:

1.3.1 A chemical reaction network ${\mathcal{A}}$ (for alembic) is defined by
a triple

${\mathcal{A}}=\\{G,{\boldsymbol{\kappa}}\in{\mathbb{M}}^{n}_{n}({\mathbb{R}}),{\bf
Y}\in{\mathbb{M}}^{s}_{n}({\mathbb{Z}})\\}$

where

*) $G$ is a directed graph with a set $V$ of $n$ ordered vertices, called reactions or nodes, with edge set $E\subset\\{(k,l)\in V\times V\;|\;k\neq l\\}$ (i.e. without closed loops),

*) ${\boldsymbol{\kappa}}=[\kappa^{k}_{l}]$ is a matrix of reaction rates satisfying $\kappa^{k}_{l}>0$ if $k\neq l$, extended by defining

$\kappa^{l}_{l}:=-\sum_{k\neq l}\kappa^{k}_{l}$

(making each row $l$ sum to zero, so $\kappa$ becomes the negative of the
Laplacian matrix for the weighted directed graph $G$), and

*) ${\bf Y}=[Y^{i}_{k}]\in{\mathbb{M}}^{s}_{n}({\mathbb{Z}}_{n\geq 0})$ is a non-negative integer matrix, with $s$ the number of chemical ‘species’ involved in the network.

A toric dynamical system is a chemical reaction network which admits a
strictly positive constant solution ${\bf c}=(c_{1},\dots,c_{m})$ of the
algebraic system ${\boldsymbol{\kappa}}\cdot{\bf c}^{\bf Y}=0$: this is a
special kind of steady state of the system

$\dot{{\bf c}}={\bf Y}\cdot{\boldsymbol{\kappa}}\cdot{\bf c}^{\bf Y}\;,$

i.e.

$\frac{dc_{i}(t)}{dt}=\sum_{1\leq k,l\leq
n}Y^{i}_{k}\kappa^{k}_{l}\cdot\prod_{1\leq j\leq s}c_{j}^{Y_{j}^{l}}=0$

of ordinary differential equations, a complex balancing steady state in the
sense of [28].

Following [48], a toric ideal in a polynomial ring is a prime ideal generated
by differences of monomials. Theorem 7 of [15] defines a variety
$V({\mathcal{A}}):=V(M_{G,Y})$ associated to a toric ideal in
${\mathbb{Q}}[\kappa^{i}_{j}]$ generated by the reaction rates of
${\mathcal{A}}$, and shows that points $\kappa\in V_{>0}({\mathcal{A}})$
characterize toric dynamical systems, whose complex balancing states are the
positive points of a variety $V(T_{G,Y})$ defined by a toric ideal in
${\mathbb{Q}}[c,\kappa]$. An outline of this construction follows:

1.3.2 Suppose at first that $G$ is strongly connected (i.e. there is a
directed path between any pair of distinct vertices); the matrix
${\boldsymbol{\kappa}}$ then has rank $n-1$. Let $(-1)^{n-1}K_{i}$ denote the
determinant of a minor of ${\boldsymbol{\kappa}}$ defined by deleting the
$i$th row and any column - by Stanley’s matrix-tree theorem [47](§2.2.1), it
doesn’t matter which - and it follows [15](Lemma 5) that
${\mathbb{Q}}[K_{1},\dots,K_{n}]\subset{\mathbb{Q}}[\kappa_{*}]$ is a
polynomial subalgebra. When $G=\coprod_{1\leq q\leq r}G(r)$ is the disjoint
union of strongly connected subgraphs, with $G(q)$ having $n(q)$ ordered nodes
(so $n=\sum n(q)$), the existence of the polynomial algebra
${\mathbb{Q}}[K_{*}]$ generalizes.

The Cayley matrix [15](p 7),[46]

${\rm Cay}_{G}(Y):=\left[\begin{array}[]{c}[Y(*)]\\\ {\bf
1}(*)\end{array}\right]\in M^{n}_{r+s}({\mathbb{Z}}_{\geq 0})$

is defined by the row of submatrices $[Y(q)^{k}_{l}]\in
M^{n(q)}_{s}({\mathbb{Z}}_{\geq 0})$ of ${\bf Y}$ (indexed by nodes of $G(q)$;
but note the reversal of rows and columns) across the top, filled in below by
appropriate rows of zeroes and ones. The ideal $M_{G,Y}$ of
${\mathbb{Q}}[K_{*}]$ generated by the elements

$\prod K_{i}^{u_{0,i}}-\prod K_{i}^{u_{1,i}}\;,$

indexed by vectors ${\bf u}_{0},{\bf u}_{1}\in{\mathbb{N}}^{n}$ such that
${\rm Cay}_{G}(Y)\cdot({\bf u}_{0}-{\bf u}_{1})=0$, then [15](Th 9 p
8),[47](§2.2.1 p 16) defines the toric variety $V({\mathcal{A}})$. The moment
map identifies the interior of the convex hull of the columns of ${\rm
Cay}_{G}(Y)$ with $V_{>0}({\mathcal{A}})\subset{\mathbb{R}}^{n}$; its
codimensions $\delta$ is the deficiency of the reaction network. Thus when
$\delta=0$, the moment polytope is an $n$-simplex.

Interesting applications to Hopf bifurcation theory are considered in [22].

1.3.3 The real positive points on such a complex toric variety can be regarded
as asymptotic limits of states of dissipative biochemical systems. As
suggested in [17], the short term mechanics of living organisms are
essentially metabolic, but in the long run they are low-energy information
management processes : the egg is the leading term in the asymptotic expansion
of the chicken. The complex points of these varieties, and their topology,
will thus not be assigned any immediate physical or biological interpretation.

1.4 The classical characteristic numbers

1.4.1 The Thom spectrum

$M{\rm U}:S^{2}M{\rm U}(k)\to M{\rm U}(k+1)$

(for the stable unitary group ${\rm U}=\cup_{k\geq 1}{\rm U}(k)$) is defined
by the Thom spaces $M{\rm U}(k)$ of the canonical ${\mathbb{C}}^{k}$-bundles
$\xi_{k}$ over the classifying spaces $B{\rm U}(k)$. The classifying map

$\nu:M\to B{\rm U}(N-n),\;N\gg 0$

of the stable normal bundle of a complex-oriented $2n$-dimensional manifold
$M\subset{\mathbb{C}}^{N}$ defines a class

$[{\mathbb{C}}^{N}_{+}\to M^{\nu}\to M{\rm U}(N-n)]\in\pi_{2n}M{\rm U}:=M{\rm
U}_{2n}$

in the complex bordism ring $M{\rm U}_{*}$ (with $MU_{\rm odd}=0$). The Thom
isomorphism

$H_{*}(M{\rm U};{\mathbb{Z}})\to H_{*}(B{\rm U};{\mathbb{Z}})$

(of comodules) identifies Hurewicz’s ring monomorphism

$\pi_{*}M{\rm U}\to{\rm Hom}^{-*}(H^{*}(M{\rm U}),H^{*}(S^{0}))\cong
H_{*}(B{\rm U})$

with the characteristic number homomorphism which sends $M$ to the evaluation

$c^{I}(\nu)[M]:=(\prod_{1\leq k\leq r}c_{k}^{i_{k}}(\nu))[M]$

of a degree $2n$ word444where $I=1^{i_{1}}2^{i_{2}}\dots r^{i_{r}},\;|I|=\sum
ki_{k}=n$ is a partition of $n$ with $r$ parts in the Chern classes of the
stable normal bundle of $M$ on the fundamental class $[M]\in
H_{2n}(M;{\mathbb{Z}})$. This map becomes an isomorphism after tensoring with
${\mathbb{Q}}$: rationally, complex cobordism theory and the theory of Chern
numbers coincide.

The group completion

$\coprod_{k\geq 0}B{\rm U}(k)\to\Omega B(\coprod_{k\geq 0}B{\rm
U}(k))\simeq{\mathbb{Z}}\times B{\rm U}\supset 0\times B{\rm U}$

defines an $H$-space structure on the classifying space for stable complex
vector bundles. Following Borel and Hirzebruch, we identify its cohomology

$H^{*}(B{\rm U};{\mathbb{Z}})\cong{\mathbb{Z}}_{k\geq 1}[e_{k}]:={\sf
S}^{*}\subset{\mathbb{Z}}_{k\geq
1}[x_{k}]\cong\lim_{r\to\infty}H^{*}(B(T^{r});{\mathbb{Z}})$

$(|x_{k}|=2,\;|e_{k}|=2k)$, with the graded polynomial algebra of symmetric
functions [35](I §2.7) generated by elementary symmetric functions,
identifying $e_{k}$ with the Chern class $c_{k}$. In terms of generating
functions,

$E(t):=\prod_{i\geq 1}(1+x_{i}t)=\sum_{j\geq 0}e_{j}t^{j}\;,$

and the complete symmetric functions $h_{j},\;H(t)=\sum_{j\geq 0}h_{j}t^{j}$
are defined by $H(t)=E(-t)^{-1}$; thus, for example,
$e_{0}=h_{0}=1,\;e_{1}=h_{1}$. The generating function

$tH^{\prime}(t)H(t)^{-1}=\sum_{k\geq 1}p_{k}t^{k}$

defines the power sums $p_{k}=\sum_{i\geq 1}x_{i}^{k}$.

The ring of symmetric functions has a canonical pair of involutions generating
an action of the Klein four-group: the first is defined by $x_{i}\mapsto-
x_{i}$, sending $e_{k}$ to $(-1)^{k}e_{k}$, while the second maps $e_{k}$ to
$-h_{k}$ (so $p_{k}\mapsto-p_{k}$). These correspond to the operations which
send a stable complex vector $V$ respectively to its complex conjugate $V^{*}$
and to its $H$-space inverse $-V$. It follows that the $i$th Chern class of
the stable normal bundle $\nu$ of a stably almost-complex manifold corresponds
to $-h_{i}$ of its stably almost-complex tangent bundle $T_{M}$ [13].

The algebra of symmetric functions is moreover self-dual with respect to the
Hall inner product [35](I §4.5), with an isomorphism

$H_{*}(B{\rm U},{\mathbb{Z}})\cong{\sf S}_{*}$

defined by sending a word $h_{I}$ in the complete symmetric functions $h_{i}$
to the word $m_{I}$ written in terms of the corresponding monomial symmetric
functions $m_{i}$; the Schur symmetric functions form an an orthonormal basis.
Note finally that the polynomial algebra $H^{*}(B{\rm
U}(1),{\mathbb{Z}})=H^{*}({\mathbb{C}}P^{\infty},{\mathbb{Z}})$ is dual to the
Hopf algebra
$H_{*}({\mathbb{C}}P^{\infty},{\mathbb{Z}})\cong{\mathbb{Z}}_{k\geq 1}[b_{k}]$
of divided powers $b_{(k)}=b^{k}/k!\in{\mathbb{Q}}[b]$.

1.4.2 The Landweber-Novikov Hopf algebra $S_{*}={\mathbb{Z}}_{i\geq 1}[t_{i}]$
of formal diffeomorphisms of the line at the origin (with $|t_{i}|=2i$ and
$t(T)=\sum_{i\geq 0}t_{i}T^{i+1}$, and an antipode given by power series
inversion) is defined by the coproduct

$(\Delta_{S}t)(T)=(t\otimes 1)((1\otimes t)(T))\in(S\otimes S)_{*}[[T]]$

($|T|=-2$); Miščenko’s logarithm

$\log_{M{\rm U}}(T)\;=\;\sum_{n\geq 1}\frac{{\mathbb{C}}P_{n-1}}{n}T^{n}\in
M{\rm U}^{*}_{\mathbb{Q}}[[T]]$

for the formal group law on $M{\rm U}^{*}({\mathbb{C}}P^{\infty})\cong M{\rm
U}^{*}[[c]]$ defines a coaction

$\psi_{M{\rm U}}(\log_{M{\rm U}}(T))=\log_{M{\rm U}}(t(T))$

of $S_{*}$ on $M{\rm U}_{*}$, yielding an isomorphism

$(M{\rm U}_{*},M{\rm U}_{*}\otimes S_{*})\cong(M{\rm U}_{*},M{\rm U}_{*}M{\rm
U})$

of Hopf algebroids [44]. The closely related Faa di Bruno Hopf algebra

$S_{*}\otimes{\mathbb{Q}}={\mathbb{Q}}_{k\geq 1}[t_{(k)}]$

is presented in terms of the formal derivatives $t_{(k)}=k!t_{k-1}$ of $t(T)$
at 0; this is a confused issue in the literature. Extending either by
adjoining an invertible degree zero element $t_{0}^{\pm 1}$ can be used to
encode the grading.

The generators $b_{i}\in M{\rm U}_{*}({\mathbb{C}}P^{\infty})$ (Kronecker dual
to $c^{i}\in M{\rm U}^{*}[[c]]$) define a formal series

$b(T)=\sum_{i\geq 0}b_{i}T^{i}=\exp(b\log_{M{\rm U}}(T))\in(M{\rm
U}^{\mathbb{Q}}_{*}{\mathbb{C}}P^{\infty})[[T]]$

of degree zero [38], with coproduct $\Delta b(T)=b(T)\otimes_{M{\rm U}}b(T)$,
making $M{\rm U}_{*}({\mathbb{C}}P^{\infty})$ an $M{\rm U}_{*}M{\rm
U}$-comodule coalgebra with

$\psi_{M{\rm U}}b(T)=b(t(T))\;.$

1.4.3 The Hurewicz homomorphism

${\mathcal{B}}_{*}\cong M{\rm U}_{*}B{\mathbb{T}}\to H_{*}(B{\rm
U};{\mathbb{Z}})\otimes H_{*}(B{\mathbb{T}};{\mathbb{Z}})\cong{\sf
S}_{*}\otimes{\mathbb{Z}}_{i\geq 1}[b_{(i)}]$

maps VL Ginzburg’s cobordism ring (of integral symplectic manifolds under non-
degenerate cobordism (as in §1.1.2 [38]) injectively, sending a
$2n$-dimensional manifold $(V,\omega)$ to the system

$I\mapsto(-1)^{\sum i_{k}}(\prod
h_{k}^{i_{k}}(T^{\mathbb{C}}_{V})\cdot[\omega]^{n-i})[V]\in{\mathbb{Z}}$

of characteristic number homomorphisms indexed by unordered partitions $I$ of
$i\leq n$. Relaxing the integrality constraint on the form $[\omega]$ in
effect embeds ${\mathcal{B}}_{*}$ in ${\mathcal{B}}_{*}\otimes{\mathbb{R}}$.

Quillen’s conventions [43] associate to a compact complex $2n$-dimensional
$G$-manifold, a class in $M{\rm U}^{-2n}(BG)$; similarly, an integrally
symplectic $2n$-dimensional ${\sf T}$-manifold defines an element of
${\mathcal{B}}^{-2n}(B{\mathbb{T}}^{n})$, with characteristic numbers in

$H^{-2n}(B{\mathbb{T}}^{n};H_{*}(M{\rm U}\wedge B{\mathbb{T}}))\;,$

generalizing the characteristic numbers for formal Hamiltonian cobordism
defined in [1](H.2 p 304). The group homomorphism

${\mathbb{T}}^{n}\times{\mathbb{T}}^{n}\to{\mathbb{T}}^{n}\to{\rm U}(n)$

defined by multiplication of abelian groups induces a map from
$B{\mathbb{T}}^{n}\times B{\mathbb{T}}^{n}$ to $B{\rm U}(n)$, and pulling back
(the stable inverse of) the universal bundle on $B{\rm U}$ defines a morphism

$B{\mathbb{T}}^{n}_{+}\wedge M{\rm U}(1)^{\wedge n}\to M{\rm U}(n)$

of spectra, adjoint to a morphism

$M{\rm U}(1)^{\wedge n}\to{\rm Maps}(B{\mathbb{T}}^{n}_{+},M{\rm U}(n))\;.$

Taking limits as $n\to\infty$ defines (see .1.2) a morphism from the homology
xgroups of $M\xi$ to those of ${\rm Maps}(B{\mathbb{T}}^{n}_{+},M{\rm U})$,
relating formal Hamiltonian characteristic classes to noncommutative symmetric
functions.

§II The Baker-Richter spectrum

2.1 New kinds of characteristic classes and numbers

2.1.1 The $2^{n-1}$ ordered partitions

${\bf n}:=n_{1}+\cdots+n_{k}$

of $n=\sum n_{i}$ into nonempty parts define the quasisymmetric function

${\langle}{\bf n}{\rangle}(x_{*}):=\sum_{0<n_{1}<\dots<n_{k}}\prod
x_{i_{j}}^{n_{j}}\in{\mathbb{Z}}[x_{*}]$

(denoted $[n_{1},\dots,n_{k}]$ in [8], cf. [27](§4)). The subring of
${\mathbb{Z}}[x_{*}]$ generated by such sums is, by Baker and Richter or
Hazewinkel’s proof of Ditters’ conjecture, an evenly graded (‘quasi-shuffle’)
Hopf algebra $\sf{QSymm}^{*}$ over ${\mathbb{Z}}$, generated over
${\mathbb{Q}}$ by certain Lyndon partitions [8](§2.2); for brevity it will
here be denoted ${\sf Q}^{*}$. The dual Hopf algebra of noncommutative
symmetric functions

${\sf N}_{*}={\sf NSymm}_{*}={\mathbb{Z}}_{i\geq 1}{\langle}Z_{i}{\rangle}$

is free associative, with coproduct $\Delta Z_{i}=\sum_{i=j+k}Z_{j}\otimes
Z_{k}$; the word $Z_{\bf n}:=\prod_{1\leq i\leq k}Z_{n_{i}}\in{\sf N}_{2n}$ is
dual to ${\langle}{\bf n}{\rangle}\in{\sf Q}^{2n}$. We will identify the
generators $Z_{i}$ with Cartier’s elements $\Lambda_{i}$, cf. [12](§4.1F eq
155). Abelianization defines dual maps

${\sf N}_{*}\to{\sf S}_{*},\;\;{\sf S}^{*}\to{\sf Q}^{*}$

of Hopf algebras555Convention: $-\otimes_{\mathbb{Z}}{\mathbb{Q}}$ sends the
graded module $M_{*}=M^{-*}$ to $M^{\mathbb{Q}}_{*}=M^{-*}_{\mathbb{Q}}$.; the
second sends $h_{k}\mapsto{\langle}{\boldsymbol{1^{k}}}{\rangle}$; see further
[35](§2).]

IM James’ construction provides a stable splitting

$\Omega\Sigma B{\rm U}(1)\sim\bigvee_{n\geq 0}B{\rm U}(1)_{+}^{\wedge n}$

and thus Hopf algebra isomorphisms

$H_{*}(\Omega\Sigma{\mathbb{C}}P^{\infty};{\mathbb{Z}})\cong{\sf
N}_{*},\;\;H^{*}(\Omega\Sigma{\mathbb{C}}P^{\infty};{\mathbb{Z}})\cong{\sf
Q}^{*}\;.$

2.1.2 Regarding a complex line bundle as a complex vector bundle defines a map
from $B{\rm U}(1)$ to $B{\rm U}$; composing with the stable inverse map of
§1.4.1 above defines a morphism

$\textstyle{\Omega\Sigma B{\rm
U}(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B{\rm
U}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{V\mapsto-V}$$\textstyle{B{\rm
U}}$

of loop-spaces. Pulling the canonical stable bundle over $B{\rm U}$ back
defines the $A_{\infty}$ spectrum $M\xi$ of [9]. A class in $\pi_{2n}M\xi$ can
then be interpreted as the cobordism class of a complex-oriented $2n$-manifold
$M$, together with a preferred isomorphism

$T_{M}\simeq\oplus L_{i}$

of its stable tangent bundle as a sum of complex line bundles; omnioriented
manifolds provide examples. Forgetting the splitting defines an abelianization
homomorphism $M\xi_{*}\to M{\rm U}_{*}$; note that expressing this in terms of
characteristic numbers will involve the second involution mentioned above.

The Hurewicz (ring) homomorphism

$M\xi_{*}:=\pi_{*}M\xi\to H_{*}(M\xi;{\mathbb{Z}})\cong
H_{*}(\Omega\Sigma{\mathbb{C}}P^{\infty};{\mathbb{Z}})\cong{\sf N}_{*}$

is injective, and becomes an isomorphism after tensoring with ${\mathbb{Q}}$
[9],[27](§2.4); it sends $M$ to the homomorphism defined by

$({\langle}{\bf n}{\rangle}([v_{*}]))[M]:=\sum_{0<n_{1}<\dots<n_{k}}(\prod
c_{1}(L_{i_{j}})^{n_{j}})[M]:{\sf Q}^{2n}\to{\mathbb{Z}}\;,$

taking the quasisymmetric function ${\langle}{\bf n}{\rangle}$ (of the Chern
classes $[v_{i}]$) to its evaluation on the fundamental class of $M$.

Similarly, a symplectic $2n$-manifold $(V,\omega)$ with a compatible
quasitoric structure (for example, a Hamiltonian ${\sf T}$-manifold as in
§1.1.2) has characteristic numbers

$(V,\omega)\mapsto[{\bf i}\mapsto({\langle}{\bf
i}(T_{V}){\rangle}[\omega]^{n-i})[V]\cdot b_{(i)}]\in{\rm Hom}({\sf
Q}^{2n},{\mathbb{R}}[b])\cong{\sf N}_{2n}\otimes{\mathbb{R}}[b]$

indexed by partitions of $i\leq n$. If $(V,\omega)$ is projective toric,
Guillemin’s theorem [§1.1.3] expresses this invariant in terms of the face
parameters $\lambda_{i}$.

The morphism

$M\xi\wedge B{\mathbb{T}}\to M{\rm U}\wedge B{\mathbb{T}}$

underlying this map has interesting arithmetic properties [12](§4.4) over
${\mathbb{R}}$.

2.1.3 Remarks:

1) A better understanding of toric varieties as limits of toric manifolds
would be very useful. A toric variety associated to a lattice polytope in
$\pi_{1}(T)\subset({\rm Lie\;}T)^{*}$ is smooth [2](5.2 p 184), [16](§7.1) if
the primitive lattice vectors along the edges meeting at a given vertex form a
basis for the lattice. Alexeev [3](§2) has constructed a moduli space for
suitably polarized smooth toric varieties, but the polarization data restricts
their allowed degenerations, making it difficult to assess the degree to which
smooth toric varieties are generic. As noted above, the theory of integral
characteristic numbers of toric manifolds extends nicely to a theory of
rational characteristic numbers for more general toric varieties.

2) By the construction in §1.2.2, we can associate to quasisymmetric
functions, characteristic classes in the face rings of simplicial spheres, and
evaluate them to define characteristic numbers for such spheres, which are
essentially the same as the characteristic numbers for their associated
(toric) moment-angle manifolds. Since joins map to products under that
correspondence, this defines a homomorphism from the noncommutative graded
ring (with respect to joins) of isomorphism classes of simplicial spheres, to
${\sf N}_{*}$. For example,

$\partial\Delta^{n}\mapsto\sum_{|{\bf n}|=n}Z_{\bf n}\;.$

2) The collection ${\mathcal{A}}$ defining a chemical reaction network, as in
§1.3, has an associated projective toric variety $V({\mathcal{A}})$; when
smooth, this is a Hamiltonian ${\sf T}$-manifold, which defines an element in
$M\xi_{*}B{\mathbb{T}}$ and thus an element of ${\mathcal{B}}_{*}$. In general
it defines an element of $M\xi_{*}B{\mathbb{T}}\otimes{\mathbb{Q}}$, with
image in ${\mathcal{B}}_{*}\otimes{\mathbb{R}}$. Following §1.3.2, the remark
above identifies the class in $M\xi_{2n}$ of a network with deficiency
$\delta=0$; see also $\S 1.1.3i$.

2.2 Formal diffeomorphisms of the noncommutative line

2.2.1 Although composition of formal power series with noncommuting
coefficients is not associative, Baker and Richter [9](Prop 3.1) show that,
with suitable care, $M\xi^{*}{\mathbb{C}}P^{\infty}$ can be understood as
having a formal one-dimensional Lie group structure defined by a coproduct

$\Delta_{\xi}x_{\xi}=\sum a_{i,j}x_{\xi}^{i+1}\otimes x_{\xi}^{j+1}$

where, however, $x_{\xi}\in M\xi^{2}{\mathbb{C}}P^{\infty}$ is not a central
element; note that $M\xi$ possesses a Thom isomorphism for complex vector
bundles even though it is not an $M{\rm U}$-algebra [8](§7). Nevertheless,
their Prop 2.3 uses the Hurewicz homomorphism to define an algebra
monomorphism

$x_{\xi}\mapsto\Theta(x_{\xi}):=\sum_{i\geq
0}Z_{i}x_{H}^{i+1}=Z(x_{H}):M\xi^{*}{\mathbb{C}}P^{\infty}\to{\sf
N}_{*}[[x_{H}]]$

with $x_{H}$ central. Moreover, the rationalization
$\Theta\otimes{\mathbb{Q}}$ is an isomorphism, and we have a commutative
diagram

$\textstyle{M\xi^{*}_{\mathbb{Q}}{\mathbb{C}}P^{\infty}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Theta_{\mathbb{Q}}}$$\scriptstyle{\cong}$$\scriptstyle{\Delta_{\xi}}$$\textstyle{M\xi^{*}_{\mathbb{Q}}({\mathbb{C}}P^{\infty}\times{\mathbb{C}}P^{\infty})\cong
M\xi^{*}_{\mathbb{Q}}{\mathbb{C}}P^{\infty}\otimes_{M\xi}M\xi^{*}_{\mathbb{Q}}{\mathbb{C}}P^{\infty}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Theta_{\mathbb{Q}}\otimes\Theta_{\mathbb{Q}}}$$\scriptstyle{\cong}$$\textstyle{{\sf
N}_{*}^{\mathbb{Q}}[[x_{H}]]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Delta_{\sf
N}}$$\textstyle{{\sf N}_{*}^{\mathbb{Q}}[[x_{H}\otimes 1,1\otimes x_{H}]]}$

defining a formal group law over ${\sf N}_{*}^{\mathbb{Q}}:={\sf
N}_{*}\otimes{\mathbb{Q}}$ with central coordinate $x_{H}$. Similarly,
$M\xi^{\mathbb{Q}}_{*}{\mathbb{C}}P^{\infty}\cong{\sf
N}^{\mathbb{Q}}_{*}[b^{H}_{i}]_{i\geq 1}$ is a Hopf algebra with binomial
coproduct

$b^{H}(T)=\sum_{i\geq 0}b^{H}_{i}T^{i},\;\Delta b^{H}(T)=b^{H}(T)\otimes_{{\sf
N}}b^{H}(T)\;,$

where $b^{H}_{i}$ is Kronecker dual to $x_{H}^{i}$.

PropositionThe Hurewicz characteristic number construction defines a
commutative diagram

$\textstyle{M\xi^{*}{\mathbb{C}}P^{\infty}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{*}({\mathbb{C}}P^{\infty},H_{*}M\xi)={\sf
N}_{*}[[x_{H}]]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M{\rm
U}^{*}{\mathbb{C}}P^{\infty}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{*}({\mathbb{C}}P^{\infty},H_{*}M{\rm
U})={\sf S}_{*}[[c]]}$

sending $\Theta(x_{\xi})$ to $\exp_{M{\rm U}}(c)$.

2.2.3 I am indebted to Michiel Hazewinkel for drawing attention to work of
Brouder, Frabetti, and Krattenthaler on a (neither commutative nor
cocommutative) Hopf algebra ${\bf{N}_{*}}:=({\sf N}_{*},\Delta_{\bf N})$ of
formal diffeomorphisms of the noncommutative line, generalizing
$(S_{*},\Delta_{S})$. It is defined on generators in terms of a formal residue
homomorphism

${\rm res}_{T=0}(\prod_{i\in{\mathbb{Z}}}a_{i}T^{i}):=a_{-1}\;,$

(related to the Stieltjes transform [29](§1.2) used in free probability
theory) by

$\Delta_{\bf N}Z(T)=\sum_{n\geq 0}{\rm res}_{U=0}(Z(U)\otimes
Z(T)^{n}U^{-n-1})\in({\sf N}\otimes{\sf N})_{*}[[T]]$

where $T$ and $U$ are central formal indeterminates of degree -2.
Noncommutative Lagrangian inversion [10] (Th 2.14) defines a (nonsymmetric)
antipode $\chi_{\bf N}$, i.e. $\Delta_{\bf N}\circ\Delta_{\bf N}\neq{\it
id_{\bf N}}$. It is an exercise, using the formal analog of Cauchy’s theorem,
to see that the BFK Hopf algebra abelianizes to the Landweber-Novikov algebra.

It will be helpful to provide a concordance of their notation. They work with
a complexification ${{\mathcal{H}}^{dif}}:={\mathbb{C}}_{i\geq 1}\langle
a_{i}\rangle$ of ${\bf{N}_{*}}$, with $a_{i}$ corresponding to our $Z_{i}$, as
well as with a cocommutative Hopf algebra
${{\mathcal{H}}^{inv}}:={\mathbb{C}}_{i\geq 1}\langle b_{i}\rangle$ (analogous
to ${\sf N}_{*}\otimes{\mathbb{C}}$, on generators $b_{i}$ with binomial
coproduct $\Delta b(T)=b(T)\otimes_{\mathbb{C}}b(T)$); for example, Cartier’s
primitive elements

$\Psi_{k}\in{\sf N}_{2k},\;\Psi(T):=TZ^{\prime}(T)\cdot Z(T)^{-1}$

define such generators by $b(T):=\exp(\Psi(T))$. The algebra homomorphism

$\psi:{{\mathcal{H}}^{inv}}\to{{\mathcal{H}}^{inv}}\otimes_{\mathbb{C}}{{\mathcal{H}}^{dif}}$

defined by

$(\psi b)(T)=\sum_{n\geq 0}{\rm res}_{U=0}(b(U)\otimes a(T)^{n}U^{-n-1})$

is, by their lemma 3.3, a morphism of coalgebras, defining (a family of) Hopf
algebroid(s)
$({{\mathcal{H}}^{inv}},{{\mathcal{H}}^{inv}}\otimes_{\mathbb{C}}{{\mathcal{H}}^{dif}})$.
The choice above defines a Hopf algebroid $({\sf N}^{\mathbb{Q}}_{*},{\sf
N}^{\mathbb{Q}}_{*}\otimes{\bf{N}_{*}})$, conjecturally sending $\Psi_{k-1}$
to $k^{-1}{\mathbb{C}}P_{k-1}\in M{\rm U}^{\mathbb{Q}}_{2(k-1)}$.

Conjecture Setting $b^{H}(T)=\exp(\beta\Psi(T))$ in §2.2.1 defines an
isomorphism

$({\sf N}^{\mathbb{Q}}_{*}[\beta],{\sf
N}^{\mathbb{Q}}_{*}[\beta]\otimes{\bf{N}_{*}})\to(M\xi^{\mathbb{Q}}_{*}{\mathbb{C}}P^{\infty},M\xi^{\mathbb{Q}}_{*}({\mathbb{C}}P^{\infty}\wedge
M\xi))$

of Hopf algebroids, which specializes at $\beta=1$ to an isomorphism

$({\sf N}^{\mathbb{Q}}_{*},{\sf
N}^{\mathbb{Q}}_{*}\otimes{\bf{N}_{*}})\to(M\xi^{\mathbb{Q}}_{*},M\xi^{\mathbb{Q}}_{*}M\xi)\;.$

2.3.1 The cumulant generating function

$\log\mathcal{E}(\exp(tX))=\sum_{n\geq 1}\kappa_{n}\frac{t^{n}}{n!}$

of a random variable $X$ in classical statistical mechanics is the logarithm
of its partition function, i.e. of the trace or expectation of Boltzmann’s
factor $\exp(tX)$; up to a factor of $t$ and a normalization, this is the
Helmholtz free energy of physical chemistry, which Schrödinger identified with
negative entropy [45](Ch 6 p 74). Its Legendre transform [36](§4),[37] is
roughly Cramér’s risk function, which measures the density of excursions from
the mean along the normal bundle transverse to an evolutionary trajectory
[46].

2.3.2 In the theory of free (noncommutative) probability, a state (i.e. a
unital ${\mathbb{C}}$-valued linear functional $\phi$ on a not necessarily
commutative algebra) associates to an element $a$ of that algebra, a moment
distribution $\\{\phi(a^{n}),\;n\geq 0\\}$ (with $\phi(a^{0})=1$). Since ${\sf
N}_{*}$ is free associative over ${\mathbb{Z}}$, we can take $a^{n}$ to be the
image under a ring homomorphism of Cartier’s $\Sigma_{n}$ (defined in terms of
generating functins by the relation $\Sigma(T)\cdot Z(-T)=1$).

Definition Following Voiculescu [41](Ch XI), let $A$ be a commutative ring,
with

$g(z)=z^{-1}+\sum_{n\geq 1}g_{n}z^{-n-1}\in A[[z^{-1}]]\;;$

then $g(z)=\gamma(z^{-1})$ defines $\gamma(z)\in A[[z]]$, with formal inverse
$\gamma^{-1}(z)=z+\cdots$ such that

$g(\frac{1}{\gamma^{-1}(z)})=z\;,$

and we can write

$K(z)=\sum_{n\geq 0}k_{n}z^{n}=\frac{z}{\gamma^{-1}(z)}=1+\dots\;.$

The free moment generating function for the distribution defined by $a$ is
then

$g(z)=\sum_{n\geq 0}\phi(a^{n})z^{-n-1}=z^{-1}+\dots\;,$

and its free cumulant generating function $\kappa(z)$ is defined by
$K(z)=1+z\kappa(z)$.

Remarkable work [20],[21](prop 4.2) of Friedrich and McKay (going back to
[26],[30](§1.3)) interprets the relation between one-dimensional formal group
laws and their associated Hirzebruch multiplicative sequences - in particular

$H(z)=\frac{z}{\log^{-1}_{M{\rm U}}(z)}$

in terms of the antipode of the Landweber-Novikov algebra (via Lagrange
inversion) to construct a state on ${\sf N}_{*}\otimes{\mathbb{C}}$ with
$\log_{M{\rm U}}$ as its free cumulant. More recent work [29](Th 3.1),
[42](§5.5.4) lifts this to a version

$K(x_{H})=\chi_{\bf N}(Z(-x_{H}))\in{\bf{N}_{*}}[[x_{H}]]\;.$

of a noncommutative cumulant generating function for the distribution defined
by $\Sigma_{n}\mapsto a^{n}$. xs

§III Afterword

In the 1960s V I Arnol’d interpreted Euler’s equations for fluid mechanics on
a Riemannian manifold as the geodesic flow on the infinite-dimensional Lie
group ${\rm SDiff}(M)$ of its volume-preserving diffeomorphisms [5, 6, 18,
31], with associated Hamiltonian systems defined by coadjoint orbits as in
§$1.1.3ii$ above: the presence of viscosity makes such finite-dimensional
analogies reasonable.

On the other hand D’Arcy Thompson’s ‘simple and most beautiful [inkdrop]
experiment’ (On falling drops, On Growth and Form p 395-6) suggests that the
work [4, 31, 33, 40] of Kolmogorov, Arnol’d, Moser and others on the stability
of quasi-periodic solutions of certain Hamiltonian systems (originating in
celestial mechanics, but generalized (as above, to hydrodynamics, or to
chemical reaction networks as in §1.3.3)) might have applications to the
analysis of self-replicating systems in biology.

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