#  $SU$-linear operations in complex cobordism and the $c_{1}$-spherical
bordism theory

Georgy Chernykh  Faculty of Mathematics and Mechanics, Moscow State
University, Russia;
Steklov Mathematical Institute of the Russian Academy of Sciences, Moscow,
Russia<EMAIL_ADDRESS>and  Taras Panov  Faculty of Mathematics and
Mechanics, Moscow State University, Russia;
Institute for Information Transmission Problems, Russian Academy of Sciences,
Moscow<EMAIL_ADDRESS>http://higeom.math.msu.su/people/taras/

###### Abstract.

We study the $SU$-linear operations in complex cobordism and prove that they
are generated by the well-known geometric operations $\partial_{i}$. For the
theory $W$ of $c_{1}$-spherical bordism, we describe all $SU$-linear
multiplications on $W$ and projections $\mathit{MU}\to W$. We also analyse
complex orientations on $W$ and the corresponding formal group laws $F_{W}$.
The relationship between the formal group laws $F_{W}$ and the coefficient
ring $W_{*}$ of the $W$-theory was studied by Buchstaber in 1972. We extend
his results by showing that for any $SU$-linear multiplication and orientation
on $W$, the coefficients of the corresponding formal group law $F_{W}$ do not
generate the ring $W_{*}$, unlike the situation with complex bordism.

###### Key words and phrases:

complex bordism, SU-bordism, cohomological operations, formal group laws

###### 2020 Mathematics Subject Classification:

55N22, 57R77

The authors are supported by the Russian Foundation for Basic Research (grant
20-01-00675). The first author is also supported by Theoretical Physics and
Mathematics Advancement Foundation “BASIS”

## Introduction

_Complex bordism_ , or _$U$ -bordism_, is the bordism theory of stably complex
manifolds. Geometrically, a stably complex structure (a $U$-structure) on a
manifold $M$ is a choice of a complex structure on its stable tangent bundle,
or a reduction of the structure group of the stable tangent bundle to the
group $U(N)$. Homotopically, a stably complex structure is the homotopy class
of a lift of the map $M\to BO(2N)$ classifying the stable tangent bundle to a
map $M\to BU(N)$. The bordism classes of stably complex manifolds form a
graded ring with respect to the operations of disjoint union and cartesian
product, called the _complex bordism ring_ and denoted by $MU_{*}$. It is the
coefficient ring of the _complex bordism theory_ , the generalised
(co)homology theory defined by the _Thom spectrum_
$\mathit{MU}=\\{\mathit{MU}(n)\\}$, where $MU(n)$ is the Thom space of the
universal $U(n)$-bundle $EU(n)\to BU(n)$. Given a CW-pair $(X,A)$, its bordism
and cobordims groups are defined by

$\displaystyle MU_{n}(X,A)$
$\displaystyle=\lim_{k\to\infty}\pi_{2k+n}\bigl{(}(X/A)\wedge\mathit{MU}(k)\bigr{)},$
$\displaystyle MU^{n}(X,A)$
$\displaystyle=\lim_{k\to\infty}\bigl{[}\varSigma^{2k-n}(X/A),\mathit{MU}(k)\bigr{]}\quad\text{for
a finite CW-pair $(X,A)$}.$

In particular,
$MU_{*}=\pi_{*}(\mathit{MU})=MU_{*}(\mathit{pt})=\lim_{k\to\infty}\pi_{2k+*}\bigl{(}\mathit{MU}(k)\bigr{)}$.
We also denote $MU^{*}=MU^{*}(\mathit{pt})$, the _complex cobordism ring_ ,
graded nonpositively.

_$SU$ -bordism_ is the bordism theory of smooth manifolds with a special
unitary structure in the stable tangent bundle. Geometrically, an
$SU$-structure on a manifold $M$ is defined by a reduction of the structure
group of the stable tangent bundle of $M$ to the group $SU(N)$. Homotopically,
an $SU$-structure is the homotopy class of a lift of the map $M\to BO(2N)$
classifying the stable tangent bundle to a map $M\to BSU(N)$. A manifold $M$
admits an $SU$-structure whenever it admits a stably complex structure with
$c_{1}(\mathcal{T}M)=0$. The _$SU$ -bordims ring_
$\mathit{MSU}_{*}=\pi_{*}(\mathit{MSU})$ is the coefficient ring of the
_$\mathit{SU}$ -bordism theory_, defined by the Thom spectrum
$\mathit{MSU}=\\{\mathit{MSU}(n)\\}$.

The details of the construction of the $\mathit{SU}$-bordism and the
description of the coefficient ring $\mathit{MSU}_{*}$ can be found in [14,
17, 11].

A (stable) _operation_ $f$ of degree $n$ in complex cobordism is a family of
additive maps

$f\colon\mathit{MU}^{k}(X,A)\to\mathit{MU}^{k+n}(X,A)$

which are functorial with respect to $(X,A)$ and commute with the suspension
isomorphisms. The set of all operations is an $\mathit{MU}^{*}$-algebra,
denoted by $A^{U}$. It can be identified with the self-maps of the
$\mathit{MU}$ spectrum:

$A^{U}\cong[\mathit{MU},\mathit{MU}]_{*}=MU^{*}(\mathit{MU})=\lim\limits_{\longleftarrow}\mathit{MU}^{*+2N}(\mathit{MU}(N)).$

There is an isomorphism of left $\mathit{MU}^{*}$-modules

$A^{U}\cong\mathit{MU}^{*}\mathbin{\widehat{\otimes}}S,$

where $S$ is the _Landweber–Novikov algebra_ , generated by the operations
$S_{\omega}=\varphi^{*}(s^{\scriptscriptstyle U}_{\omega})$ corresponding via
the Thom isomorphism $\varphi^{*}$ to the universal characteristic classes
$s^{\scriptscriptstyle U}_{\omega}\in\mathit{MU}^{*}(\mathit{BU})$ defined by
symmetrising the monomials $t_{1}^{i_{1}}\cdots t_{k}^{i_{k}}$ indexed by
partitions $\omega=(i_{1},\ldots,i_{k})$. Therefore, any element $a\in A^{U}$
can be written uniquely as an infinite series
$a=\sum_{\omega}\lambda_{\omega}S_{\omega}$ where
$\lambda_{\omega}\in\mathit{MU}^{*}$. The Hopf algebra structure of $S$ is
described in [10] and [14, §5].

The spectrum $\mathit{MU}$ is an $\mathit{MSU}$-module via the forgetful
morphism $\mathit{MSU}\to\mathit{MU}$, and an operation
$f\colon\mathit{MU}\to\mathit{MU}$ is _$\mathit{MSU}$ -linear_ if it is an
$\mathit{MSU}$-module map. By the standard property of spectra with torsion-
free homotopy and homology groups, the $\mathit{MSU}$-linearity of an
operation $f\colon\mathit{MU}\to\mathit{MU}$ can be detected by its action on
the coefficients $MU_{*}=\pi_{*}(\mathit{MU})$. Namely, an operation $f$ is
$\mathit{SU}$-linear if and only if it satisfies $f(ab)=af(b)$ for any
$a\in\mathit{MSU}_{*}$, $b\in\mathit{MU}_{*}$ (see Theorem 1.3).

A family of geometric operations
$\partial_{i}\in[\mathit{MU},\mathit{MU}]_{-2i}=[\mathit{MU},\Sigma^{2i}\mathit{MU}]$
was defined by Conner and Floyd [8] and studied further by Novikov [14]. The
operation $\partial_{i}$ assigns to a complex bordism class
$[M]\in\mathit{MU}_{2n}$ the bordism class of the submanifold $M_{i}\subset M$
dual to $(\det\mathcal{T}M)^{\oplus i}$ (the $i$-fold direct sum of the
determinant of the tangent bundle). In particular,
$\partial_{1}=\partial\colon\mathit{MU}_{2n}\to\mathit{MU}_{2n-2}$ is the
“boundary operation” that takes $[M]$ to the bordims class of the submanifold
dual to $c_{1}(\mathcal{T}M)$. Clearly, $\partial[M]$ belongs to the image of
the forgetful map $\mathit{MSU}_{*}\to\mathit{MU}_{*}$. Furthermore, the
operations $\partial_{i}$ are $SU$-linear by inspection.

In Section 1 we describe the algebra of $\mathit{SU}$-linear operations in
complex cobordism. We show that the operations $\partial_{i}$, $i=1,2,\ldots$,
form a topological basis of the left $MU^{*}$-module of $\mathit{SU}$-linear
operations. That is, any $SU$-linear operation
$f\in[\mathit{MU},\mathit{MU}]_{\mathit{MSU},*}$ can be written uniquely as a
series $f=\sum_{i\geqslant 0}\mu_{i}\partial_{i}$ with
$\mu_{i}\in\mathit{MU}^{-2i-*}$, see Theorem 1.12. In Theorem 1.14 we describe
the product structure of $\mathit{SU}$-linear operations under composition via
the coefficients of the formal group law in complex cobordism.

Conner and Floyd [8] and Stong [17] defined _$c_{1}$ -spherical bordism_ $W$,
an intermediate theory between the $\mathit{SU}$\- and $U$-bordism, following
a similar construction of Wall in oriented bordism. The theory $W$ was a key
technical tool in Conner and Floyd’s calculation of torsion in $SU$-bordism.
Both [8] and [17] defined a multiplicative structure on $W$ using an
$\mathit{SU}$-linear projection $\pi\colon\mathit{MU}\to W$. Stong [17] showed
that the coefficient ring of the theory $W$ is polynomial with respect to the
multiplication defined by his projection. Although Conner–Floyd and Stong
defined their projections differently, in the subsequent literature on
$\mathit{SU}$\- and $c_{1}$-spherical bordism the two projections were used
interchangeably, suggesting that the two may coincide. As shown in [11, §6],
the Conner–Floyd and Stong projections are different, despite defining the
same multiplication on $W$ (see Example 2.20).

In Section 2 we give several descriptions of projections
$\pi\colon\mathit{MU}\to W$ and give the conditions specifying $SU$-linear
projections. We express the $SU$-linear Stong projection
$\pi_{0}\colon\mathit{MU}\to W$ as a series in operations $\partial_{i}$ via
the coefficients of the formal group law in complex cobordism (Proposition
2.9), and show that any other projection $\pi\colon\mathit{MU}\to W$ has the
form $\pi_{0}(1+f\varDelta)$ for some operation
$f\in[\mathit{MU},\Sigma^{-4}\mathit{MU}]$ (Theorem 2.14), where
$\varDelta\in[\mathit{MU},\Sigma^{4}\mathit{MU}]$ is the Conner–Floyd
operation satisfying $W=\mathop{\rm Ker}\varDelta$. In this description,
$\mathit{SU}$-linear projections correspond to $\mathit{SU}$-linear operations
$f$. In Theorem 2.21 we describe all $\mathit{SU}$-linear multiplications on
$c_{1}$-spherical bordism $W$, and specify the condition for the
multiplication to be defined by an $\mathit{SU}$-linear projection.

In Section 3 we study complex orientations of the theory $W$ and the
corresponding formal group laws. Any complex orientation
$w\in\widetilde{W}^{2}(\mathbb{C}P^{\infty})$ is obtained by applying an
$SU$-linear projection to an orientation
$\widetilde{u}\in\widetilde{\mathit{MU}}\vphantom{MU}^{2}(\mathbb{C}P^{\infty})$
of complex cobordism (Proposition 3.1). A multiplication on $W$ together with
a complex orientation $w$ define a formal group law $F_{W}\in W_{*}[[u,v]]$.
This formal group law was studied by Buchstaber in [6], where it was stated
that the coefficients of $F_{W}$ do not generate the ring $W_{*}$, unlike the
situation with complex cobordism. We give a complete proof of this claim in
Theorem 3.12. It is shown that the coefficients of $F_{W}$ do not generate
$W_{*}$ for any multiplication and complex orientation of $W$, not just for
the standard ones defined by the Stong projection. We also prove another
statement from [6]: after inverting $2$, the ring generated by the
coefficients of $F_{W}$ for some orientation $w$ coincides with
$W_{*}[\frac{1}{2}]$ (Theorem 3.13).

It would be interesting to give a geometric construction for the
multiplicative transformation (genus) $\mathit{MU}\to W$ classifying the
formal group law $F_{W}$, for example, in terms of the geometric generators of
$W_{*}$ and $\mathit{MSU}_{*}[\frac{1}{2}]$ described in [11, Part II]. A
related construction of polynomial generators of
$\mathit{MSU}_{*}[\frac{1}{2}]$ associated with the classifying maps for the
Abel, Buchstaber and Krichever formal group laws was recently given by
Bakuradze [3] (the Krichever formal group law gives rise to the
Krichever–Hoehn complex elliptic genus).

######  Acknowledgments.

The authors benefitted greatly from the advice and encouragement of Victor
Buchstaber. We thank Tom Bachmann for the inspiring discussion of
$\mathit{SU}$-linear operations in complex cobordism, and in particular for
his question on whether the geometric operations of Conner and Floyd form a
topological basis of $\mathit{SU}$-linear operations, which resulted in our
Theorem 1.12.

##  1\. $\mathit{SU}$-linear operations in complex cobordism

We are interested in cohomology operations and their linearity properties with
respect to module pairings in cohomology theories. Therefore, we work in the
stable homotopy category (see [1, 18, 12, 16, 4]) and _do not_ use any strict
models for spectra. All modules, rings and linear properties (as in Definition
1.1) below are considered in the homotopy sense. We only use the monoidal
structure of the stable homotopy category and _do not_ use any strict monoidal
model categories of spectra (see however the remark after Proposition 1.8).
Whenever we refer to a spectrum or a map (between spectra) it means an object
or a morphism in the stable homotopy category.

###  1.1. Equivalent definitions of $\mathit{SU}$-linearity

Given spectra $X$ and $Y$, we denote by $[X,Y]$ the set of morphisms (in the
stable homotopy category) between them, which is an abelian group, and denote
by $\pi_{*}(X)$ the homotopy groups of $X$.

######  Definition 1.1.

Let $R$ be a commutative ring spectrum, and let $E$ and $F$ be $R$-modules.
Consider the following properties of a morphism $f\in[E,F]$:

  * (a)

$f$ is _$R$ -linear_, that is, the following square is commutative (in the
stable homotopy category)

$\textstyle{R\wedge
E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1\wedge
f}$$\textstyle{R\wedge
F\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{F}$

  * (b)

$f$ is linear with respect to the multiplication by elements of $\pi_{*}(R)$,
that is, for any $r\in\pi_{k}(R)$ the following diagram of spectra is
commutative (in the stable homotopy category)

$\textstyle{\Sigma^{k}E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r\cdot}$$\scriptstyle{\Sigma^{k}f}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\Sigma^{k}F\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r\cdot}$$\textstyle{F}$

  * (c)

$\pi_{*}(f)\in\mathop{\mathrm{Hom}}\nolimits(\pi_{*}(E),\pi_{*}(F))$ is
$\pi_{*}(R)$-linear.

######  Proposition 1.2.

Property (a) implies (b), and (b) implies (c).

###### Proof.

(a)$\Rightarrow$(b): The diagram of (b) is expanded as follows:

$\textstyle{S^{k}\wedge
E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r\wedge
1}$$\scriptstyle{1\wedge f}$$\textstyle{R\wedge
E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1\wedge
f}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{S^{k}\wedge
F\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r\wedge
1}$$\textstyle{R\wedge
F\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F}$

The left square commutes, and the commutativity of the right square is
asserted by (a). So the whole diagram also commutes, which implies (b).

(b)$\Rightarrow$(c): For $r\in\pi_{k}(R)$ and $a\in\pi_{n}(E)$, the condition
$\pi_{*}(f)(ra)=r\pi_{*}(f)(a)$ is expressed as the commutativity of the outer
diagram:

---
$\textstyle{R\wedge
E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{S^{k}\wedge
S^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r\wedge
a}$$\scriptstyle{1\wedge a}$$\textstyle{F}$$\textstyle{S^{k}\wedge
E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r\wedge
f}$$\scriptstyle{r\wedge 1}$$\textstyle{R\wedge
F\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

Here the left triangle is commutative, and the commutativity of the right part
is asserted by (b). ∎

Alongside with $[E,F]$ we consider the graded abelian group $[E,F]_{*}$ with
graded components given by
$[E,F]_{k}=[\Sigma^{k}E,F]=[E,\Sigma^{-k}F]=F^{-k}(E)$, $k\in\mathbb{Z}$.

We are interested in $\mathit{SU}$-linear operations
$f\in[\mathit{MU},\mathit{MU}]_{*}$ in complex cobordism, which correspond to
$R=\mathit{MSU}$, $E=\mathit{MU}$ and $F=\Sigma^{k}\mathit{MU}$ in the
notation of Definition 1.1. In this case the three versions of $SU$-linearity
agree:

######  Theorem 1.3.

The three conditions in Definition 1.1 are equivalent for $\mathit{SU}$-linear
operations $f\in[\mathit{MU},\mathit{MU}]_{*}$ in complex cobordism.

The proof consists of three lemmata. First, there is the following useful
result relating a morphism of spectra to its action on the homotopy groups.

######  Lemma 1.4 ([16, Lemma VII.3.2]).

Let $E$ and $F$ be connective spectra of finite type such that $H_{*}(E)$ and
$\pi_{*}(F)$ are torsion-free. Then the natural map

$p\colon[E,F]\to\mathop{\mathrm{Hom}}\nolimits(\pi_{*}(E),\pi_{*}(F))$

is injective.

Lemma 1.4 can be generalised to morphisms involving three spectra. Given three
spectra $E,F,G$, there is a natural map $q\colon[E\wedge
F,G]\to\mathop{\mathrm{Hom}}\nolimits(\pi_{*}(E)\otimes\pi_{*}(F),\pi_{*}(G))$
defined as follows: for $f\in[E\wedge F,G]$ and
$a\in\pi_{k}(E),\,b\in\pi_{n}(F)$, the element $q(f)(a,b)\in\pi_{k+n}(G)$ is
represented by the map $S^{k}\wedge S^{n}\xrightarrow{a\wedge b}E\wedge
F\xrightarrow{f}G$.

######  Lemma 1.5.

Let $E$, $F$ and $G$ be connective spectra of finite type such that
$H_{*}(E)$, $H_{*}(F)$ and $\pi_{*}(G)$ are torsion-free. Then the natural map

$q\colon[E\wedge
F,G]\to\mathop{\mathrm{Hom}}\nolimits(\pi_{*}(E)\otimes\pi_{*}(F),\pi_{*}(G))$

is injective.

###### Proof.

Applying Lemma 1.4 to the spectra $E\wedge F$ and $G$ we obtain that the map

$[E\wedge F,G]\to\mathop{\mathrm{Hom}}\nolimits(\pi_{*}(E\wedge
F),\pi_{*}(G))$

is injective. We therefore need to prove the injectivity of the map

$\mathop{\mathrm{Hom}}\nolimits(\pi_{*}(E\wedge
F),\pi_{*}(G))\to\mathop{\mathrm{Hom}}\nolimits(\pi_{*}(E)\otimes\pi_{*}(F),\pi_{*}(G))$

induced by the natural map $\pi_{*}(E)\otimes\pi_{*}(F)\to\pi_{*}(E\wedge F)$.

We have the following commutative square

$\textstyle{\mathop{\mathrm{Hom}}\nolimits(\pi_{*}(E\wedge
F),\pi_{*}(G))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathop{\mathrm{Hom}}\nolimits(\pi_{*}(E)\otimes\pi_{*}(F),\pi_{*}(G))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathop{\mathrm{Hom}}\nolimits(\pi_{*}(E\wedge
F)\otimes\mathbb{Q},\pi_{*}(G)\otimes\mathbb{Q})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathop{\mathrm{Hom}}\nolimits(\pi_{*}(E)\otimes\pi_{*}(F)\otimes\mathbb{Q},\pi_{*}(G)\otimes\mathbb{Q})}$

Here the left map is the composite of the map
$\mathop{\mathrm{Hom}}\nolimits(\pi_{*}(E\wedge
F),\pi_{*}(G))\to\mathop{\mathrm{Hom}}\nolimits(\pi_{*}(E\wedge
F),\pi_{*}(G)\otimes\mathbb{Q})$ induced by the homomorphism
$\pi_{*}(G)\to\pi_{*}(G)\otimes\mathbb{Q}$, which is injective since
$\pi_{*}(G)$ is torsion-free, and the natural isomorphism
$\mathop{\mathrm{Hom}}\nolimits(\pi_{*}(E\wedge
F),\pi_{*}(G)\otimes\mathbb{Q})\cong\mathop{\mathrm{Hom}}\nolimits(\pi_{*}(E\wedge
F)\otimes\mathbb{Q},\pi_{*}(G)\otimes\mathbb{Q})$. Hence, the left map in the
diagram above is injective.

The bottom map is induced by the map
$\pi_{*}(E)\otimes\pi_{*}(F)\otimes\mathbb{Q}\to\pi_{*}(E\wedge
F)\otimes\mathbb{Q}$, which is an isomorphism. Indeed, it is a natural
transformation of homology theories
$\pi_{*}(-)\otimes\pi_{*}(F)\otimes\mathbb{Q}$ and $\pi_{*}(-\wedge
F)\otimes\mathbb{Q}$, which is an isomorphism on the sphere spectrum. Hence,
the bottom map in the diagram above is an isomorphism.

It follows that the top map in the commutative square above is injective, as
needed. ∎

######  Remark.

By induction, the statement of Lemma 1.5 can be generalised to the smash
product of an arbitrary number of spectra.

######  Lemma 1.6.

In the notation of Definition 1.1, let $R$, $E$ and $F$ be connective spectra
of finite type such that $H_{*}(R)$, $H_{*}(E)$ and $\pi_{*}(F)$ are torsion-
free. Then property (c) implies (a).

###### Proof.

Suppose $f\in[E,F]$ satisfies property (c) of Definition 1.1. Consider the
morphisms $\varphi_{1}\colon R\wedge E\xrightarrow{1\wedge f}R\wedge F\to F$,
$\varphi_{2}\colon R\wedge E\to E\xrightarrow{f}F$ and set
$\varphi=\varphi_{1}-\varphi_{2}$. We need to prove that $\varphi=0$.

Applying the map $q\colon[R\wedge
E,F]\to\mathop{\mathrm{Hom}}\nolimits(\pi_{*}(R)\otimes\pi_{*}(E),\pi_{*}(F))$
to $\varphi\colon R\wedge E\to F$ we obtain
$q(\varphi)(r,a)=rf_{*}(a)-f_{*}(ra)$, which is zero for any $r\in\pi_{*}(R)$,
$a\in\pi_{*}(E)$ by property (c). Then Lemma 1.5 implies that $\varphi=0$. ∎

###### Proof of Theorem 1.3.

The result follows by setting $R=\mathit{MSU}$, $E=\mathit{MU}$ and
$F=\Sigma^{k}\mathit{MU}$ in Lemma 1.6. ∎

###  1.2. Description of $\mathit{SU}$-linear operations in complex cobordism

The following proposition is contained implicitly in Conner and Floyd [8,
Chapter III], and its real version in Atiyah [2, Proposition 4.1].

######  Proposition 1.7.

There is an equivalence of $\mathit{MSU}$-modules

$\mathit{MU}\simeq\mathit{MSU}\wedge\Sigma^{-2}\mathbb{C}P^{\infty}.$

###### Proof.

Since $\mathbb{C}P^{\infty}=\mathit{MU}(1)$, we have an $\mathit{MSU}$-module
map

$\mathit{MSU}\wedge\Sigma^{-2}\mathbb{C}P^{\infty}=\mathit{MSU}\wedge\Sigma^{-2}MU(1)\xrightarrow{1\wedge
i}\mathit{MSU}\wedge\mathit{MU}\to\mathit{MU}.$

It is a map of Thom spectra induced by the map of spaces

$\mathit{BSU}\times\mathit{BU}(1)\to\mathit{BU}\times\mathit{BU}\xrightarrow{\oplus}\mathit{BU},$

This composite map is a homotopy equivalence, with the homotopy inverse given
by

$\mathit{BU}\to\mathit{BSU}\times\mathit{BU}(1),\quad\eta\mapsto(\eta-\det\eta)\times\det\eta.$

Hence, it induces an equivalence of the corresponding Thom spectra. ∎

Given $R$-module spectra $E,F$, denote by $[E,F]_{R}$ the abelian group of
$R$-linear morphisms. For a free $R$-module $R\wedge X$, there is a natural
isomorphism between $[R\wedge X,F]_{R}$ and $[X,F]$, which is defined as
follows. A morphism $X\xrightarrow{f}F$ corresponds to an $R$-linear morphism
$R\wedge X\xrightarrow{1\wedge f}R\wedge F\to F$. Conversely, an $R$-linear
morphism $R\wedge X\xrightarrow{g}F$ corresponds to a morphism $X\simeq
S\wedge X\xrightarrow{e\wedge 1}R\wedge X\xrightarrow{g}F$, where $e\colon
S\to R$ is the unit of $R$.

We have $\mathit{MU}^{*}(\mathbb{C}P^{\infty})=\mathit{MU}^{*}[[u]]$, where
$\mathit{MU}^{*}=\mathit{MU}^{*}(\mathit{pt})$ is the cobordism ring of point,
$u=c^{\scriptscriptstyle
U}_{1}\in\widetilde{\mathit{MU}}\vphantom{MU}^{2}(\mathbb{C}P^{\infty})$ is
the canonical orientation (the universal first Conner–Floyd class) defined by
the hyperplane section $\mathbb{C}P^{\infty-1}\subset\mathbb{C}P^{\infty}$.
Elements of $\widetilde{\mathit{MU}}\vphantom{MU}^{*}(\mathbb{C}P^{\infty})$
are represented by power series $f(u)$ in $u$ with zero constant term.

######  Proposition 1.8.

The abelian group $[\mathit{MU},\mathit{MU}]_{\mathit{MSU},\,k}$ of
$\mathit{SU}$-linear operations is isomorphic to
$\widetilde{\mathit{MU}}\vphantom{MU}^{2-k}(\mathbb{C}P^{\infty})$. More
precisely, if
$u\in\widetilde{\mathit{MU}}\vphantom{MU}^{2}(\mathbb{C}P^{\infty})$ is the
canonical orientation in complex cobordism, then the map

$[\mathit{MU},\mathit{MU}]_{*}\to\widetilde{\mathit{MU}}\vphantom{MU}^{2-*}(\mathbb{C}P^{\infty}),\quad
f\mapsto f(u),$

becomes an isomorphism when restricted to the subgroup
$[\mathit{MU},\mathit{MU}]_{\mathit{MSU},*}$ of $SU$-linear operations.

###### Proof.

We have

$[\mathit{MU},\mathit{MU}]_{\mathit{MSU},*}\simeq[\mathit{MSU}\wedge\Sigma^{-2}\mathit{MU}(1),\mathit{MU}]_{\mathit{MSU},*}\simeq[\Sigma^{-2}\mathit{MU}(1),\mathit{MU}]_{*}=\widetilde{\mathit{MU}}\vphantom{MU}^{2-*}(\mathbb{C}P^{\infty}),$

where the first isomorphism follows from Proposition 1.7, and the second by
the discussion preceding Proposition 1.8. Under these isomorphisms, an
$\mathit{SU}$-linear operation $f\colon\mathit{MU}\to\Sigma^{-k}\mathit{MU}$
is mapped to the composite

$\Sigma^{-2}\mathit{MU}(1)\simeq
S\wedge\Sigma^{-2}\mathit{MU}(1)\xrightarrow{e\wedge
1}\mathit{MSU}\wedge\Sigma^{-2}\mathit{MU}(1)\xrightarrow{1\wedge
i}\mathit{MSU}\wedge\mathit{MU}\to\mathit{MU}\xrightarrow{f}\Sigma^{-k}MU$

Since $f$ is $\mathit{SU}$-linear, we can rewrite this composite by changing
the last two maps as follows:

$S\wedge\Sigma^{-2}\mathit{MU}(1)\xrightarrow{e\wedge
1}\mathit{MSU}\wedge\Sigma^{-2}\mathit{MU}(1)\xrightarrow{1\wedge
i}\mathit{MSU}\wedge\mathit{MU}\xrightarrow{1\wedge
f}\mathit{MSU}\wedge\Sigma^{-k}\mathit{MU}\to\Sigma^{-k}\mathit{MU}$

Since $e$ is the unit, the composite above is
$\Sigma^{-2}\mathit{MU}(1)\xrightarrow{i}\mathit{MU}\xrightarrow{f}\Sigma^{-k}\mathit{MU}$.
Now the map $\Sigma^{-2}\mathit{MU}(1)\xrightarrow{i}\mathit{MU}$ represents
the canonical orientation
$u\in\widetilde{\mathit{MU}}\vphantom{MU}^{2}(\mathbb{C}P^{\infty})$, so the
composite represents
$f(u)\in\widetilde{\mathit{MU}}\vphantom{MU}^{2-k}(\mathbb{C}P^{\infty})$, as
claimed. ∎

######  Remark.

Although we work in the stable homotopy category, the bordism spectra such as
$\mathit{MU}$ and $\mathit{MSU}$ have strictly commutative models. More
precisely, there is a symmetric monoidal category of strict
$\mathit{MSU}$-modules $\mathcal{M}_{\mathit{MSU}}$ and its homotopy (derived)
category $\mathcal{D}_{\mathit{MSU}}$ (see [9]). The same argument as in
Proposition 1.7 shows that there is an equivalence of strict
$\mathit{MSU}$-modules
$\mathit{MU}\cong\mathit{MSU}\wedge\Sigma^{-2}\mathbb{C}P^{\infty}$, which
together with the adjunction equivalence of [9, Proposition III.4.1] gives an
equivalence

$\mathcal{D}_{\mathit{MSU}}(\mathit{MU},\mathit{MU})\cong\mathcal{D}_{\mathit{MSU}}(\mathit{MSU}\wedge\Sigma^{-2}\mathbb{C}P^{\infty},\mathit{MU})\\\
\cong\mathcal{D}_{S}(\Sigma^{-2}\mathbb{C}P^{\infty},\mathit{MU})=[\Sigma^{-2}\mathbb{C}P^{\infty},\mathit{MU}].$

Hence, the result of Proposition 1.8 holds for the set of homotopy classes of
strict $\mathit{MSU}$-module self-maps of $\mathit{MU}$. Nevertheless, since
our primary motivation lies in the study of cohomology operations that are
$\mathit{SU}$-linear in the sense of pairings in cohomology theories, we
consider self-maps of $\mathit{MU}$ that are $\mathit{SU}$-linear only up to
homotopy.

By Proposition 1.8, a series
$f(u)\in\widetilde{\mathit{MU}}\vphantom{MU}^{*}(\mathbb{C}P^{\infty})$
determines uniquely an $\mathit{SU}$-linear operation $f$. Therefore, to
complete the description of $\mathit{SU}$-linear operations in complex
cobordism we need to describe the $\mathit{SU}$-linear operations acting on
$\mathit{MU}^{*}(\mathbb{C}P^{\infty})$ by $u\mapsto\sum_{i\geqslant
0}\lambda_{i}u^{i+1}$, where $\sum_{i\geqslant
0}\lambda_{i}u^{i+1}\in\widetilde{\mathit{MU}}\vphantom{MU}^{2-2k}(\mathbb{C}P^{\infty})$
is an arbitrary series with $\lambda_{i}\in\mathit{MU}^{-2i-2k}$. The required
operations were introduced by Novikov in [14]. We recall their definition
following [11].

The stable Thom isomorphism

$\varphi^{*}\colon\mathit{MU}^{n}(\mathit{BU})\xrightarrow{\cong}\mathit{MU}^{n}(\mathit{MU})=[\mathit{MU},\mathit{MU}]_{-n}$

identifies universal characteristic classes with cohomological operations in
complex cobordism.

Given two complex line bundles $\xi$, $\eta$ with $u=c^{\scriptscriptstyle
U}_{1}(\xi)$, $v=c^{\scriptscriptstyle U}_{1}(\eta)$, the first Conner–Floyd
class of their tensor product is expressed as a power series in $u,v$ with
coefficients in $\mathit{MU}^{*}$, known as the _formal group law of geometric
cobordisms_ [14], [7], or simply the _formal group law in complex cobordism_ :

(1.1) $F_{U}(u,v)=c^{\scriptscriptstyle
U}_{1}(\xi\otimes\eta)=u+v+\sum_{i\geqslant 1,\,j\geqslant
1}\alpha_{ij}u^{i}v^{j},\quad\alpha_{ij}\in\mathit{MU}^{-2(i+j-1)}.$

We denote by $\overline{u}$ the inverse series of $u$ with respect to $F_{U}$;
it satisfies $F_{U}(u,\overline{u})=0$.

######  Construction 1.9 (operations $\varDelta_{(k_{1},k_{2})}$).

Consider the universal characteristic class $d^{\scriptscriptstyle
U}\in\mathit{MU}^{2}(BU)$ given by $d^{\scriptscriptstyle
U}(\xi)=c^{\scriptscriptstyle U}_{1}(\det\xi)$. We also set
$\overline{d}\phantom{d}\\!\\!\\!^{\scriptscriptstyle U}=c^{\scriptscriptstyle
U}_{1}(\overline{\det\xi})$, which can be identified with the inverse series
of $d^{\scriptscriptstyle U}$ and satisfies $F_{U}(d^{\scriptscriptstyle
U},\overline{d}\phantom{d}\\!\\!\\!^{\scriptscriptstyle U})=0$.

Given non-negative integers $k_{1}$, $k_{2}$, define the operation

$\varDelta_{(k_{1},k_{2})}=\varphi^{*}\bigl{(}(\overline{d}\phantom{d}\\!\\!\\!^{\scriptscriptstyle
U})^{k_{1}}(d^{\scriptscriptstyle
U})^{k_{2}}\bigr{)}\in[\mathit{MU},\mathit{MU}]_{-2(k_{1}+k_{2})}$

Geometrically, the operation $\varDelta_{(k_{1},k_{2})}$ takes a bordism class
$[M]\in\mathit{MU}_{*}$ to the class $[M_{k_{1},k_{2}}]$ of a submanifold dual
to $(\det\mathcal{T}M)^{\oplus
k_{1}}\oplus(\overline{\det\mathcal{T}M})^{\oplus k_{2}}$.

The action of $\varDelta_{(k_{1},k_{2})}$ on the canonical orientation
$u\in\widetilde{\mathit{MU}}\vphantom{MU}^{2}(\mathbb{C}P^{\infty})$ is given
by

(1.2) $\varDelta_{(k_{1},k_{2})}u=u\,\overline{u}^{k_{1}}u^{k_{2}}$

######  Proposition 1.10.

The operation $\varDelta_{(k_{1},k_{2})}$ is $\mathit{SU}$-linear.

###### Proof.

By Theorem 1.3 it is enough to check that $\varDelta_{(k_{1},k_{2})}$ is
$SU$-linear on the homotopy groups $\pi_{*}(\mathit{MU})=\mathit{MU}_{*}$.
Take bordism classes $[M]\in\mathit{MU}_{m}$, $[N]\in\mathit{MSU}_{n}$. We
need to prove that $\varDelta_{(k_{1},k_{2})}([N\times
M])=[N]\cdot\varDelta_{(k_{1},k_{2})}([M])$.

By definition,

$\varDelta_{(k_{1},k_{2})}([N\times M])=\varepsilon
D_{U}\bigl{(}\bigl{(}c^{\scriptscriptstyle U}_{1}(\det\mathcal{T}(N\times
M))\bigr{)}^{k_{1}}\bigl{(}c^{\scriptscriptstyle
U}_{1}(\overline{\det\mathcal{T}(N\times M)})\bigr{)}^{k_{2}}\bigr{)},$

where $D_{U}\colon\mathit{MU}^{*}(N\times
M)\xrightarrow{\simeq}\mathit{MU}_{n+m-*}(N\times M)$ is the Poincaré–Atiyah
duality isomorphism and $\varepsilon\colon\mathit{MU}_{*}(N\times
M)\to\mathit{MU}_{*}(\mathit{pt})$ is the augmentation. We have
$\det\mathcal{T}(N\times
M)=\det\mathcal{T}N\otimes\det\mathcal{T}M=\det\mathcal{T}M$, as $N$ is an
$\mathit{SU}$-manifold. Hence,

$\varDelta_{(k_{1},k_{2})}([N\times M])=\varepsilon
D_{U}\bigl{(}\bigl{(}c^{\scriptscriptstyle
U}_{1}(\det\mathcal{T}(M))\bigr{)}^{k_{1}}\bigl{(}c^{\scriptscriptstyle
U}_{1}(\overline{\det\mathcal{T}(M)})\bigr{)}^{k_{2}}\bigr{)}.$

Clearly, the submanifold dual to $\bigl{(}c^{\scriptscriptstyle
U}_{1}(\det\mathcal{T}(M))\bigr{)}^{k_{1}}\bigl{(}c^{\scriptscriptstyle
U}_{1}(\overline{\det\mathcal{T}(M)})\bigr{)}^{k_{2}}$ in the product $N\times
M$ is $N\times\varDelta_{(k_{1},k_{2})}([M])$, which implies the statement. ∎

We denote

$\varDelta=\varDelta_{(1,1)},\quad\partial_{k}=\varDelta_{(k,0)},\quad\partial=\partial_{1},\quad\overline{\partial}_{k}=\varDelta_{(0,k)}.$

Note that $\partial_{0}$ is the identity operation. It follows from definition
that each operation $\varDelta_{(k_{1},k_{2})}$ can be expressed as a series
in $\partial_{k}$ (as well as in $\overline{\partial}_{k}$).

######  Lemma 1.11.

Under the isomorphism of Proposition 1.8, a power series $\sum_{i\geqslant
0}\lambda_{i}u^{i+1}\in\widetilde{\mathit{MU}}\vphantom{MU}^{2-2k}(\mathbb{C}P^{\infty})$
corresponds to the operation $\sum_{i\geqslant
0}\lambda_{i}\overline{\partial}_{i}$, $\lambda_{i}\in\mathit{MU}^{-2i-2k}$.

###### Proof.

Indeed, we have
$\sum\lambda_{i}\overline{\partial}_{i}(u)=\sum\lambda_{i}u^{i+1}$ by formula
(1.2). ∎

Now we can formulate the main result of the section.

######  Theorem 1.12.

Any $\mathit{SU}$-linear operation
$f\in[\mathit{MU},\mathit{MU}]_{\mathit{MSU},*}$ can be written uniquely as a
series $f=\sum_{i\geqslant 0}\mu_{i}\partial_{i}$, where
$\mu_{i}\in\mathit{MU}^{-2i-*}$.

###### Proof.

It follows from Proposition 1.8 and Lemma 1.11 that any $\mathit{SU}$-linear
operation $f$ can be represented as a series
$\sum\lambda_{i}\overline{\partial}_{i}$. Since $\overline{\partial}_{i}$ is a
series in $\partial_{i}$, we can also write $f$ as a series
$\sum\mu_{i}\partial_{i}$. The coefficients $\lambda_{i}$ and $\mu_{i}$ are
determined uniquely by the action of $f$ on the canonical orientation
$u\in\widetilde{\mathit{MU}}\vphantom{MU}^{2}(\mathbb{C}P^{\infty})$. Namely,
$f(u)=\sum_{i\geqslant 0}\lambda_{i}u^{i+1}=\sum_{i\geqslant
0}\mu_{i}u\overline{u}^{i}$ by formula (1.2). ∎

Theorem 1.12 is readily extended to $\mathit{SU}$-multilinear operations:

######  Theorem 1.13.

Any $\mathit{SU}$-multilinear operation in complex cobordism can be written
uniquely as a series
$\sum\mu_{i_{1},\ldots,i_{k}}\partial_{i_{1}}\cdot\ldots\cdot\partial_{i_{k}}$.

This result will be used when we study $\mathit{SU}$-bilinear multiplications
in Subsection 2.3.

The product structure (with respect to the composition) of
$\mathit{SU}$-linear operations $f=\sum_{i\geqslant 0}\beta_{i}\partial_{i}$
can be described via the coefficients of the formal group law in complex
cobordism, as described next.

For an integer $k\in\mathbb{Z}$, the _$k$ -th power_ $[k](u)$ in the formal
group law $F_{U}$ is defined inductively:

$[0](u)=0,\quad[k](u)=F\bigl{(}[k-1](u),u\bigr{)}\text{ for
}k>0,\quad[k](u)=F\bigl{(}[k+1](u),\overline{u}\bigr{)}\text{ for }k<0.$

For a complex line bundle $\xi$, we have $[k](c^{\scriptscriptstyle
U}_{1}(\xi))=c^{\scriptscriptstyle U}_{1}(\xi^{\otimes k})$ and
$[-k](c^{\scriptscriptstyle U}_{1}(\xi))=c^{\scriptscriptstyle
U}_{1}(\overline{\xi}^{\otimes k})$, $k\geqslant 0$.

We can now describe the product structure of $\mathit{SU}$-linear operations.

######  Theorem 1.14.

Given non-negative integers $k,m$, consider the two power series

$(F(u,v))^{k}=\sum_{i,j\geqslant
0}\alpha_{ij}^{(k)}u^{i}v^{j},\quad\bigl{(}[1-m](u)\bigr{)}^{k}u^{m}=\sum_{i\geqslant
0}\beta_{i}^{(k,m)}u^{i}.$

Then

(1.3) $\partial_{k}(a\cdot
b)=\sum_{i,j}\alpha_{ij}^{(k)}\,\partial_{i}a\cdot\partial_{j}b,\quad\partial_{k}\partial_{m}=\sum_{i}\beta_{i}^{(k,m)}\partial_{i},$

where $a,b\in[\mathit{MU},\mathit{MU}]_{*}$ are arbitrary operations, and
$a\cdot b\in[\mathit{MU}\wedge\mathit{MU},\mathit{MU}]_{*}$ denotes the
exterior product.

###### Proof.

By Lemmata 1.4 and 1.5 it is enough to verify the identities on the homotopy
groups of the $\mathit{MU}$-spectrum, that is, on elements of
$\mathit{MU}_{*}=\pi_{*}(\mathit{MU})$.

Let $a=[M],\,b=[N]\in\mathit{MU}_{*}$. We denote $u=c^{\scriptscriptstyle
U}_{1}(\det\mathcal{T}M)$, $v=c^{\scriptscriptstyle U}_{1}(\det\mathcal{T}N)$.
The first identity follows by calculation

$\partial_{k}([M\times N])=\varepsilon
D_{U}\bigl{(}\bigl{(}c^{\scriptscriptstyle U}_{1}(\det\mathcal{T}(M\times
N))\bigr{)}^{k}\bigr{)}=\varepsilon D_{U}\bigl{(}\bigl{(}c^{\scriptscriptstyle
U}_{1}(\det\mathcal{T}M\otimes\det\mathcal{T}N)\bigr{)}^{k}\bigr{)}\\\
=\varepsilon D_{U}\bigl{(}(F(u,v))^{k}\bigr{)}=\varepsilon
D_{U}\Bigl{(}\sum_{i,j}\alpha^{(k)}_{ij}u^{i}v^{j}\Bigr{)}=\sum_{i,j}\alpha^{(k)}_{ij}\partial_{i}([M])\partial_{j}([N]),$

where the last equality follows from the fact that the submanifold
$\partial_{i}(M)\times\partial_{j}(N)$ is dual to $u^{i}v^{j}$ in the product
$M\times N$.

For the second identity, let $\partial_{m}([M])=[N]$, where $i\colon
N\hookrightarrow M$ is an embedding. The normal bundle of $N$ is isomorphic to
$i^{*}(\det\mathcal{T}M)^{\oplus m}$, therefore,

$\det\mathcal{T}N\otimes i^{*}(\det\mathcal{T}M)^{\otimes
m}=\det\bigl{(}\mathcal{T}N\oplus i^{*}(\det\mathcal{T}M)^{\oplus
m}\bigr{)}\cong i^{*}\det\mathcal{T}M.$

Hence, $\det\mathcal{T}N=i^{*}(\overline{\det\mathcal{T}M})^{\otimes(m-1)}$.
Denoting $u=c^{\scriptscriptstyle U}_{1}(\det\mathcal{T}M)$, we obtain
$c^{\scriptscriptstyle U}_{1}(\det\mathcal{T}N)=i^{*}[1-m](u)$. On the other
hand, the submanifold $N$ is dual to $u^{m}$, that is,
$i_{*}[N]=D_{U}(u^{m})=u^{m}\frown[M]$. Then the following calculation proves
the second identity:

$\partial_{k}[N]=\bigl{\langle}\bigl{(}c^{\scriptscriptstyle
U}_{1}(\det\mathcal{T}N)\bigr{)}^{k},[N]\bigr{\rangle}=\bigl{\langle}i^{*}\bigl{(}[1-m](u)\bigr{)}^{k},[N]\bigr{\rangle}=\bigl{\langle}\bigl{(}[1-m](u)\bigr{)}^{k},i_{*}[N]\bigr{\rangle}\\\
=\bigl{\langle}\bigl{(}[1-m](u)\bigr{)}^{k},u^{m}\frown[M]\bigr{\rangle}=\bigl{\langle}\bigl{(}[1-m](u)\bigr{)}^{k}u^{m},[M]\bigr{\rangle}=\sum_{i}\beta_{i}^{(k,m)}\partial_{i}([M]).\qed$

The second relation in (1.3) implies $\partial_{k}\partial=0$, as
$\beta_{i}^{(k,1)}=0$. The identity $\partial_{k}\partial=0$ follows also from
the geometric definition of $\partial_{k}$.

Relations (1.3) can be used to express an arbitrary composition of operations
$f=\sum_{i}\lambda_{i}\partial_{i}$ as an operation of the same form.

##  2\. $c_{1}$-spherical bordism $W$, projections and multiplications

Here we consider the $c_{1}$-spherical bordism theory $W$, describe
$\mathit{SU}$-linear multiplications on it and $\mathit{SU}$-linear
projections $\mathit{MU}\to W$.

###  2.1. Definition and the $\mathit{MSU}$-module structure.

The theory $W$ of $c_{1}$-spherical bordism is defined geometrically as
follows (see [17, Chapter VIII]). We consider closed manifolds $M$ with a
_$c_{1}$ -spherical structure_, which consists of

  * –

a stably complex structure on the tangent bundle $\mathcal{T}M$;

  * –

a _$\mathbb{C}P^{1}$ -reduction_ of the determinant bundle, that is, a map
$f\colon M\to\mathbb{C}P^{1}$ and an equivalence
$f^{*}(\eta)\cong\det\mathcal{T}M$, where $\eta$ is the tautological bundle
over $\mathbb{C}P^{1}$.

This is a natural generalisation of an $\mathit{SU}$-structure, which can be
thought of as a “$\mathbb{C}P^{0}$-reduction”, that is, a trivialisation of
the determinant bundle. The corresponding bordism theory is called _$c_{1}$
-spherical bordism_ and is denoted by $W$.

As in the case of stable complex structures, a $c_{1}$-spherical complex
structure on the stable tangent bundle is equivalent to such a structure on
the stable normal bundle. There are forgetful transformations $\mathit{MSU}\to
W\to\mathit{MU}$.

Homotopically, a $c_{1}$-spherical structure on a stable complex bundle
$\xi\colon M\to\mathit{BU}$ is defined by a choice of lifting to a map $M\to
X$, where $X$ is the (homotopy) pullback:

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{C}P^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\xi}$$\textstyle{\mathit{BU}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\det}$$\textstyle{\mathbb{C}P^{\infty}}$

As $\mathbb{C}P^{\infty}$ is a topological abelian group, we can change the
pullback square above to the following:

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ast\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\xi}$$\textstyle{\mathit{BU}\times\mathbb{C}P^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\det-\,i}$$\textstyle{\mathbb{C}P^{\infty}}$

The Thom spectrum corresponding to the map $X\to\mathit{BU}$ defines the
bordism theory of manifolds with a $\mathbb{C}P^{1}$-reduction of the stable
normal bundle, that is, the theory $W$. We shall also denote this spectrum by
$W$.

######  Remark.

In order to get a $\mathbb{C}P^{1}$-reduction of the stable tangent bundle, we
need to replace the inclusion
$i\colon\mathbb{C}P^{1}\hookrightarrow\mathbb{C}P^{\infty}$ in the pullback
squares above by $-i$. Replacing the basepoint inclusion
$*\to\mathbb{C}P^{\infty}$ by the fibration
$S^{\infty}\to\mathbb{C}P^{\infty}$ we obtain the definition given in [17,
Chapter 8]:

$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S^{\infty}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathit{BU}\times\mathbb{C}P^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\det+i}$$\textstyle{\mathbb{C}P^{\infty}}$

The spectrum $W$ has a natural $\mathit{MSU}$-module structure. The forgetful
morphisms $\mathit{MSU}\to W\to\mathit{MU}$ are $\mathit{MSU}$-module maps.
The following description of the $\mathit{MSU}$-module $W$ is contained
implicitly in the works of Atiyah [2], Conner and Floyd [8] and Stong [17]
(compare Proposition 1.7).

######  Proposition 2.1.

There is an equivalence of $\mathit{MSU}$-modules

$W\simeq\mathit{MSU}\wedge\Sigma^{-2}\mathbb{C}P^{2}.$

Under this equivalence, the forgetful maps $\mathit{MSU}\to W\to\mathit{MU}$
are identified with the free $\mathit{MSU}$-module maps

$\mathit{MSU}=\mathit{MSU}\wedge\Sigma^{-2}\mathbb{C}P^{1}\to\mathit{MSU}\wedge\Sigma^{-2}\mathbb{C}P^{2}\to\mathit{MSU}\wedge\Sigma^{-2}\mathbb{C}P^{\infty}.$

###### Proof.

Consider the commutative diagram

$\textstyle{\mathit{BSU}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{C}P^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{-i}$$\textstyle{\mathit{BSU}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathit{BU}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\det}$$\textstyle{\mathbb{C}P^{\infty}}$

where the rows are fibrations and the right square is pullback. The bottom
fibration is split (see Proposition 1.7). Hence, the top fibration is also
split, and we obtain a homotopy equivalence
$X\simeq\mathit{BSU}\times\mathbb{C}P^{1}$. The map $X\to\mathit{BU}$ is
identified with
$\mathop{\mathrm{id}}\times(-i)\colon\mathit{BSU}\times\mathbb{C}P^{1}\to\mathit{BSU}\times\mathbb{C}P^{\infty}$.
It induces an equivalence of the corresponding Thom spectra
$W\simeq\mathit{MSU}\wedge\Sigma^{-2}\mathbb{C}P^{2}$, as $\mathbb{C}P^{2}$ is
the Thom space of the tautological line bundle over $\mathbb{C}P^{1}$. ∎

######  Proposition 2.2.

The spectrum $W$ is the cofibre of the multiplication
$\Sigma\mathit{MSU}\xrightarrow{\cdot\theta}\mathit{MSU}$ by the nontrivial
element $\theta\in\mathit{MSU}_{1}\cong\mathbb{Z}_{2}$. The right arrow in the
resulting cofibre sequence

$\Sigma\mathit{MSU}\xrightarrow{\cdot\theta}\mathit{MSU}\to W$

is the forgetful map.

###### Proof.

The element $\theta$ is given by the Hopf map
$S^{3}\xrightarrow{\eta}S^{2}=\mathit{MSU}(1)$. The multiplication by $\theta$
is given by the map of spectra

$\mathit{MSU}\wedge\Sigma^{-2}S^{3}\xrightarrow{1\wedge\Sigma^{-2}\eta}\mathit{MSU}\wedge\Sigma^{-2}S^{2}.$

The cofibre of $S^{3}\xrightarrow{\eta}S^{2}$ is $\mathbb{C}P^{2}$. Hence, the
cofibre of the map above is equivalent to
$\mathit{MSU}\wedge\Sigma^{-2}\mathbb{C}P^{2}$, which coincides with the
spectrum $W$ by Proposition 2.1. ∎

######  Proposition 2.3.

The spectra $W$ and $\mathit{MSU}$ are Bousfield equivalent, that is,
$\mathit{MSU}_{*}(X)=0$ if and only if $W_{*}(X)=0$, and a map $X\to Y$
induces an isomorphism
$\mathit{MSU}_{*}(X)\xrightarrow{\cong}\mathit{MSU}_{*}(Y)$ if and only if it
induces an isomorphism $W_{*}(X)\xrightarrow{\cong}W_{*}(Y)$.

###### Proof.

This is a well-known property of the cofibre of a nilpotent map (see, for
example, the same argument in [15, Theorem 8.14] for the case of $K$-theory).
Consider the homology exact sequence of the cofibration from Proposition 2.2:

$\cdots\to\mathit{MSU}_{*-1}(X)\xrightarrow{\cdot\theta}\mathit{MSU}_{*}(X)\to
W_{*}(X)\to\mathit{MSU}_{*-2}(X)\xrightarrow{\cdot\theta}\mathit{MSU}_{*-1}(X)\to\cdots$

Clearly, $\mathit{MSU}_{*}(X)=0$ implies $W_{*}(X)=0$. For the opposite
direction, if $W_{*}(X)=0$, then
$\mathit{MSU}_{*-1}(X)\xrightarrow{\cdot\theta}\mathit{MSU}_{*}(X)$ is an
isomorphism. As $\theta^{3}\in MSU_{3}=0$ (see, for instance, [11, Example
5.7]), we obtain that $\mathit{MSU}_{*}(X)=0$.

The second assertion (about an isomorphism in homology) follows from the first
one by considering the homology exact sequence of the map $X\to Y$. ∎

######  Proposition 2.4.

The integral homology $H_{*}(W)$ is concentrated in even dimensions, and there
are the short exact sequences

$0\to H_{2k}(\mathit{MSU})\to H_{2k}(W)\to H_{2k-2}(\mathit{MSU})\to 0$

In particular, the homology $H_{*}(W)$ has no torsion.

###### Proof.

Consider the integral homology exact sequence of the cofibration from
Proposition 2.2:

$\ldots\to H_{2k-1}(\mathit{MSU})\to H_{2k}(\mathit{MSU})\to H_{2k}(W)\to
H_{2k-2}(\mathit{MSU})\to H_{2k-1}(\mathit{MSU})\to\ldots$

As the homology $H_{*}(\mathit{MSU})$ is concentrated in even dimensions and
has no torsion, the same is true for $W$. ∎

Recall from Section 1 that the operation
$\partial\colon\mathit{MU}\to\Sigma^{2}\mathit{MU}$ assigns to a bordism class
$[M]\in\mathit{MU}_{*}$ the bordism class of a submanifold dual to $c_{1}(M)$.

######  Proposition 2.5.

The composite

$W\xrightarrow{\partial^{\prime}}\Sigma^{2}\mathit{MSU}\to\Sigma^{2}\mathit{MU}$

of the connecting map in the cofibre sequence of Proposition 2.2 and the
forgetful map coincides with $-\partial\colon W\to\Sigma^{2}\mathit{MU}$.

###### Proof.

By Lemma 1.4, it is enough to verify the claim on the homotopy groups of
spectra, that is, to check that
$W_{2n}\xrightarrow{\partial^{\prime}}\mathit{MSU}_{2n-2}\to\mathit{MU}_{2n-2}$
coincides with $-\partial$. This is proved in [8, (17.3)] (Conner and Floyd
define $W_{*}$ as $\ker\varDelta$, see Proposition 2.10 below). ∎

Combining Proposition 2.2 and Proposition 2.5 we obtain the exact sequence

$\cdots\to\Sigma\mathit{MSU}\xrightarrow{\cdot\theta}\mathit{MSU}\to
W\xrightarrow{\partial^{\prime}}\Sigma^{2}\mathit{MSU}\xrightarrow{\cdot\theta}\Sigma\mathit{MSU}\to\cdots$

of Conner and Floyd [8]. For the homotopy groups of spectra we obtain a 5-term
exact sequence [8, (18.1)]

$0\longrightarrow\mathit{MSU}_{2n-1}\stackrel{{\scriptstyle\cdot\theta}}{{\longrightarrow}}\mathit{MSU}_{2n}\longrightarrow
W_{2n}\stackrel{{\scriptstyle\partial^{\prime}}}{{\longrightarrow}}\mathit{MSU}_{2n-2}\stackrel{{\scriptstyle\cdot\theta}}{{\longrightarrow}}\mathit{MSU}_{2n-1}\longrightarrow
0.$

###  2.2. Relation to the operation $\varDelta$

Recall that $W_{*}=\pi_{*}(W)$ denotes the homotopy groups (coefficients) of
the spectrum $W$.

######  Construction 2.6 ([17, Chapter VIII]).

Define a homomorphism $\pi_{0}\colon\mathit{MU}_{*}\to W_{*}$ sending a
bordism class $[M]$ to the class of the submanifold
$N\subset\mathbb{C}P^{1}\times M$ dual to
$\overline{\eta}\otimes\det\mathcal{T}M$. We have $\det\mathcal{T}N\cong
i^{*}\overline{\eta}$, where $i$ is the embedding
$N\hookrightarrow\mathbb{C}P^{1}\times M$, so $N$ has a natural
$c_{1}$-spherical stably complex structure.

######  Proposition 2.7 ([17, Chapter VIII]).

The composite $W_{*}\to\mathit{MU}_{*}\xrightarrow{\pi_{0}}W_{*}$ is the
identity map. In particular, the image of the forgetful homomorphism
$W_{*}\to\mathit{MU}_{*}$ is a direct summand of $\mathit{MU}_{*}$.

######  Corollary 2.8.

The groups $W_{*}$ are concentrated in even dimensions and are torsion-free.

We can also view $\pi_{0}$ as an idempotent homomorphism of abelian groups
$\mathit{MU}_{*}\to\mathit{MU}_{*}$ and refer to it as the _Stong projection_.

######  Proposition 2.9.

For any $a\in\mathit{MU}_{*}$ we have

(2.1) $\pi_{0}(a)=a+\sum_{k\geqslant 2}\alpha_{1k}\partial_{k}a,$

where $\alpha_{1k}$ are the coefficients of the formal group law $F_{U}$ in
complex cobordism (1.1). Furthermore,

$\partial\pi_{0}=\pi_{0}\partial=\partial.$

######  Remark.

By Lemma 1.4, formula (2.1) extends $\pi_{0}$ uniquely to a cohomological
operation from $[\mathit{MU},\mathit{MU}]$.

###### Proof of Proposition 2.9.

Let $a=[M]$. By the definition of $\pi_{0}$,

$\pi_{0}(a)=\varepsilon D_{U}(c^{\scriptscriptstyle
U}_{1}(\overline{\eta}\otimes\det\mathcal{T}M)),$

where $D_{U}\colon\mathit{MU}^{2}(\mathbb{C}P^{1}\times
M^{n})\xrightarrow{\cong}\mathit{MU}_{n}(\mathbb{C}P^{1}\times M^{n})$ is the
Poincaré–Atiyah duality isomorphism and
$\varepsilon\colon\mathit{MU}_{*}(X)\to\mathit{MU}_{*}(\mathit{pt})$ is the
augmentation.

Let $u=c^{\scriptscriptstyle U}_{1}(\overline{\eta})$,
$v=c^{\scriptscriptstyle
U}_{1}(\det\mathcal{T}M)\in\mathit{MU}^{2}(\mathbb{C}P^{1}\times M)$. Then

$\varepsilon D_{U}(c^{\scriptscriptstyle
U}_{1}(\overline{\eta}\otimes\det\mathcal{T}M))=\varepsilon
D_{U}(F(u,v))=\varepsilon D_{U}(u)+\varepsilon D_{U}(v)+\sum_{i,j\geqslant
1}\alpha_{ij}\varepsilon D_{U}(u^{i}v^{j})\\\
=[M]+[\mathbb{C}P^{1}]\partial[M]+\sum_{j\geqslant
1}\alpha_{1j}\partial_{j}[M],$

where we used the identities $u^{2}=0$, $\varepsilon
D_{U}(uv^{j})=\partial_{j}[M]$ and $\varepsilon
D_{U}(v)=[\mathbb{C}P^{1}]\partial[M]$. Formula (2.1) follows by noting that
$\alpha_{11}=-[\mathbb{C}P^{1}]$.

The identity $\pi_{0}\partial=\partial$ is obtained by applying (2.1) to
$\partial a$ and using the identity $\partial_{k}\partial=0$.

It remains to prove $\partial\pi_{0}=\partial$. Let $\pi_{0}[M]=[N]$. We need
to show that $\partial[N]=\partial[M]$. We have
$\det\mathcal{T}N=i^{*}\overline{\eta}$, where $i\colon
N\hookrightarrow\mathbb{C}P^{1}\times M$, and

$i_{*}[N]=D_{U}(c^{\scriptscriptstyle
U}_{1}(\overline{\eta}\otimes\det(\mathcal{T}M)))=D_{U}(F_{U}(u,v))=F_{U}(u,v)\frown[M\times\mathbb{C}P^{1}].$

Then the required identity follows by calculation:

$\partial[N]=\varepsilon D_{U}(c^{\scriptscriptstyle
U}_{1}(\det\mathcal{T}N))=\varepsilon D_{U}(i^{*}u)=\langle
i^{*}u,[N]\rangle=\langle u,i_{*}[N]\rangle\\\ =\langle
u,F_{U}(u,v)\frown[M\times\mathbb{C}P^{1}])=\langle
uF_{U}(u,v),[M\times\mathbb{C}P^{1}]\rangle=\langle
uv,[M\times\mathbb{C}P^{1}]\rangle=\partial[M].\qed$

######  Remark.

Formula (2.1) implies that the projection $\pi_{0}$ is $\mathit{SU}$-linear,
which is also clear from its geometric definition.

######  Proposition 2.10.

The image of the forgetful homomorphism $W_{*}\to\mathit{MU}_{*}$ coincides
with $\ker\varDelta$.

###### Proof.

The operation $\varDelta$ sends a bordism class $[M]$ to the class of the
submanifold $[N]$ dual to $\det\mathcal{T}M\oplus\overline{\det\mathcal{T}M}$.
For a $c_{1}$-spherical manifold $M$, the bundle $\det\mathcal{T}M$ is induced
from the tautological bundle $\eta$ over $\mathbb{C}P^{1}$. Since
$\eta\oplus\overline{\eta}$ is trivial over $\mathbb{C}P^{1}$, the operation
$\varDelta$ vanishes on the image of $W_{*}$.

Conversely, let $a\in\ker\varDelta$. By [11, Corollary 6.4], $\partial_{k}a=0$
for $k\geqslant 2$. Then (2.1) implies $\pi_{0}(a)=a$, so $a$ is in the image
of the forgetful homomorphism $W_{*}\to\mathit{MU}_{*}$. ∎

######  Remark.

The coefficient group $W_{*}$ was originally introduced by Conner and Floyd
[8] precisely as $\ker\varDelta$.

There is the following homotopical description of the spectrum $W$.

######  Proposition 2.11.

The spectrum $W$ is the fibre of the map
$\mathit{MU}\xrightarrow{\varDelta}\Sigma^{4}\mathit{MU}$.

###### Proof.

Denote the fibre by $F$. We have a long exact sequence of homotopy groups

$\cdots\to\pi_{*-3}(\mathit{MU})\to\pi_{*}(F)\to\pi_{*}(\mathit{MU})\xrightarrow{\varDelta}\pi_{*-4}(\mathit{MU})\to\pi_{*-1}(F)\to\cdots$

The operation $\varDelta$ has a right inverse (see [11, Lemma 4.3] and Example
2.20 below), so it is surjective. Therefore, the long exact sequence above
splits:

$0\to\pi_{*}(F)\to\pi_{*}(\mathit{MU})\xrightarrow{\varDelta}\pi_{*-4}(\mathit{MU})\to
0$

Proposition 2.10 implies that there are similar short exact sequences for
$W_{*}$:

$0\to\pi_{*}(W)\to\pi_{*}(\mathit{MU})\xrightarrow{\varDelta}\pi_{*-4}(\mathit{MU})\to
0$

This short exact sequence together with the fact that $H_{*}(W)$ is torsion-
free and Proposition 1.4 implies that the composite
$W\to\mathit{MU}\xrightarrow{\varDelta}\Sigma^{4}\mathit{MU}$ is homotopic to
zero. Hence, there is a morphism $W\to F$ making the following diagram
commutative

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{*}(W)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{*}(\mathit{MU})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varDelta}$$\textstyle{\pi_{*-4}(\mathit{MU})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{*}(F)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{*}(\mathit{MU})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varDelta}$$\textstyle{\pi_{*-4}(\mathit{MU})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

It follows that the map $W\to F$ induces an isomorphism of homotopy groups,
and therefore it is an equivalence of the spectra. ∎

######  Proposition 2.12.

For any space (or spectrum) $X$, the forgetful map
$W_{*}(X)\to\mathit{MU}_{*}(X)$ is injective and its image coincides with
$\ker\varDelta$. The same holds for $W^{*}(X)$.

###### Proof.

By Proposition 2.11 we obtain a long exact sequence

$\cdots\to
W_{*}(X)\to\mathit{MU}_{*}(X)\xrightarrow{\varDelta}\mathit{MU}_{*-4}(X)\to\cdots$

Since $\varDelta$ has a right inverse, this long exact sequence splits into
short ones

$\qquad\qquad 0\to
W_{*}(X)\to\mathit{MU}_{*}(X)\xrightarrow{\varDelta}\mathit{MU}_{*-4}(X)\to
0.\qquad\qquad\qed$

Propositions 2.10 and 2.12 were proved in [17, Chapter VIII] by geometric
methods.

######  Remark.

Proposition 2.11 also implies that the Stong projection
$\pi_{0}\in[\mathit{MU},\mathit{MU}]$ lifts uniquely to an operation
$\pi_{0}\in[\mathit{MU},W]$.

###  2.3. $\mathit{SU}$-linear projections onto $W$ and $\mathit{SU}$-linear
multiplications

A morphism $\mathit{MU}\to W$ is called a _projection onto $W$_ if it is
identical on $W$, where $W$ is viewed as a submodule of $\mathit{MU}$ via the
forgetful morphism $W\to\mathit{MU}$. We often view such projections as
idempotent morphisms $\mathit{MU}\to\mathit{MU}$ with image $W$. An example is
given by the Stong projection $\pi_{0}\colon\mathit{MU}\to W$.

Any such projection maps $\mathit{MU}_{*}(X)$ onto the direct summand
$W_{*}(X)=\ker\varDelta$, and similarly for $W^{*}(X)$. Moreover, Proposition
2.11 implies that any projection $\mathit{MU}\to W$ defines a splitting of the
complex cobordism spectrum $\mathit{MU}\simeq W\vee\Sigma^{4}\mathit{MU}$, and
the fibre sequence from Proposition 2.11 also splits.

The Stong projection is $SU$-linear, and therefore can be written as a series
in $\partial_{k}$. The coefficients of this series are given by (2.9). More
generally, we have

######  Proposition 2.13.

Any $\mathit{SU}$-linear projection $\mathit{MU}\to W$ has the form
$\pi=1+\sum_{i\geqslant 2}\lambda_{i}\partial_{i}$ with
$\lambda_{i}\in\mathit{MU}^{-2i}$.

###### Proof.

We write $\pi=\sum_{i\geqslant 0}\lambda_{i}\partial_{i}$ according to Theorem
1.12. Then $\pi(1)=1$ and $\pi([\mathbb{C}P^{1}])=[\mathbb{C}P^{1}]$, because
$[\mathbb{C}P^{1}]\in W_{2}$. Since $\partial[\mathbb{C}P^{1}]=2$ and
$\partial_{i}[\mathbb{C}P^{1}]=0$ for $i\geqslant 2$, we obtain
$\lambda_{0}=1$ and $\lambda_{1}=0$, as needed. ∎

######  Theorem 2.14.

Let $\pi\colon\mathit{MU}\to W$ be a projection onto $W$. Then any other
projection $\mathit{MU}\to W$ has the form $\pi(1+f\varDelta)$ for some
operation $f\in[\mathit{MU},\Sigma^{-4}\mathit{MU}]$. Furthermore, if $\pi$ is
$\mathit{SU}$-linear, then any other $\mathit{SU}$-linear projection has the
form $\pi(1+f\varDelta)$ with $SU$-linear $f$.

###### Proof.

The (split) fibre sequence of Proposition 2.11 gives the exact sequence

$\cdots\leftarrow[\Sigma^{3}\mathit{MU},W]\leftarrow[W,W]\leftarrow[\mathit{MU},W]\leftarrow[\Sigma^{4}\mathit{MU},W]\leftarrow\cdots$

A projection $\mathit{MU}\to W$ is an element of $[\mathit{MU},W]$ that maps
to the identity in $[W,W]$. Such a projection exists because
$[\Sigma^{3}\mathit{MU},W]=0$ (the homotopy groups $W_{*}$ are concentrated in
even dimensions). Furthermore, any two projections $\mathit{MU}\to W$ differ
by the image of an element from $[\Sigma^{4}\mathit{MU},W]$. That is, any
projection has the form $\pi+g\varDelta$, where
$g\in[\Sigma^{4}\mathit{MU},W]$. It remains to note that any
$g\in[\Sigma^{4}\mathit{MU},W]$ can be written as $\pi f$ for
$f\in[\Sigma^{4}\mathit{MU},\mathit{MU}]$. This proves the first statement.

Now suppose $\pi$ is an $SU$-linear projection. Then $SU$-linear operations
$f$ give $SU$-linear projections $\pi(1+f\varDelta)$. Conversely, if a
projection $\pi(1+f\varDelta)$ is $\mathit{SU}$-linear, then the operation
$\pi f\varDelta$ is also $SU$-linear. Denote by
$f^{\prime}\in[\Sigma^{4}\mathit{MU},\mathit{MU}]$ the composite of $\pi
f\in[\Sigma^{4}\mathit{MU},W]$ and the forgetful map $W\to\mathit{MU}$. Then
the operation $f^{\prime}\varDelta$ is $\mathit{SU}$-linear. Since $\varDelta$
has a right inverse, $f^{\prime}$ is also $SU$-linear. Now $\pi
f^{\prime}\varDelta=\pi f\varDelta$, so we can replace $f$ in
$\pi(1+f\varDelta)$ by an $SU$-linear operation $f^{\prime}$. ∎

######  Lemma 2.15.

The following three groups of $\mathit{SU}$-linear operations coincide:

  * (1)

$\mathit{SU}$-linear operations vanishing on $W$;

  * (2)

operations of the form $g\varDelta$ with $\mathit{SU}$-linear $g$;

  * (3)

operations $\sum_{i\geqslant 2}\lambda_{i}\partial_{i}$, $\lambda_{i}\in
MU_{*}$.

###### Proof.

Any operation $g\varDelta$ vanishes on $W$ by Proposition 2.11. On the other
hand, $\sum_{i\geqslant 2}\lambda_{i}\partial_{i}$ vanishes on $W$ by [11,
Corollary 6.4].

Conversely, if an operation vanishes on $W$, then it has the form $g\varDelta$
by Proposition 2.11. If $g\varDelta$ is $\mathit{SU}$-linear, then $g$ is also
$\mathit{SU}$-linear, as $\varDelta$ has a right inverse.

Finally, by Theorem 1.12, any $\mathit{SU}$-linear operation has the form
$\sum_{i\geqslant 0}\lambda_{i}\partial_{i}$. If it vanishes on $W$, then
calculating at $1\in\pi_{0}(W)$ and $[\mathbb{C}P^{1}]\in\pi_{2}(W)$ we obtain
$\lambda_{0}=\lambda_{1}=0$. ∎

######  Theorem 2.16.

Any projection $\mathit{MU}\to W$ has the form $1-f\varDelta$, where $f$ is an
arbitrary operation satisfying $\varDelta f=1$. Furthermore, different
projections correspond to different $f$, and $\mathit{SU}$-linear projections
correspond to $\mathit{SU}$-linear $f$.

###### Proof.

The split fibration sequence of Proposition 2.11 gives rise to a short exact
sequence

$0\leftarrow[W,\mathit{MU}]\leftarrow[\mathit{MU},\mathit{MU}]\leftarrow[\Sigma^{4}\mathit{MU},\mathit{MU}]\leftarrow
0,$

because $[\Sigma W,\mathit{MU}]=0$ and
$[\Sigma^{3}\mathit{MU},\mathit{MU}]=0$.

Let $p\in[\mathit{MU},\mathit{MU}]$ be a projection onto $W$. Since it is
identical on $W$, it maps to the forgetful morphism $W\to\mathit{MU}$. The
identity $1\in[\mathit{MU},\mathit{MU}]$ also maps to the forgetful morphism,
so we obtain $1-p=f\varDelta$ for some
$f\in[\Sigma^{4}\mathit{MU},\mathit{MU}]$, and different $f$ correspond to
different $p$. Hence, $p=1-f\varDelta$. It is a projection onto $W$ if and
only if $\varDelta(1-f\varDelta)=0$. As $\varDelta$ has a right inverse, we
obtain $1-\varDelta f=0$. Conversely, the latter condition implies that
$\varDelta(1-f\varDelta)=0$. The existence of a right inverse for $\varDelta$
also implies that $p$ is $\mathit{SU}$-linear if and only if so is $f$. ∎

An $SU$-linear projection $\pi\colon\mathit{MU}\to W$ defines an
$\mathit{SU}$-bilinear multiplication on $W$ by the formula

(2.2) $W\wedge
W\to\mathit{MU}\wedge\mathit{MU}\xrightarrow{m_{\mathit{MU}}}\mathit{MU}\xrightarrow{\pi}W.$

As $\pi$ is a projection, this multiplication has a unit, obtained from the
unit of $\mathit{MSU}$ by the forgetful morphism.

Given elements $a,b\in W_{*}$, we denote by $ab$ the product of their images
in $\mathit{MU}_{*}$ under the forgetful morphism.

######  Proposition 2.17.

The multiplication (2.2) corresponding to an $\mathit{SU}$-linear projection
$\pi=1+\sum_{i\geqslant 2}\lambda_{i}\partial_{i}$ is given by

$a*b=ab+2\lambda_{2}\partial a\partial b,$

This formula can be understood as an identity of the operations from $[W\wedge
W,W]_{*}$, or as an identity for cohomology classes $a,b\in[E,W]_{*}$, where
$E$ is an arbitrary spectrum.

In particular, the multiplication defined by the Stong projection
$\pi_{0}=1+\sum_{k\geqslant 2}\alpha_{1k}\partial_{k}$ is given by

$a*b=ab+2[V]\partial a\partial b.$

where $[V]=\alpha_{12}\in\mathit{MU}_{4}$ is the cobordism class
$[\mathbb{C}P^{1}]^{2}-[\mathbb{C}P^{2}]$.

###### Proof.

It is enough to verify the identity on the elements of $W_{*}=[S,W]_{*}$. We
use the formula from Proposition 1.14 and the fact that $\partial_{i}$
vanishes on $W_{*}$ for $i\geqslant 2$ (see [11, Corollary 6.4]):

$a*b=\pi(ab)=ab+\lambda_{2}\partial_{2}(ab)+\sum_{i\geqslant
3}\lambda_{i}\partial_{i}(ab)=ab+\lambda_{2}\alpha^{(2)}_{11}\partial
a\partial b=ab+2\lambda_{2}\partial a\partial b.\qed$

######  Lemma 2.18 (see [11, Lemma 6.5]).

For any elements $a,b\in W_{*}$,

$\displaystyle\partial(ab)=a\,\partial b+\partial
a\,b-[\mathbb{C}P^{1}]\partial a\,\partial b,$
$\displaystyle\varDelta(ab)=-2\partial a\,\partial b.$

Here is an alternative way to describe multiplications on $W$ corresponding to
$\mathit{SU}$-linear projections.

######  Proposition 2.19.

The multiplication (2.2) corresponding to an $\mathit{SU}$-linear projection
$\pi$ is given by the formula

$a*b=ab+2([V]-\omega)\partial a\partial b,$

where $[V]=\alpha_{12}=[\mathbb{C}P^{1}]^{2}-[\mathbb{C}P^{2}]$ and
$\omega=\pi[V]\in W_{4}$. Furthermore, any element of $W_{4}$ can be taken as
$\omega$ for some $\pi$.

###### Proof.

By Theorem 2.14, $\pi=\pi_{0}+\pi_{0}f\varDelta$ for an $SU$-linear
$f\in[\mathit{MU},\Sigma^{-4}\mathit{MU}]$. Then we use the formulae from
Proposition 2.17 and Lemma 2.18 to calculate

$a*b=\pi_{0}(ab)+\pi_{0}f\varDelta(ab)=ab+2[V]\partial a\partial
b-2\pi_{0}f(1)\partial a\partial b.$

In the last identity we used the fact that $\pi_{0}f$ is $SU$-linear. Clearly,
any element $\omega\in W_{4}$ can be taken as $\pi_{0}f(1)$, proving the
identity. Now we have

$a*b=\pi(a*b)=\pi(ab)+2\pi([V]-\omega)\partial a\partial
b=a*b+2\pi([V]-\omega)\partial a\partial b.$

Therefore, $\pi([V]-\omega)=0$ and $\pi[V]=\pi(\omega)=\omega$. ∎

######  Example 2.20.

Conner and Floyd [8] defined geometrically a right inverse to the operation
$\varDelta$ on the complex bordism groups $\mathit{MU}_{*}$. Novikov [8]
extended it to a cohomological operation
$\varPsi\in[\Sigma^{4}\mathit{MU},\mathit{MU}]$ (see [11, Construction 4.2]).
We therefore obtain an example of a projection of the form described in
Theorem 2.16, the _Conner–Floyd projection_ $1-\varPsi\varDelta$. As observed
in [11] this projection is different from the Stong projection $\pi_{0}$,
although the two projections define the same multiplication on $W$. This
reflects the fact that both Stong and Conner–Floyd projections have the same
coefficient of $\partial_{2}$ in their expansions $1+\sum_{i\geqslant
2}\lambda_{i}\partial_{i}$.

######  Theorem 2.21.

Any $\mathit{SU}$-bilinear multiplication on $W$ with the standard unit
(obtained by the forgetful map from the unit of $\mathit{MSU}$) has the form

$a*b=ab+(2[V]-\omega)\partial a\partial b$

for $\omega\in W_{4}$. Any such multiplication is associative and commutative.
Furthermore, the multiplications obtained from $\mathit{SU}$-linear
projections are those with $\omega=2\widetilde{\omega}$,
$\widetilde{\omega}\in W_{4}$.

###### Proof.

Let $m(a,b)$ be an arbitrary $\mathit{SU}$-bilinear operation on $W$. Given an
$\mathit{SU}$-linear projection $\pi\colon\mathit{MU}\to W$, we obtain an
$\mathit{SU}$-bilinear operation on $\mathit{MU}$ by composing
$m(\pi(x),\pi(y))$ with the forgetful homomorphism $W\to\mathit{MU}$. By
Theorem 1.13, such an operation can be written as a series in products of
$\partial_{i}$. Restricting back to $W$ and using the fact that $\partial_{i}$
vanishes on $W$ for $i\geqslant 2$, we obtain

$m(a,b)=\alpha\,ab+\beta\,\partial a\,b+\gamma\,a\,\partial b+\delta\,\partial
a\,\partial b.$

The unit identity $m(a,1)=a$ implies $\alpha a+\beta\partial a=a$.
Substituting $a=1$ and $a=[\mathbb{C}P^{1}]$ we obtain $\alpha=1$, $\beta=0$.
Similarly, $\gamma=0$. Finally, the multiplication takes values in $W$ if and
only if

$0=\varDelta m(a,b)=\varDelta(ab+\delta\,\partial a\,\partial b)=-2\partial
a\,\partial b+\varDelta\delta\,\partial a\,\partial b.$

Hence, $\varDelta\delta=2$. Since $\varDelta[V]=1$, this is equivalent to
$\delta=2[V]-\omega$, $\omega\in W_{4}$.

The multiplication $a*b$ is clearly commutative. For the associativity, we
have

$(a*b)*c=(ab+\delta\,\partial a\,\partial b)*c=(ab+\delta\,\partial
a\,\partial b)c+\delta\,\partial(ab+\delta\,\partial a\,\partial b)\,\partial
c.$

The operation $\partial$ is $\mathit{SU}$-linear and vanishes on $MU_{4}$,
hence, $\partial(\delta\,\partial a\,\partial b)=\partial\delta\,\partial
a\,\partial b=0$. We also have $\partial(ab)=a\,\partial b+b\,\partial
a-[\mathbb{C}P^{1}]\partial a\,\partial b$ by Lemma 2.18. Therefore,

$(a*b)*c=abc+\delta\,\partial a\,\partial b\,c+\delta\,a\,\partial b\,\partial
c+\delta\,b\,\partial a\,\partial c-\delta[\mathbb{C}P^{1}]\partial
a\,\partial b\,\partial c=a*(b*c).$

Finally, Proposition 2.19 implies that a multiplication obtained from an
$\mathit{SU}$-linear projection has $\omega$ divisible by $2$. ∎

We refer to [5] for a general algebraic approach to multiplications in complex
cobordism resulting from projections.

##  3\. Complex orientations of $W$ and formal group laws

This last section is motivated by the work [6] of Buchstaber. We start by an
observation that the $c_{1}$-spherical bordism theory $W$ is complex oriented
for any multiplication (2.2). Furthermore, any complex orientation of $W$ is
obtained from a complex orientation of $\mathit{MU}$ by applying an
$\mathit{SU}$-linear projection $\pi$. A complex orientation $w$ of $W$
defines a formal group law $F_{W}(u,v)$ in the theory $W$. Unlike the
situation with the complex bordism, the coefficients of the formal group law
$F_{W}$ _do not_ generate the coefficient ring $W_{*}$ of the theory $W$, for
any choice of $w$ and $\pi$. This result is stated in [6] with a short sketch
of proof (more details are provided in the case of the Stong projection
$\pi_{0}$). We give a complete proof using the technique developed in the
previous sections.

By Theorem 2.21, an $\mathit{SU}$-bilinear multiplication on $W$ is given by
the formula

(3.1) $a*b=ab+\delta\partial a\partial b,$

where $\delta=2[V]-\omega$, $\omega\in W_{4}$. As $\partial\delta=0$, we also
have

(3.2) $\partial(a*b)=\partial(ab)=a\,\partial b+b\,\partial
a-[\mathbb{C}P^{1}]\partial a\,\partial b.$

In what follows we fix an $\mathit{SU}$-bilinear multiplication on $W$.

######  Proposition 3.1.

The theory $W$ is complex orientable. For any $\mathit{SU}$-linear projection
$\pi\colon\mathit{MU}\to W$ and a complex orientation
$\widetilde{u}\in\widetilde{\mathit{MU}}\vphantom{MU}^{2}(\mathbb{C}P^{\infty})$,
the element $\pi(\widetilde{u})\in\widetilde{W}^{2}(\mathbb{C}P^{\infty})$ is
a complex orientation for $W$. Furthermore, for any complex orientation $w$ of
$W$ and any $\mathit{SU}$-linear projection $\pi\colon\mathit{MU}\to W$, there
exists a complex orientation
$\widetilde{u}\in\widetilde{\mathit{MU}}\vphantom{MU}^{2}(\mathbb{C}P^{\infty})$
such that $w=\pi(\widetilde{u})$.

###### Proof.

Let $u\in\widetilde{\mathit{MU}}\vphantom{MU}^{2}(\mathbb{C}P^{\infty})$ be
the canonical orientation of complex cobordism. Then
$\widetilde{u}|_{\mathbb{C}P^{1}}=u|_{\mathbb{C}P^{1}}$. Hence,
$\pi(\widetilde{u})|_{\mathbb{C}P^{1}}=\pi(u)|_{\mathbb{C}P^{1}}=u|_{\mathbb{C}P^{1}}$
by Proposition 2.13, as $\partial_{i}u|_{\mathbb{C}P^{1}}=0$ for $i\geqslant
1$. This implies that $\pi(\widetilde{u})$ is a complex orientation of $W$.

Conversely, given an orientation
$w\in\widetilde{W}^{2}(\mathbb{C}P^{\infty})$, its image under the forgetful
map
$\widetilde{W}^{2}(\mathbb{C}P^{\infty})\to\widetilde{\mathit{MU}}\vphantom{MU}^{2}(\mathbb{C}P^{\infty})$
is a complex orientation $\widetilde{u}$ of $\mathit{MU}$. Therefore,
$w=\pi(\widetilde{u})$ for any $\mathit{SU}$-linear projection
$\pi\colon\mathit{MU}\to W$. ∎

######  Remark.

As one can see from the proof, the statement holds because a complex
orientation of a cohomology theory is defined via the unit of the theory. The
forgetful map $W\to\mathit{MU}$ and projections $\mathit{MU}\to W$ preserve
the standard unit (from $\mathit{MSU}$) and therefore send orientations to
orientations.

A complex orientation $w\in\widetilde{W}^{2}(\mathbb{C}P^{\infty})$ defines a
formal group law with coefficients in $W^{*}$, which we denote by $F_{W}(u,v)$
(it depends on both the multiplication and $w$, but we do not reflect this in
the notation). For example, we may take $w=\pi_{0}(u)$, where
$u\in\widetilde{\mathit{MU}}\vphantom{MU}^{2}(\mathbb{C}P^{\infty})$ is the
canonical orientation and $\pi_{0}$ is the Stong projection. The formal group
law $F_{W}$ is classified by a multiplicative transformation
$\psi\colon\mathit{MU}\to W$ sending $u$ to $w$. However, even in the case
$w=\pi_{0}(u)$, the transformation $\psi$ does not coincide with the
projection $\pi_{0}\colon\mathit{MU}\to W$, as the latter is _not_
multiplicative. To study the formal group law $F_{W}$ we map $W$ further to a
one-parameter extension of the $U$-theory, as described next.

######  Construction 3.2.

Following [6], consider the multiplicative cohomology theory $\varGamma$
defined by

$\varGamma^{*}(X)=\mathit{MU}^{*}(X)[t]/(t^{2}=-[\mathbb{C}P^{1}]t+\delta)$

for any CW-complex $X$. Additively, $\varGamma^{*}(X)$ is a free
$\mathit{MU}^{*}(X)$-module with basis $\\{1,t\\}$, and the multiplication is
defined by the relation $t^{2}=-[\mathbb{C}P^{1}]t+\delta$.

There is a natural transformation $\varphi\colon W\to\varGamma$ given by
$\varphi(x)=x+t\partial x$.

######  Proposition 3.3 ([6, Lemma 2]).

The transformation $\varphi\colon W\to\varGamma$ is multiplicative.

###### Proof.

Given $a,b\in W_{*}$, we use (3.1) and (3.2) to calculate

$(a+t\partial a)(b+t\partial b)=ab+t(a\partial b+b\partial a)+t^{2}\partial
a\partial b=\\\ =a*b+t\partial(a*b)+(t^{2}+t[\mathbb{C}P^{1}]-\delta)\partial
a\partial b=a*b+t\partial(a*b).\qed$

There is a canonical orientation of the theory $\varGamma$ given by the image
of the canonical orientation
$u\in\widetilde{\mathit{MU}}\vphantom{MU}^{2}(\mathbb{C}P^{\infty})$ under the
natural inclusion $\mathit{MU}\hookrightarrow\varGamma$. This orientation of
$\varGamma$ will be also denoted by $u$.

The map $\varphi$ sends the orientation $w$ to an orientation $\varphi(w)$ of
the theory $\varGamma$. Hence, $\varphi(w)$ can be written as a power series
$\gamma(u)$ in $u$ with coefficients in $\varGamma^{*}=\varGamma^{*}(pt)$.

######  Proposition 3.4 ([6, Lemma 3]).

We have

$\varphi_{*}F_{W}(u,v)=\gamma F_{U}(\gamma^{-1}(u),\gamma^{-1}(v)),$

where $F_{U}(u,v)$ is the formal group law in complex cobordism, considered as
a formal group law over $\varGamma^{*}$ via the natural inclusion
$\mathit{MU}\hookrightarrow\varGamma$.

###### Proof.

Since $\varphi$ is multiplicative, the formal group law corresponding to the
orientation $\varphi(w)$ is $\varphi_{*}F_{W}$. Similarly, since the inclusion
$\mathit{MU}\hookrightarrow\varGamma$ is also multiplicative, the formal group
law corresponding to the orientation $u$ is $F_{U}$, regarded as a formal
group law over $\varGamma^{*}$. Now $\varphi(w)=\gamma(u)$ implies the
required identity. ∎

######  Construction 3.5.

Denote by $J=\mathit{MU}^{<0}\subset\mathit{MU}^{*}$ the ideal of elements of
non-zero degree. Then $J^{2}$ is the ideal of decomposables in
$\mathit{MU}^{*}$. Clearly, $J^{2}+tJ$ is an ideal in $\varGamma^{*}$.

Consider the quotient ring $R=\varGamma^{*}/(J^{2}+tJ)$. As a graded abelian
group, $R=(\mathit{MU}^{*}\\!/J^{2})\oplus\mathbb{Z}\langle t\rangle$, $\deg
t=-2$. The multiplication in $R$ is defined by $ab=0$, $at=0$ for $a,b\in
J/J^{2}$ and $t^{2}=\delta$, so that $t^{3}=0$. This implies
$R^{<-2}R^{<0}=0$.

We write

$F_{W}(u,v)=u+v+\sum_{i\geqslant 1,\,j\geqslant 1}\omega_{ij}u^{i}v^{j}.$

In order to compare the subring generated by the coefficients $\omega_{ij}$
with the whole ring $W^{*}$ we need to identify the characteristic
$s_{k}$-numbers of $\omega_{ij}$. (Recall that $s_{k}$ is the Chern
characteristic number corresponding to the symmetric polynomial
$t_{1}^{k}+\cdots+t_{n}^{k}$ in Chern roots; it vanishes on decomposables
$J^{2}\subset\mathit{MU}^{*}$.)

We shall calculate the formal group law
$\varphi_{*}F_{W}(u,v)=u+v+\sum(\omega_{ij}+t\partial\omega_{ij})u^{i}v^{j}$
over the ring $R$ (that is, reducing the coefficents mod $J^{2}+tJ$), using
the formula from Proposition 3.4. As the $s_{k}$-numbers vanish on $J^{2}$, in
this way we obtain information about the $s_{k}$-numbers of the coefficients
of the formal group law $F_{W}$.

######  Lemma 3.6.

The following identity holds in $\varGamma^{*}$:

$\gamma(u)=u-(\lambda+(2\ell+1)t)u^{2}+\sum_{i\geqslant
2}\gamma_{i+1}u^{i+1}\mod J^{2}+tJ,$

where $\lambda\in\mathit{MU}^{-2}=W^{-2}$, $2\ell=\partial\lambda$,
$\ell\in\mathbb{Z}$, and $\gamma_{i+1}=(-1)^{i}\alpha_{1i}+\omega_{i}$,
$\omega_{i}\in W^{-2i}$. Furthermore, any $\lambda$ and $\omega_{i}$ can be
obtained from some orientation $w\in\widetilde{W}^{2}(\mathbb{C}P^{\infty})$.

###### Proof.

By Proposition 3.1, every complex orientation $w$ has the form
$\pi_{0}(\widetilde{u})$ for some orientation $\widetilde{u}$ in
$\mathit{MU}$. Writing
$\widetilde{u}\in\widetilde{\mathit{MU}}\vphantom{MU}^{2}(\mathbb{C}P^{\infty})$
as a power series $f(u)$ in the standard orientation $u$ and observing that
$u^{i+1}=\overline{\partial}_{i}u$, we obtain

(3.3) $\widetilde{u}=f(u)=u+\sum_{i\geqslant
1}\lambda_{i}u^{i+1}=\bigl{(}1+\sum_{i\geqslant
1}\lambda_{i}\overline{\partial}_{i}\bigr{)}u=(1+\lambda\partial+g\varDelta)u.$

In the last identity we used Theorem 1.12 and Lemma 2.15 to write the
$\mathit{SU}$-linear operation $f=1+\sum_{i\geqslant
1}\lambda_{i}\overline{\partial}_{i}$ as $1+\lambda\partial+g\varDelta$, for a
certain $\mathit{SU}$-linear operation $g$. Note that any $\lambda$ and $g$
can be obtained from an orientation $\widetilde{u}$.

Now we calculate

(3.4) $\gamma(u)=\varphi(w)=w+t\partial
w=\pi_{0}(\widetilde{u})+t\partial\pi_{0}(\widetilde{u})=\pi_{0}f(u)+t\partial
f(u)\\\
=(\pi_{0}+t\partial)(1+\lambda\partial+g\varDelta)u=\pi_{0}(u)+\lambda\partial
u+\pi_{0}g\varDelta u+(2\ell+1)t\partial u+t\partial g\varDelta u\\\
=\pi_{0}(u)+(\lambda+(2\ell+1)t)\partial u+\pi_{0}g\varDelta u+t\partial
g\varDelta u,$

where we used the identity $\partial\pi_{0}=\partial$ from Proposition 2.9,
and the identities
$\pi_{0}(\lambda\partial)=\pi_{0}(\lambda)\partial=\lambda\partial$,
$t\partial(\lambda\partial)=t(\partial\lambda)\partial=2\ell t\partial$, which
are valid because both $\pi_{0}$ and $\partial$ are $\mathit{SU}$-linear.

Next we consider each of the four summands on the right hand side of (3.4)
separately.

Note that $\partial_{i}u=u\overline{u}^{i}=(-1)^{i}u^{i+1}\mod J$. Using the
formula from Proposition 2.9 we obtain

(3.5) $\pi_{0}(u)=u+\sum_{i\geqslant
2}\alpha_{1i}\partial_{i}u=u+\sum_{i\geqslant 2}(-1)^{i}\alpha_{1i}u^{i+1}\mod
J^{2}.$

Similarly,

(3.6) $(\lambda+(2\ell+1)t)\partial u=-(\lambda+(2\ell+1)t)u^{2}\mod
J^{2}+tJ.$

By Lemma 2.15, we have $g\varDelta(u)=\sum_{i\geqslant 2}\mu_{i}u^{i+1}$. It
follows that $g\varDelta(u)$ is an element of
$\widetilde{\mathit{MU}}\vphantom{MU}^{2}(\mathbb{C}P^{\infty})$ with the
property $g\varDelta(u)|_{\mathbb{C}P^{2}}=0$. Furthermore, by Lemma 2.15 any
$v\in\widetilde{\mathit{MU}}\vphantom{MU}^{2}(\mathbb{C}P^{\infty})$ with
$v|_{\mathbb{C}P^{2}}=0$ has the form $g\varDelta(u)$ for some
$\mathit{SU}$-linear operation $g$. Applying the projection $\pi_{0}$ we
obtain the element
$\pi_{0}g\varDelta(u)\in\widetilde{W}^{2}(\mathbb{C}P^{\infty})$ with the same
property. Therefore, $\pi_{0}g\varDelta(u)$ can be written as a power series
in $w$ without quadratic part with respect to the product $*$:

$\pi_{0}g\varDelta(u)=\sum_{i\geqslant 2}\omega_{i}*w^{*(i+1)},$

where $\omega_{i}\in W^{-2i}$ can be arbitrary. By (3.1),
$\omega_{i}*w=\omega_{i}w+\delta\partial\omega_{i}\partial w=\omega_{i}w\mod
J^{2}$, where the last identity holds because $\delta$ and
$\partial\omega_{i}$ are in $J$ for $i\geqslant 2$. Furthermore,

$w=\pi_{0}(\widetilde{u})=\pi_{0}(u)+\lambda\partial(u)+\pi_{0}g\varDelta(u)=u\mod
J.$

It follows that

(3.7) $\pi_{0}g\varDelta(u)=\sum_{i\geqslant 2}\omega_{i}u^{i+1}\mod J^{2}.$

Finally, by Lemma 2.15, $\partial g\varDelta=\sum\limits_{i\geqslant
2}\lambda_{i}\partial_{i}$, where $\lambda_{i}\in\mathit{MU}^{2-2i}\subset J$.
It follows that $\partial g\varDelta(u)=0\mod J$ and $t\partial
g\varDelta(u)=0\mod tJ$. Substituting this together with (3.5), (3.6), (3.7)
in (3.4) we obtain the required identity $\mod J^{2}+tJ$. ∎

The proof of Lemma 3.6 implies the following condition on the coefficients of
the series expansion of $\mathit{SU}$-linear projections $\pi$ in Proposition
2.13:

######  Proposition 3.7.

Let $\pi=1+\sum_{i\geqslant 2}\lambda_{i}\partial_{i}$ be an
$\mathit{SU}$-linear projection $\mathit{MU}\to W$. Then
$\lambda_{i}=\alpha_{1i}+\omega_{i}$ modulo decomposables in
$\mathit{MU}^{*}$, where $\omega_{i}$ can be an arbitrary element of
$W^{-2i}$.

###### Proof.

We have

$\pi(u)=u+\sum_{i\geqslant 2}\lambda_{i}\partial_{i}u=u+\sum_{i\geqslant
2}\lambda_{i}u\overline{u}^{i}=u+\sum_{i\geqslant
2}(-1)^{i}\lambda_{i}u^{i+1}\mod J^{2}.$

On the other hand, using Theorem 2.14, (2.1) and (3.7) we obtain

$\pi(u)=\pi_{0}(u)+\pi_{0}f\varDelta(u)=u+\sum_{i\geqslant
2}\alpha_{1i}\partial_{i}u+\sum_{i\geqslant
2}\omega_{i}u^{i+1}=u+\sum_{i\geqslant
2}((-1)^{i}\alpha_{1i}+\omega_{i})u^{i+1}\mod J^{2}.$

Comparing the coefficients in the above two expressions for $\pi(u)$ we obtain
the result. ∎

Now we return to the formal group law
$\varphi_{*}F_{W}(u,v)=u+v+\sum(\omega_{ij}+t\partial\omega_{ij})u^{i}v^{j}$
of the theory $\varGamma$.

######  Lemma 3.8.

In the notation of Lemma 3.6,

$\varphi_{*}F_{W}(u,v)=u+v-2\bigl{(}\lambda+(2\ell+1)t\bigr{)}uv-2\delta(2\ell+1)^{2}(uv^{2}+vu^{2})+\\\
+\sum_{i\geqslant 1,\,j\geqslant 1}\alpha_{ij}u^{i}v^{j}+\sum_{i\geqslant
3}\gamma_{i}\bigl{(}(u+v)^{i}-u^{i}-v^{i}\bigr{)}\mod J^{2}+tJ.$

###### Proof.

We have $\varphi_{*}F_{W}(u,v)=\gamma F_{U}(\gamma^{-1}(u),\gamma^{-1}(v))$ by
Proposition 3.4. Furthermore,
$\gamma(u)=u-(\lambda+(2\ell+1)t)u^{2}+\sum_{i\geqslant 3}\gamma_{i}u^{i}\mod
J^{2}+tJ$ by Lemma 3.6. The identity $(F_{U}(x,y))^{i}=(x+y)^{i}\mod J$
implies

(3.8)
$\gamma(F_{U}(x,y))=F_{U}(x,y)-\bigl{(}\lambda+(2\ell+1)t\bigr{)}(x+y)^{2}+\sum\limits_{i\geqslant
3}\gamma_{i}(x+y)^{i}\mod J^{2}+tJ.$

Next we need to calculate $x=\gamma^{-1}(u)$. We denote

$\gamma^{-1}(u)=u+\sum_{j\geqslant
2}\varepsilon_{j}u^{j},\quad\gamma(u)=u+\sum_{i\geqslant
2}\gamma_{i}u^{i},\text{ where }\gamma_{2}=-\lambda-(2\ell+1)t.$

All subsequent calculations will be carried out in the ring
$R=\varGamma^{*}/(J^{2}+tJ)$, that is, $\mod J^{2}+tJ$, see Construction 3.2.
We have $\varepsilon_{j}\in R^{2-2j}$ and $R^{<0}R^{<-2}=0$, which implies
$\varepsilon_{2}(u+\sum_{i\geqslant
2}\gamma_{i}u^{i})=\varepsilon_{2}(u-(2\ell+1)tu^{2})$ and
$\varepsilon_{j}(u+\sum_{i\geqslant 2}\gamma_{i}u^{i})=\varepsilon_{j}u$ for
$j\geqslant 3$. Hence,

$u=\gamma^{-1}(\gamma(u))=u+\sum_{i\geqslant
2}\gamma_{i}u^{i}+\sum_{j\geqslant 2}\varepsilon_{j}(u+\sum_{i\geqslant
2}\gamma_{i}u^{i})^{j}\\\ =u+\sum_{i\geqslant
2}\gamma_{i}u^{i}+\varepsilon_{2}(u-(2\ell+1)tu^{2})^{2}+\sum_{j\geqslant
3}\varepsilon_{j}u^{j}.$

Comparing the coefficients of $u^{j}$ we obtain

$\displaystyle\varepsilon_{2}$ $\displaystyle=-\gamma_{2}=\lambda+(2\ell+1)t,$
$\displaystyle\varepsilon_{3}$
$\displaystyle=2\varepsilon_{2}(2\ell+1)t-\gamma_{3}=2(\lambda+(2\ell+1)t)(2\ell+1)t-\gamma_{3}=2(2\ell+1)^{2}\delta-\gamma_{3},$
$\displaystyle\varepsilon_{j}$ $\displaystyle=-\gamma_{j}\quad\text{for
}j\geqslant 4.$

Therefore,

$\gamma^{-1}(u)=u+\bigl{(}\lambda+(2\ell+1)t\bigr{)}u^{2}+\bigl{(}2(2\ell+1)^{2}\delta-\gamma_{3}\bigr{)}u^{3}-\sum_{j\geqslant
4}\gamma_{j}u^{j}.$

It remains to substitute $x=\gamma^{-1}(u)$ and $y=\gamma^{-1}(v)$ in (3.8).
We have
$F_{U}(x,y)=x+y+\sum\alpha_{ij}x^{i}y^{j}=x+y+\sum\alpha_{ij}u^{i}v^{j}$ in
$R$, and similarly $\sum_{i\geqslant 3}\gamma_{i}(x+y)^{i}=\sum_{i\geqslant
3}\gamma_{i}(u+v)^{i}$. For the remaining summand of (3.8), we calculate

$\bigl{(}\lambda+(2\ell+1)t\bigr{)}(x+y)^{2}=\bigl{(}\lambda+(2\ell+1)t\bigr{)}\bigl{(}u+v+(\lambda+(2\ell+1)t)(u^{2}+v^{2})\bigr{)}^{2}\\\
=\bigl{(}\lambda+(2\ell+1)t\bigr{)}\bigr{(}(u+v)^{2}+2(\lambda+(2\ell+1)t)(u+v)(u^{2}+v^{2})\bigr{)}\\\
=(\lambda+(2\ell+1)t)(u+v)^{2}+2(2\ell+1)^{2}\delta(u+v)(u^{2}+v^{2}).$

Substituting these expressions in (3.8) we obtain

$\gamma(F_{U}(\gamma^{-1}(u),\gamma^{-1}(v)))=\gamma^{-1}(u)+\gamma^{-1}(v)+\sum\alpha_{ij}u^{i}v^{j}+\sum\limits_{i\geqslant
3}\gamma_{i}(u+v)^{i}\\\
-(\lambda+(2\ell+1)t)(u+v)^{2}-2(2\ell+1)^{2}\delta(u+v)(u^{2}+v^{2})=u+(\lambda+(2\ell+1)t)u^{2}\\\
+(2(2\ell+1)^{2}\delta-\gamma_{3})u^{3}-\sum\limits_{i\geqslant
4}\gamma_{i}u^{i}+v+(\lambda+(2\ell+1)t)v^{2}+(2(2\ell+1)^{2}\delta-\gamma_{3})v^{3}-\sum\limits_{i\geqslant
4}\gamma_{i}v^{i}\\\ +\sum\alpha_{ij}u^{i}v^{j}+\sum\limits_{i\geqslant
3}\gamma_{i}(u+v)^{i}-(\lambda+(2\ell+1)t)(u+v)^{2}-2(2\ell+1)^{2}\delta(u^{3}+uv^{2}+vu^{2}+v^{3})\\\
=u+v-2(\lambda+(2\ell+1)t)uv-2\delta(2\ell+1)^{2}(uv^{2}+vu^{2})+\sum\alpha_{ij}u^{i}v^{j}+\sum\limits_{i\geqslant
3}\gamma_{i}((u+v)^{i}-u^{i}-v^{i}),$

as claimed. ∎

######  Lemma 3.9.

The coefficients of the formal group law
$F_{W}(u,v)=u+v+\sum\omega_{ij}u^{i}v^{j}$ satisfy

(3.9)
$\sum\limits_{i+j=k+1}\omega_{ij}u^{i}v^{j}=\sum\limits_{i+j=k+1}\alpha_{ij}u^{i}v^{j}+\gamma_{k+1}\bigl{(}(u+v)^{k+1}-u^{k+1}-v^{k+1}\bigr{)}\mod
J^{2}$

for $k\geqslant 3$.

###### Proof.

We have $t\partial\omega_{ij}=0\mod tJ$ for $i+j>2$, which implies
$\varphi_{*}F_{W}(u,v)=u+v+(\omega_{11}+t\partial\omega_{11})uv+\sum_{i+j>2}\omega_{ij}u^{i}v^{j}\mod
J^{2}+tJ$. Now the required identity follows from the identity of Lemma 3.8. ∎

For an integer $k\geqslant 1$ let

$m_{k}=\gcd\biggl{\\{}{k+1\choose i},\,1\leqslant i\leqslant
k\biggr{\\}}=\begin{cases}1&\text{if $k+1\neq p^{\ell}$ for any prime $p$,}\\\
p&\text{if $k+1=p^{\ell}$ for some prime $p$ and integer
$\ell>0$.}\end{cases}$

######  Theorem 3.10 (see [17, Chapter X] or [11, Theorem 6.10]).

With respect to the multiplication defined by the Stong projection $\pi_{0}$,
the ring $W_{*}$ is polynomial on generators in every positive even degree
except $4$:

$W_{*}\cong\mathbb{Z}[x_{1},x_{k}\colon k\geqslant 3],\quad
x_{1}=[\mathbb{C}P^{1}],\quad x_{k}\in W_{2k}.$

The polynomial generators $x_{k}$ are specified by the condition
$s_{k}(x_{k})=\pm m_{k}m_{k-1}$ for $k\geqslant 3$.

######  Lemma 3.11 ([6]).

For the coefficients of the formal group law $F_{W}(u,v)$, we have

$\gcd\bigl{\\{}s_{i+j-1}(\omega_{ij})\colon
i+j=k+1\bigr{\\}}=m_{k}\bigl{(}1+(-1)^{k}(k+1)+c_{k}m_{k}m_{k-1}\bigr{)}$

for $k\geqslant 3$, where $c_{k}$ can be an arbitrary integer depending on the
orientation $w$.

###### Proof.

Recall that $\mathit{MU}_{*}\cong\mathbb{Z}[a_{1},a_{2},\ldots]$,
$a_{k}\in\mathit{MU}_{2k}$ and $s_{k}(a_{k})=m_{k}$.

There is the following formula for the coefficients of the formal group law
$F_{U}$ modulo decomposables:

$\sum\limits_{i+j=k+1}\alpha_{ij}u^{i}v^{j}=-a_{k}\frac{(u+v)^{k+1}-u^{k+1}-v^{k+1}}{m_{k}}\mod
J^{2},$

see, e. g., [1]. Hence, $\alpha_{ij}=-\frac{{i+j\choose
i}}{m_{i+j-1}}a_{i+j-1}\mod J^{2}$ and, in particular,
$\alpha_{1j}=-\frac{j+1}{m_{j}}a_{j}\mod J^{2}$. This implies that
$\gamma_{k+1}=(-1)^{k}\alpha_{1k}+\omega_{k}=(-1)^{k+1}\frac{k+1}{m_{k}}a_{k}+\omega_{k}\mod
J^{2}$. Substituting this in (3.9) we obtain

$\sum\limits_{i+j=k+1}\omega_{ij}u^{i}v^{j}=-\bigl{(}a_{k}+(-1)^{k}(k+1)a_{k}-m_{k}\omega_{k}\bigr{)}\frac{(u+v)^{k+1}-u^{k+1}-v^{k+1}}{m_{k}}\mod
J^{2}.$

It follows that

$\gcd\bigl{\\{}s_{i+j-1}(\omega_{ij})\colon i+j=k+1\bigr{\\}}\\\
=s_{k}\bigl{(}a_{k}+(-1)^{k}(k+1)a_{k}-m_{k}\omega_{k}\bigr{)}=m_{k}\bigl{(}1+(-1)^{k}(k+1)-s_{k}(\omega_{k})\bigr{)}.$

Formula (3.1) implies that if an element $x\in W_{2i}$, $i\geqslant 3$, is
decomposable in $W_{*}$ (with respect to an arbitrary multiplication), then
its forgetful image in $\mathit{MU}_{*}$ is also decomposable. Hence,
$\omega_{k}=c_{k}x_{k}\mod J^{2}$ for an integer $c_{k}$. Therefore,
$s_{k}(\omega_{k})=c_{k}m_{k}m_{k-1}$, and the result follows. ∎

######  Theorem 3.12.

For any complex orientation of $W$, the coefficients of the corresponding
formal group law $F_{W}$ do not generate the whole ring $W_{*}$.

###### Proof.

Consider a polynomial generator $x_{k}$ from Theorem 3.10 for $k\geqslant 3$.
Supppose $x_{k}$ lies in the ring generated by the coefficients of $F_{W}$.
Since an element decomposable in $W_{*}$ is also decomposable in
$\mathit{MU}_{*}$ (in dimensions $\geqslant 6$), we get

$s_{k}(x_{k})=\pm\gcd\bigl{\\{}s_{i+j-1}(\omega_{ij})\colon
i+j=k+1\bigr{\\}}=\pm m_{k}(1+(-1)^{k}(k+1)+c_{k}m_{k}m_{k-1}).$

On the other hand, by Theorem 3.10 we have $s_{k}(x_{k})=\pm m_{k}m_{k-1}$. We
show that there is $k\geqslant 3$ such that the two numbers do not agree even
up to a sign.

Indeed, recall that $m_{k}=p$ if $k+1=p^{s}$ for some prime $p$, and $m_{k}=1$
otherwise. Therefore, if $k=2^{\ell}$ and, in addition, $k+1=p^{s}$ for an odd
prime $p$, then $m_{k}m_{k-1}=2p$, whereas
$m_{k}(1+(-1)^{k}(k+1)+c_{k}m_{k}m_{k-1})=2p+p(2^{\ell}+2c_{k}p)=2p(1+2^{\ell-1}+c_{k}p)$.
Suppose $\pm 2p=2p(1+2^{\ell-1}+c_{k}p)$ or, equivalently,
$1+2^{\ell-1}+c_{k}p=\pm 1$. Since $p$ is odd, $2^{\ell-1}+c_{k}p\neq 0$ for
any $c_{k}$. So, $1+2^{\ell-1}+c_{k}p\neq 1$. If $1+2^{\ell-1}+c_{k}p=-1$,
then $-2c_{k}p=4+2^{\ell}=3+p^{s}$. This is impossible for $p>3$. As a result,
we obtain a contradiction in dimensions of the form $k=2^{\ell}=p^{s}-1$ for
$\ell>1$. ∎

We can also prove the following result stated in [6].

######  Theorem 3.13.

Let $A$ be the subring of $W_{*}$ generated by the coefficients of the formal
group law $F_{W}$. Then there is an orientation of $W$ such that
$A[\frac{1}{2}]=W_{*}[\frac{1}{2}]$.

######  Remark.

The proof given below would be simpler if we knew that $W_{*}$ is a polynomial
ring _for arbitrary $\mathit{SU}$-linear multiplication_ on $W$. However, the
description of Theorem 3.10 is valid only for the multiplication defined by
the Stong projection.

The proof is based on three lemmata. The first lemma says that the case
$k=2^{\ell}=p^{s}-1$ considered in the proof of Theorem 3.12 is the only case
when the gcd of the $s$-numbers of the coefficients of the formal group law
$F_{W}$ does not agree with $m_{k}m_{k-1}$:

######  Lemma 3.14.

If $k$ is not of the form $k=2^{\ell}=p^{s}-1$ for some odd prime $p$, then
$\gcd\bigl{\\{}s_{i+j-1}(\omega_{ij})\colon i+j=k+1\bigr{\\}}=m_{k}m_{k-1}$
for some value of $c_{k}$ (see Lemma 3.11).

###### Proof.

By Lemma 3.11, we need to find $c_{k}$ such that
$1+(-1)^{k}(k+1)+c_{k}m_{k}m_{k-1}=m_{k-1}$.

If $m_{k-1}=1$, then we set $c_{k}=(-1)^{k+1}\frac{k+1}{m_{k}}$, which is an
integer as $m_{k}$ always divides $k+1$.

If $m_{k-1}=2$, then $k=2^{\ell}$. By assumption, $k\neq p^{s}-1$, so
$m_{k}=1$. The required identity becomes $1+(2^{\ell}+1)+2c_{k}=2$, which is
satisfied for $c_{k}=-2^{\ell-1}$.

If $m_{k-1}=p$ is an odd prime, then $k=p^{s}$. Hence, $m_{k}=1$ or $2$. The
required identity becomes $1-(p^{s}+1)+pc_{k}m_{k}=p$, which is satisfied for
$c_{k}=\frac{p^{s-1}+1}{m_{k}}$. The latter is an integer as $p^{s-1}+1$ is
even. ∎

######  Lemma 3.15.

If $p^{s}=2^{\ell}+1$ for odd prime $p$ and positive integers $\ell,s$, then
either $s=1$ and $\ell=2^{n}$ (so $p$ is a Fermat prime), or $p=3$, $s=2$ and
$\ell=3$.

###### Proof.

_Case 1:_ $p=3$.

There are obvious solutions $s=1,\ell=1$ and $s=2,\ell=3$.

Now suppose $s>2$, so that $\ell>3$. Then $3^{s}=2^{\ell}+1\equiv 0\pmod{9}$.
It is easy to check that $2^{\ell}\equiv-1\pmod{9}$ if and only if
$\ell=6m+3$. Then
$3^{s}=2^{\ell}+1=(2^{2m+1})^{3}+1=(2^{2m+1}+1)(2^{4m+2}-2^{2m+1}+1)$. Hence,
$2^{4m+2}-2^{2m+1}+1=3^{s^{\prime}}$ and $2^{2m+1}+1=3^{s^{\prime\prime}}$.
Now $\ell>3$ implies $m>0$ and therefore $s^{\prime}>1$. Hence,
$2^{4m+2}-2^{2m+1}+1\equiv 0\pmod{9}$. Similarly, $s^{\prime\prime}>1$, hence,
$2^{2m+1}+1=3^{s^{\prime\prime}}\equiv 0\pmod{9}$. The latter implies
$2^{4m+2}-2^{2m+1}+1\equiv 1+1+1=3\not\equiv 0\pmod{9}$. A contradiction.

_Case 2:_ $p>3$.

Reducing the identity $p^{s}=2^{\ell}+1$ modulo 3 we obtain $(\pm
1)^{s}\equiv(-1)^{\ell}+1\pmod{3}$. This implies that $s$ is odd and $\ell$ is
even.

Write $p-1=a2^{q}$ with odd $a$. Then
$2^{\ell}+1=(a2^{q}+1)^{s}=a^{s}2^{qs}+\cdots+sa2^{q}+1$. Suppose $s>1$. Then
$\ell>q$ and $as+2^{q}(a^{s}2^{q(s-2)}+\cdots)=2^{\ell-q}$ is even. This is a
contradiction because $as$ is odd. Hence, $s=1$ and $p=2^{\ell}+1$.

Write $\ell=r2^{n}$ with odd $r$. Then
$p=2^{\ell}+1=(2^{2^{n}}+1)(2^{(r-1)2^{n}}-\cdots+1)$. Since $p$ is prime, we
obtain $p=2^{2^{n}}+1$ and $\ell=2^{n}$. ∎

######  Lemma 3.16.

If $[\mathbb{C}P^{1}]\in A$ and there are elements $x_{k}\in A_{2k}$,
$k\geqslant 2$, such that $s_{k}(x_{k})=m_{k}m_{k-1}$ up to a power of $2$,
then $A[\frac{1}{2}]=W_{*}[\frac{1}{2}]$.

###### Proof.

There is a polynomial subring $\mathit{MSU}_{*}[\frac{1}{2}]\subset
W_{*}[\frac{1}{2}]$. This follows from the fact that any $\mathit{SU}$-linear
multiplication (3.1) induces the standard multiplication on
$\mathit{MSU}_{*}$, because $\mathit{MSU}_{*}\subset\mathop{\rm Ker}\partial$.
Furthermore, $\mathit{MSU}_{*}[\frac{1}{2}]$ coincides with $\mathop{\rm
Ker}\partial=\mathop{\rm Im}\partial$ on $W_{*}[\frac{1}{2}]$ (see [17,
Chapter X] or [11, Theorem 5.11]). From (3.2) we obtain the identity
$x=\frac{1}{2}(\partial([\mathbb{C}P^{1}]x)+[\mathbb{C}P^{1}]\partial x)$ for
any $x\in W_{*}[\frac{1}{2}]$, which implies that $W_{*}[\frac{1}{2}]$ is a
free $\mathit{MSU}_{*}[\frac{1}{2}]$-module with basis
$\\{1,[\mathbb{C}P^{1}]\\}$. Since $[\mathbb{C}P^{1}]\in A$ by assumption, we
need to show that $\mathit{MSU}_{*}[\frac{1}{2}]\subset A[\frac{1}{2}]$.

By the theorem of Novikov [13],
$\mathit{MSU}_{*}[\frac{1}{2}]\cong\mathbb{Z}[\frac{1}{2}][y_{2},y_{3},\ldots]$,
$\dim y_{k}=2k$. The polynomial generators $y_{k}$ are specified by the
condition $s_{k}(y_{k})=\pm m_{k}m_{k-1}$ up to a power of $2$ (see [17,
Chapter X]).

Let $x_{1}=[\mathbb{C}P^{1}]\in A$. We can assume by induction that
$\mathit{MSU}_{*}[\frac{1}{2}]\subset A[\frac{1}{2}]$ in dimensions less than
$2k$. For $x_{k}\in A$, let
$\widetilde{y}_{k}=\frac{1}{2}\partial(x_{1}x_{k})=x_{k}-\frac{1}{2}x_{1}\partial
x_{k}$. We have $\widetilde{y}_{k}\in\mathit{MSU}_{*}[\frac{1}{2}]$. By the
induction hypothesis, $\partial x_{k}\in A[\frac{1}{2}]$, so
$\widetilde{y}_{k}\in A[\frac{1}{2}]$. Since $x_{1}\partial x_{k}$ is
decomposable, $s_{k}(\widetilde{y}_{k})=s_{k}(x_{k})=m_{k}m_{k-1}$ up to a
power of 2. It follows that $\widetilde{y}_{k}$ is a polynomial generator of
$\mathit{MSU}_{*}[\frac{1}{2}]$. Therefore, we obtain
$\mathit{MSU}_{*}[\frac{1}{2}]\subset A[\frac{1}{2}]$ by induction. ∎

###### Proof of Theorem 3.13..

By Lemma 3.16, we need to specify an orientation of $W$ and elements $x_{k}\in
A_{2k}$, $k\geqslant 2$, such that $s_{k}(x_{k})=m_{k}m_{k-1}$ modulo a power
of $2$.

The formula from Lemma 3.8 implies
$\omega_{11}=\alpha_{11}-2\lambda=-[\mathbb{C}P^{1}]-2\lambda$. Choosing an
orientation of $W$ with $\lambda=0$ we get $x_{1}=[\mathbb{C}P^{1}]\in A$.

Next we need to find $x_{2}\in A$ with $s_{2}(x_{2})=m_{2}m_{1}=3$ up to a
power of $2$.

The multiplication on $W_{*}$ is given by $a*b=ab+\delta\partial a\partial b$,
$\delta=2[V]+\omega$, $\omega\in W_{4}$,
$[V]=[\mathbb{C}P^{1}]^{2}-[\mathbb{C}P^{2}]$. We have
$W_{4}=\mathbb{Z}\langle 9[\mathbb{C}P^{1}]^{2}-8[\mathbb{C}P^{2}]\rangle$, so
$s_{2}(\omega)=24\alpha$ with $\alpha\in\mathbb{Z}$. Hence,
$s_{2}(\delta)=-6+24\alpha$.

The formula from Lemma 3.8 implies

$\omega_{12}=-2\delta(2\ell+1)^{2}+\alpha_{12}+3\gamma_{3}\mod J^{2}.$

Substituting here $2\ell=\partial\lambda=0$ and
$\gamma_{3}=\omega_{2}+\alpha_{12}$ (see Lemma 3.6), we obtain

$\omega_{12}=4\alpha_{12}+3\omega_{2}-2\delta\mod J^{2},$

where $\omega_{2}\in W_{4}$ can be chosen arbitrarily depending on the
orientation of $W$. Since $s_{2}(\alpha_{12})=3$ and
$s_{2}(\omega_{2})=24\beta$ with $\beta\in\mathbb{Z}$, we obtain
$s_{2}(\omega_{12})=24-48\alpha+72\beta$.

_Case 1:_ $\alpha=3n$, $n\in\mathbb{Z}$. Let $\beta=2n$. Take
$x_{2}=\omega_{12}\in A$. Then $s_{2}(x_{2})=24=3\cdot 2^{3}$.

_Case 2:_ $\alpha=3n+\varepsilon$, $n\in\mathbb{Z}$, $\varepsilon=1$ or $2$.
Let $\beta=-2n$. Take $x_{2}=\omega_{12}+x_{1}*x_{1}\in A$. We have
$x_{1}*x_{1}=(x_{1})^{2}+4\delta$, so
$s_{2}(x_{2})=s_{2}(\omega_{12})+4s_{2}(\delta)=24(3\beta+2\alpha)=3\cdot\varepsilon
2^{4}$, which is equal to $3$ up to a power of $2$.

It remains to choose $x_{k}$ for $k\geqslant 3$. By Lemma 3.14, there is an
integral linear combination $x_{k}$ of the coefficients $\omega_{ij}$,
$i+j=k+1$, such that $s_{k}(x_{k})=m_{k}m_{k-1}$, except for the case
$k+1=p^{s}=2^{\ell}+1$.

For the remaining case $k=2^{\ell}=p^{s}-1$, Lemma 3.15 implies that either
$p=3$, $s=2$ or $p=2^{2^{n}}+1$, $s=1$.

In the first case ($k=8$), Lemma 3.11 gives
$\gcd\bigl{\\{}s_{k}(\omega_{ij})\colon i+j=k+1\bigr{\\}}=3(1+9+6c_{k})$.
Setting $c_{k}=-2$, we obtain $\gcd\bigl{\\{}s_{k}(\omega_{ij})\colon
i+j=k+1\bigr{\\}}=-6=-m_{k}m_{k-1}$, as needed.

In the second case, setting $c_{k}=\frac{p-1}{2}-1$ we obtain
$\gcd\bigl{\\{}s_{k}(\omega_{ij})\colon
i+j=k+1\bigr{\\}}=p(p^{2}-2p+1)=p(p-1)^{2}=2^{2^{n+1}}p$, as needed. ∎

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