Twistors, Self-Duality, and Spinc Structures

Twistors, Self-Duality, and Spinc Structures††This paper is a contribution to
the Special Issue on Twistors from Geometry to Physics in honor of Roger
Penrose. The full collection is available at
https://www.emis.de/journals/SIGMA/Penrose.html

Claude LEBRUN

C. LeBrun

Department of Mathematics, Stony Brook University, Stony Brook, NY 11794-3651
USA<EMAIL_ADDRESS>http://www.math.stonybrook.edu/~claude/

Received August 02, 2021, in final form November 15, 2021; Published online
November 19, 2021

The fact that every compact oriented 4-manifold admits spinc structures was
proved long ago by Hirzebruch and Hopf. However, the usual proof is neither
direct nor transparent. This article gives a new proof using twistor spaces
that is simpler and more geometric. After using these ideas to clarify various
aspects of four-dimensional geometry, we then explain how related ideas can be
used to understand both spin and spinc structures in any dimension.

4-manifold; spinc structure; twistor space; self-dual 2-form

53C27; 53C28; 57R15

##  1 Twistor spaces and spinc structures

Every compact oriented 4-manifold admits spinc structures. The standard proof
of this fact is due to Hirzebruch and Hopf [3], although the result had
previously been hinted at by Whitney [9]. For readable modernized English-
language versions of the Hirzebruch–Hopf proof, see Killingback and Rees [5]
or Gompf and Stipsicz [2, Section 5.7].

However, the Hirzebruch–Hopf proof is so indirect that it does not really
involve the notion of a spinc structure at all, and it proceeds by so
completely isolating the topological issues from the geometric motivation as
to make it seem rather formal and unenlightening. The main purpose of this
article is to give a self-contained proof of this important fact that is based
on ideas from twistor theory. In the process, we will also see how this result
is inextricably related to other fundamental aspects of $4$-dimensional
geometry. The article then concludes by putting this $4$-dimensional story in
the context of a twistor approach to spin and spinc structures in other
dimensions.

Let us begin by recalling that dimension four is profoundly exceptional for
both differential topology and differential geometry. This idiosyncrasy is
largely attributable to a fluke of Lie-group theory: the rotation group ${\rm
SO}(4)$ is not a simple Lie group. Instead, its Lie algebra splits as a direct
sum

$\displaystyle{\mathfrak{so}}(4)\cong{\mathfrak{so}}(3)\oplus{\mathfrak{so}}(3),$

as a consequence of the fact that left- and right-multiplication by the unit
quaternions ${\rm Sp}(1)$ belong to different subgroups of the rotation group.
On an oriented Riemannian $4$-manifold $\big{(}M^{4},g\big{)}$, this gives
rise to an invariant direct-sum decomposition

$\displaystyle\Lambda^{2}=\Lambda^{+}\oplus\Lambda^{-}$

of the bundle of $2$-forms, because the action of ${\rm SO}(4)$ on $2$-forms
is isomorphic, via index raising, to its adjoint representation on the Lie
algebra ${\mathfrak{so}}(4)$ of skew $4\times 4$ matrices. This decomposition
in fact coincides with the decomposition of the $2$-forms into the
$(\pm)$-eigenspaces of the Hodge star operator

$\displaystyle\star\colon\ \Lambda^{2}\to\Lambda^{2}.$

We will now emphasize our choice of an orientation by focusing on the bundle
$\Lambda^{+}$ of self-dual $2$-forms $\varphi$, which are characterized by the
condition $\star\varphi=\varphi$.

While the rank-$3$ oriented vector bundle $\Lambda^{+}\to M$ depends on the
conformal class $[g]$ of the Riemannian metric $g$, the bundles
$\Lambda^{+}_{g}$ and $\Lambda^{+}_{g^{\prime}}$ associated with two different
metrics are nonetheless canonically bundle-isomorphic via the natural
identification $\Lambda^{+}_{g}=\Lambda^{2}/\Lambda^{-}_{g}$, because we
always have $\Lambda_{g^{\prime}}^{+}\cap\Lambda_{g}^{-}=0$. This algorithm
for producing an isomorphism suffers from some defects, though. First of all,
interchanging $g$ and $g^{\prime}$ does not produce the inverse isomorphism.
Second, the isomorphism produced by this algorithm does not preserve the
relevant inner products. Fortunately, however, the latter can be corrected by
applying a unique positive, self-adjoint endomorphism to $\Lambda^{+}$, and
this then allows us to identify the oriented bundles-with-inner-product
$\Lambda^{+}$ for two different metrics in a manner that is unique up to
isotopy. This will suffice to give a metric-independent meaning to the notions
that are the main focus of our discussion.

We now fix a Riemannian metric $g$ on our oriented $4$-manifold $M$, and
notice that, since ${\rm SO}(4)/\mathbb{Z}_{2}\cong{\rm SO}(3)\times{\rm
SO}(3)$, the $4$-dimensional rotation group acts transitively on the unit
sphere in $\Lambda^{+}$. For this reason, any $\omega\in\Lambda^{+}_{x}$,
$x\in M$, with $|\omega|=\sqrt{2}$ can be expressed as

$\displaystyle\omega=e^{1}\wedge e^{2}+e^{3}\wedge e^{4}$

in some oriented orthonormal basis for $T_{x}M$, and hence corresponds, via
index raising, to the endomorphism $\jmath\colon T_{x}M\to T_{x}M$ represented
by the matrix

$\displaystyle\left[\begin{array}[]{cccc}&-1&&\\\ 1&&&\\\ &&&-1\\\
&&1&\end{array}\right].$

In other words, any such $\omega$ defines an almost-complex structure $\jmath$
at $x$ that is compatible with the metric $g$ and determines the given
orientation. Conversely, if $\jmath\colon T_{x}M\to T_{x}M$ satisfies
$\jmath^{2}=-I$ and $\jmath^{*}g=g$, and also determines the given orientation
of $M$, then $\jmath$ arises, via index raising, from a unique
$\omega\in\Lambda^{+}_{x}$ with $|\omega|=\sqrt{2}$.

We now define the twistor space of our oriented Riemannian $4$-manifold
$(M,g)$ to be the total space

$\displaystyle
Z:=S_{\sqrt{2}}\big{(}\Lambda^{+}\big{)}=\big{\\{}\omega\in\Lambda^{+}\,|\,|\omega|=\sqrt{2}\big{\\}}$

of the $2$-sphere bundle $\wp\colon Z\to M$ associated with the oriented
rank-$3$ vector bundle $\Lambda^{+}\to M$ of self-dual $2$-forms. We may then
give $Z$ an almost-complex structure $J\colon TZ\to TZ$, $J^{2}=-I$, by the
following construction, which is essentially due to Atiyah–Hitchin–Singer [1],
and which provides a general Riemannian context for Penrose’s non-linear
graviton construction [7]. We begin by decomposing $TZ$ into vertical and
horizontal components,

$\displaystyle TZ=\mathsf{V}\oplus\mathsf{H},$ (1.1)

where $\mathsf{V}:=\ker{\rm d}\wp$ and $\mathsf{H}$ is induced by parallel
transport in $\Lambda^{2}$ with respect to the Riemannian connection of $g$.
Now notice that index raising gives us an alternative, conformally invariant
description

$\displaystyle Z=\big{\\{}{\mathsf{\jmath}}\colon T_{x}M\to T_{x}M,\,x\in
M\,|\,{\mathsf{\jmath}}^{2}=-I,\,{\mathsf{\jmath}}^{*}g=g,\,{\mathsf{\jmath}}>0\big{\\}}$

of the twistor space. Since the derivative ${\rm d}\wp\colon TZ\to TM$ of the
bundle projection $\wp$ induces a tautological isomorphism
$\mathsf{H}\cong\wp^{*}TM$, we may therefore define an endomorphism
$J_{\mathsf{H}}\colon\mathsf{H}\to\mathsf{H}$ whose action at
$\jmath\in\wp^{-1}(x)$ is the horizontal lift of $\jmath\colon T_{x}M\to
T_{x}M$. Meanwhile, since each fiber $\wp^{-1}(x)$ of $\wp\colon Z\to M$ is a
round $2$-sphere in an oriented $3$-dimensional inner-product space
$\Lambda^{+}_{x}$, we can therefore declare
$J_{\mathsf{V}}\colon\mathsf{V}\to\mathsf{V}$ to be $+90^{\circ}$ rotation in
the tangent space of each fiber $2$-sphere with respect to the outward-
pointing orientation. Since these recipes guarantee that
$J_{\mathsf{H}}^{2}=-I_{\mathsf{H}}$ and $J_{\mathsf{V}}^{2}=-I_{\mathsf{V}}$,
setting

$\displaystyle J:=J_{\mathsf{H}}\oplus J_{\mathsf{V}}$

now produces an almost-complex structure $J$ on $Z$, as illustrated by Figure
1.

Figure 1: The twistor space $Z$ of an oriented Riemannian $4$-manifold $(M,g)$
is the sphere bundle of the oriented rank-$3$ bundle $\Lambda^{+}\to M$ of
self-dual $2$-forms. This $6$-manifold can be given a canonical almost-complex
structure $J$ that is compatible with the decomposition of $TZ$ into vertical
and horizontal subspaces.

The almost-complex structure $J$ is actually conformally invariant, even
though the decomposition (1.1) is not; however, $J$ is only integrable [1, 7]
if the Weyl curvature of $(M,g)$ is anti-self-dual. Nonetheless, some useful
remnants of integrability persist, even in the general case; in particular,
each fiber $\wp^{-1}(x)\cong\mathbb{CP}_{1}$ is a $J$-holomorphic curve, and
the normal bundle $\nu=T^{1,0}Z/\mathsf{V}^{1,0}$ of each fiber is a
holomorphic bundle $\nu\cong\mathcal{O}(1)\oplus\mathcal{O}(1)$ over this
$\mathbb{CP}_{1}$.

Since this emphasizes the fact that $\wp\colon Z\to M$ may be thought of as a
$\mathbb{CP}_{1}$-bundle, it seems natural to ask whether this
$\mathbb{CP}_{1}$-bundle can always be expressed as the projectivization
$\mathbb{P}(\mathbb{V}_{+})$ of a rank-$2$ complex vector bundle
$\mathbb{V}_{+}\to M$. As we will see, the answer always turns out to be,
“Yes!” This assertion exactly amounts to the fact that any oriented
$4$-manifold admits spinc structures. The road that will bring us to this
realization begins with the following definition:

######  Geometric Definition.

A spinc structure on a connected oriented Riemannian $4$-manifold $(M,g)$ is a
complex line bundle ${\mathscr{L}}\to{Z}$ on the twistor space that has degree
$1$ on any ${S^{2}}$ fiber of ${Z}\to M$.

Here, two isomorphic complex line bundles on $Z$ are considered to define the
same spinc structure. However, the first Chern class $c_{1}$ defines a
bijection between equivalence classes of complex line bundles on $Z$ and
$H^{2}(Z,\mathbb{Z})$, in a manner that converts the tensor-product of line
bundles into the addition of cohomology classes. This means that the above
definition can be reformulated as saying that a spinc structure on $M$ is a
cohomology class $\mathbf{a}\in H^{2}(Z,\mathbb{Z})$ with
$\langle\mathbf{a},F\rangle=1$, where $F\in H_{2}(Z,\mathbb{Z})$ is the
homology class of a fiber $\wp^{-1}(x)\approx S^{2}$.

The above should be compared and contrasted with the following:

######  Standard Definition.

A spinc structure on an oriented Riemannian $4$-manifold $(M,g)$ is a circle
bundle $\widehat{\mathfrak{F}}\to\mathfrak{F}$ over the oriented orthonormal
frame bundle that is also compatibly endowed with the structure of a principal
${\rm Spin}^{c}(4)$-bundle, where

$\displaystyle{\rm Spin}^{c}(4):=[{\rm Sp}(1)\times{\rm Sp}(1)\times{\rm
U}(1)]/\langle(-1,-1,-1)\rangle.$

Fortunately, these competing definitions are exactly equivalent. For example,
one can pass from the geometric definition to the standard definition by first
expressing $Z$ as $\mathfrak{F}/{\rm U}(2)$. Thus, if $\mathscr{L}\to Z$ is a
complex line bundle of fiber degree $1$, we can pull the circle bundle
$S(\mathscr{L})$ back to obtain a circle bundle over $\mathfrak{F}$, and this
pull-back can then be made into principal ${\rm Spin}^{c}(4)$-bundle over $M$
using the fact that $H^{1}({\rm SO}(4),\mathcal{E}^{\times})=H^{1}({\rm
SO}(4),\mathbb{C}^{\times})=H^{2}({\rm SO}(4),\mathbb{Z})=\mathbb{Z}_{2}$.

In the opposite direction, given a principal ${\rm Spin}^{c}(4)$-bundle over
$M$ that is also a circle bundle $\widehat{\mathfrak{F}}\to{\mathfrak{F}}$, we
first construct a vector bundle $\mathbb{V}_{+}\to M$ by applying the
associated bundle construction to the representation ${\rm Spin}^{c}(4)\to{\rm
U}(2)\cong[{\rm Sp}(1)\times{\rm U}(1)]/\langle(-1,-1)\rangle$ obtained by
dropping the second ${\rm Sp}(1)$. The map
$\widehat{\mathfrak{F}}\to\mathfrak{F}$ then allows us to identify $Z$ with
$\mathbb{P}(\mathbb{V}_{+})$, and the $\mathcal{O}(1)$-bundle dual to the
tautological line bundle of $\mathbb{P}(\mathbb{V}_{+})$ then provides the
fiber-degree-$1$ line bundle featured in the geometric definition.

While every oriented $4$-manifold will turn out to admit spinc structures, the
situation is entirely different for spin structures, which are much more
restrictive:

######  Geometric Definition.

A spin structure on an oriented Riemannian $4$-manifold $(M,g)$ is a square-
root ${\mathscr{V}^{1/2}}$ of the vertical complex line bundle
$\mathscr{V}:=\mathsf{V}^{1,0}$ of the twistor space ${Z}\to M$.

Here, a square-root $\mathscr{V}^{1/2}$ of $\mathscr{V}$ means a line-bundle
$\mathscr{L}\to Z$ that is equipped with a specific isomorphism
$\Phi\colon\mathscr{L}\otimes\mathscr{L}\to\mathscr{V}$. Two such square-roots
$(\mathscr{L},\Phi)$ and $(\mathscr{L}^{\prime},\Phi^{\prime})$ are considered
to be the same if there is an isomorphism
$\Psi\colon\mathscr{L}\to\mathscr{L}^{\prime}$ of complex line bundles that
induces a commutative diagram

$\mathscr{L}\otimes\mathscr{L}$$\mathscr{L}^{\prime}\otimes\mathscr{L}^{\prime}$$\mathscr{V}$Φ${}_{\Phi^{\prime}}$Ψ⊗Ψ

######  Standard Definition.

A spin structure on an oriented Riemannian $4$-manifold $(M,g)$ is a double
cover ${\widetilde{\mathfrak{F}}}\to{\mathfrak{F}}$ of the principal ${\rm
SO}(4)$-bundle of oriented orthonormal frames by a principal ${\rm
Spin}(4)$-bundle, where

$\displaystyle{\rm Spin}(4)={\rm Sp}(1)\times{\rm Sp}(1).$

Once again, these competing definitions are exactly equivalent. Indeed, the
${\rm SO}(3)$-bundle $\mathsf{F}=\mathfrak{F}/{\rm Sp}(1)$ may be viewed as
the circle bundle $S(\mathscr{V})$ over $Z=\mathsf{F}/{\rm SO}(2)$, so a
square-root of $\mathscr{V}$ amounts to a double-cover of $\mathsf{F}$. On the
other hand, the quotient map $\mathfrak{F}\to\mathsf{F}$ induces an
isomorphism of fundamental groups, so double covers of $\mathsf{F}$ are in
bijective correspondence with double covers of $\mathfrak{F}$.

##  2 Oriented rank-3 vector bundles

We now focus on the case where our smooth oriented $4$-manifold is compact.
The Euler class $\mathsf{e}(\Lambda^{+})$ will then play the starring role in
our proofs. However, our approach will be clarified by initially working in a
more general context. Let us therefore first consider the Euler class
$\mathsf{e}(\mathsf{E})\in H^{3}(M,\mathbb{Z})$ of any oriented rank-$3$
bundle $\mathsf{E}\to M$.

Now, since the rank of $\mathsf{E}$ is odd, multiplication by $-1$ induces an
orientation-reversing self-isomorphism $\mathsf{E}\to\mathsf{E}$. Hence the
Euler class $\mathsf{e}=\mathsf{e}(\mathsf{E})\in H^{3}(M,\mathbb{Z})$
satisfies $\mathsf{e}=-\mathsf{e}$, and so is a $2$-torsion element. This
Euler class is Poincaré dual to the homology class $\in H_{1}(M,\mathbb{Z})$
of the oriented curve cut out by the zero set of any section of $\mathsf{E}\to
M$ that is transverse to the zero section $\mathsf{0}_{M}\subset\mathsf{E}$.
Of course, if $\mathsf{E}$ had a nowhere-zero section, its Euler class
$\mathsf{e}(\mathsf{E})$ would consequently vanish. However, since
$\operatorname{rank}\mathsf{E}<\dim M$, the converse is definitely false! This
common mistake seems to arise from over-familiarity with the Poincaré–Hopf
theorem, which concerns the case where the rank of the bundle equals the
dimension of the base; in that context, generic zeroes are just isolated
points, and generic zeroes of opposite sign can then be eliminated in pairs.
In our case, the zero locus of a generic section is instead a union of
oriented circles, and the vanishing of the Euler class just means that this
curve bounds an oriented surface.

######  Example.

Since $H^{3}\big{(}S^{4},\mathbb{Z}\big{)}=0$, the oriented rank-$3$ bundle
$\Lambda^{+}\to S^{4}$ certainly must have $\mathsf{e}(\Lambda^{+})=0$.
However, non-zero global sections of this bundle certainly do not exist.
Otherwise, the twistor projection $Z\to S^{4}$ would admit a smooth global
section, and we could then interpret this as an orientation-compatible almost
complex structure $J$ on $S^{4}$. However, $S^{4}$ does not admit an almost-
complex structure! For example, if it did, the index of the spinc Dirac
operator generalizing $\bar{\partial}+\bar{\partial}^{*}$ would be
$\operatorname{Td}\big{(}S^{4}\big{)}=\frac{\chi+\tau}{4}\big{(}S^{4}\big{)}=\frac{1}{2}\not\in\mathbb{Z}$;
for further discussion, see the commentary following (4.1) below. This
contradiction shows that every smooth section of $\Lambda^{+}\to S^{4}$ must
have non-empty zero locus, even through $\mathsf{e}(\Lambda^{+})=0$.

We will show in Section 3 that $\mathsf{e}(\Lambda^{+})=0$ on any smooth
oriented $4$-manifold $M$, even if $H^{3}(M,\mathbb{Z})\neq 0$. In order to do
this, however, it will help to first put the question into the broader context
of rank-$3$ oriented vector bundles on compact oriented $4$-manifolds. Here,
our approach will crucially depend on the following technical result:

######  Proposition 2.1.

Let $M$ be a smooth connected compact oriented $4$-manifold, let
$\mathsf{E}\to M$ be a real oriented rank-$3$ vector bundle, let
$\varpi\colon\mathcal{Z}\to M$ be the unit $2$-sphere bundle
$\mathcal{Z}=S(\mathsf{E})$ with respect to some positive-definite inner
product, and let $F\in H_{2}((\mathcal{Z}),\mathbb{Z})$ denote the homology
class of an $S^{2}$-fiber of $\varpi$. Then the following conditions are all
equivalent:

  1. $(i)$

The Euler class $\mathsf{e}(\mathsf{E})\in H^{3}(M,\mathbb{Z})$ vanishes;

  2. $(ii)$

There is a cohomology class $\mathbf{a}\in H^{2}(\mathcal{Z},\mathbb{Z})$ with
$\langle\mathbf{a},F\rangle=1$;

  3. $(iii)$

$H_{2}(\mathcal{Z},\mathbb{Z})\cong H_{2}(M,\mathbb{Z})\oplus\mathbb{Z}$; and

  4. $(iv)$

$|\mathfrak{T}_{2}(\mathcal{Z})|=|\mathfrak{T}_{2}(M)|$, where
$\mathfrak{T}_{k}$ is the torsion subgroup of
$H_{k}(\underline{\hphantom{M}},\mathbb{Z})$.

###### Proof.

Let us first recall that the cohomology of $\mathcal{Z}$ is related to that of
$M$ by the Gysin exact sequence [6, Section 12]

$\displaystyle\cdots
H^{k-3}(M)\stackrel{{\scriptstyle\cup\mathsf{e}}}{{\to}}H^{k}(M)\stackrel{{\scriptstyle\varpi^{*}}}{{\to}}H^{k}(\mathcal{Z})\stackrel{{\scriptstyle\varpi_{*}}}{{\to}}H^{k-2}(M)\stackrel{{\scriptstyle\cup\mathsf{e}}}{{\to}}H^{k+1}(M)\cdots,$
(2.1)

where $\mathsf{e}=\mathsf{e}(\mathsf{E})$ is the Euler class of the oriented
rank-$3$ bundle $\mathsf{E}$. This is really just a disguised form of the long
exact sequence

$\displaystyle\cdots\to
H^{k}(\mathsf{E},\mathsf{E}-M){\to}H^{k}(\mathsf{E}){\to}H^{k}(\mathsf{E}-M){\to}H^{k+1}(\mathsf{E},\mathsf{E}-M)\to\cdots$

of the pair $(\mathsf{E},\mathsf{E}-M)$, because the zero section
$M\hookrightarrow\mathsf{E}$ is a deformation retract of $\mathsf{E}$, and
$\mathcal{Z}$ is a deformation retract of $\mathsf{E}-M$; the Thom isomorphism
$H^{k-3}(M)\to H^{k}(\mathsf{E},\mathsf{E}-M)$, given by cupping with the Thom
class, therefore converts one exact sequence into the other. While this works
perfectly well with coefficients in any ring, we will actually only use it for
$\mathbb{Z}$, $\mathbb{Z}_{2}$, and $\mathbb{R}$ coefficients in this article.

First observe that the exactness of the Gysin sequence

$\displaystyle\cdots\to
H^{2}(Z,\mathbb{Z})\stackrel{{\scriptstyle\varpi_{*}}}{{\to}}H^{0}(M,\mathbb{Z})\stackrel{{\scriptstyle\mathsf{e}\cdot}}{{\to}}H^{3}(M,\mathbb{Z})\to\cdots$

implies that (i) $\Longrightarrow$ (ii), because the vanishing of
$\mathsf{e}=\mathsf{e}(\mathsf{E})$ implies the surjectivity of the map
$\varpi_{*}\colon H^{2}(\mathcal{Z},\mathbb{Z})\to
H^{0}(M,\mathbb{Z})=\mathbb{Z}$ given by
$\mathbf{a}\mapsto\langle\mathbf{a},F\rangle$.

Next, since $H^{4}(M,\mathbb{Z})=\mathbb{Z}$ is free, while
$\mathsf{e}(\mathsf{E})$ is a torsion class, observe that the terminal segment
of the Gysin sequence (2.1) breaks off as the short exact sequence

$\displaystyle 0\to
H^{4}(M,\mathbb{Z})\stackrel{{\scriptstyle\varpi^{*}}}{{\to}}H^{4}(\mathcal{Z},\mathbb{Z})\stackrel{{\scriptstyle\varpi_{*}}}{{\to}}H^{2}(M,\mathbb{Z})\to
0,$

where the image of $H^{4}(M,\mathbb{Z})$ in $H^{4}(\mathcal{Z},\mathbb{Z})$ is
generated by the Poincaré dual of the fiber class $F$. Applying Poincaré
duality in both $\mathcal{Z}$ and $M$ therefore converts this into an exact
sequence

$\displaystyle 0\to\mathbb{Z}\stackrel{{\scriptstyle\cdot
F}}{{\to}}H_{2}(\mathcal{Z},\mathbb{Z}){\to}H_{2}(M,\mathbb{Z})\to 0.$ (2.2)

If statement (ii) holds, then there is an $\mathbf{a}\in
H^{2}(\mathcal{Z},\mathbb{Z})$ with $\langle\mathbf{a},F\rangle=1$, and
pairing with $\mathbf{a}$ then provides a left inverse of
$\mathbb{Z}\hookrightarrow H_{2}(\mathcal{Z},\mathbb{Z})$. Thus (ii) implies
that the exact sequence (2.2) splits, and so yields an isomorphism

$\displaystyle H_{2}(\mathcal{Z},\mathbb{Z})\cong\mathbb{Z}\oplus
H_{2}(M,\mathbb{Z}).$ (2.3)

This shows that (ii) $\Longrightarrow$ (iii).

Next, observe that whenever (2.3) holds, one also has

$\displaystyle\mathfrak{T}_{2}(\mathcal{Z})\cong\mathfrak{T}_{2}(M),$

where $\mathfrak{T}_{k}$ denotes the torsion subgroup of the integer homology
$H_{k}$. Since this in particular implies that these torsion subgroups have
the same order, it follows that (iii) $\Longrightarrow$ (iv).

On the other hand, the universal coefficients theorem tells us that

$\displaystyle\mathfrak{T}^{3}(\mathcal{Z})\cong\mathfrak{T}_{2}(\mathcal{Z})\qquad\mbox{and}\qquad\mathfrak{T}^{3}(M)\cong\mathfrak{T}_{2}(M),$
(2.4)

where
$\mathfrak{T}^{k}=\operatorname{Ext}(\mathfrak{T}_{k-1},\mathbb{Z})\cong\mathfrak{T}_{k-1}$
denotes the torsion subgroup of the integer cohomology $H^{k}$. Thus, if
$|\mathfrak{T}_{2}(\mathcal{Z})|=|\mathfrak{T}_{2}(M)|$, it then follows that
$\big{|}\mathfrak{T}^{3}(\mathcal{Z})\big{|}=\big{|}\mathfrak{T}^{3}(M)\big{|}$.
However, the central portion of the Gysin sequence reads

$\displaystyle\cdots\to
H^{0}(M,\mathbb{Z})\stackrel{{\scriptstyle\mathsf{e}\cdot}}{{\to}}H^{3}(M,\mathbb{Z})\to
H^{3}(\mathcal{Z},\mathbb{Z})\to H^{1}(M,\mathbb{Z})\to\cdots,$

where $\mathsf{e}=\mathsf{e}(\mathsf{E})\in H^{3}(M,\mathbb{Z})$ is a
$2$-torsion class. Since $H^{1}(M,\mathbb{Z})$ is torsion-free, this means, in
particular, that

$\displaystyle\mathfrak{T}^{3}(\mathcal{Z})\cong\begin{cases}\mathfrak{T}^{3}(M)&\text{if
}\mathsf{e}(\mathsf{E})=0,\\\
\mathfrak{T}^{3}(M)/\mathbb{Z}_{2}&\text{otherwise}.\end{cases}$ (2.5)

Thus,
$\big{|}\mathfrak{T}^{3}(\mathcal{Z})\big{|}=\big{|}\mathfrak{T}^{3}(M)\big{|}$
only when $\mathsf{e}(\mathsf{E})=0$. This proves that (iv) $\Longrightarrow$
(i). We have thus shown that

$\displaystyle{\rm(i)}\Longrightarrow{\rm(ii)}\Longrightarrow{\rm(iii)}\Longrightarrow{\rm(iv)}\Longrightarrow{\rm(i)},$

and our proof of the proposition is therefore complete. ∎

##  3 Existence of spinc structures

Proposition 2.1 and some twistor geometry now imply the following:

######  Theorem 3.1.

Any smooth compact oriented $4$-manifold admits spinc structures.

###### Proof.

For any Riemannian metric $g$ on $M$, our geometric definition of a spinc
structure restates the claim as asserting the existence of a cohomology class
$\mathbf{a}\in H^{2}(Z,\mathbb{Z})$ with $\langle\mathbf{a},F\rangle=1$, where
$Z=S(\Lambda^{+})$ is the twistor space, and $F=\big{[}S^{2}\big{]}$ is the
fiber homology class. The equivalence (ii) $\Longleftrightarrow$ (iv) in
Proposition 2.1 therefore says that it suffices to show that
$|\mathfrak{T}_{2}(Z)|=|\mathfrak{T}_{2}(M)|$, where $\mathfrak{T}_{2}$
denotes the torsion subgroup of the integer homology $H_{2}$. Since (2.4) and
(2.5) also guarantee that $|\mathfrak{T}_{2}(Z)|\leq|\mathfrak{T}_{2}(M)|$, it
therefore suffices to show that the homomorphism
$\wp_{*}\colon\mathfrak{T}_{2}(Z)\to\mathfrak{T}_{2}(M)$ is surjective.

To see this, we begin by recalling that any element of $H_{2}(M,\mathbb{Z})$
can be represented by a smooth compact connected embedded oriented surface
$\Sigma^{2}\subset M^{4}$. Indeed, any homology class $[\Sigma]\in
H_{2}(M,\mathbb{Z})$ is Poincaré dual to an element of $H^{2}(M,\mathbb{Z})$,
which can then be realized as the first Chern class of a complex line bundle
on $M$. A generic smooth section of this line bundle then provides a smooth
compact oriented embedded surface ${\Sigma}_{0}$ representing $[\Sigma]$. This
representative might still not be connected, but we can then correct this by
connecting the various components of ${\Sigma}_{0}$ by narrow tubes $I\times
S^{1}$ that approximate a collection of disjoint arcs between the different
connected components.

Now suppose that $\Sigma\subset M$ is a smooth compact connected oriented
surface representing a torsion class $[\Sigma]\in\mathfrak{T}_{2}(M)\subset
H_{2}(M,\mathbb{Z})$. Since the homological self-intersection
$[\Sigma]\bullet[\Sigma]$ of a torsion class must vanish, it therefore follows
that the normal bundle $\mathfrak{N}$ of $\Sigma\subset M$ has trivial Euler
class, and is therefore trivial. We now define an orientation- and
$g$-compatible almost-complex structure $\jmath$ on
$TM|_{\Sigma}=\mathfrak{N}\oplus T\Sigma$ by declaring it to be given by
$90^{\circ}$ rotation in both $\mathfrak{N}$ and $T\Sigma$. Since this
$\jmath$ defines a section of $Z|_{\Sigma}$, its image defines an embedded
surface $\hat{\Sigma}\subset Z$ that projects diffeomorphically to $\Sigma$
via the twistor projection. However, the twistor almost-complex structure
$J_{\mathsf{H}}$ on the horizontal bundle $\mathsf{H}\subset TZ$ then
restricts to $\hat{\Sigma}$ as an exact copy of the action of $\jmath$ on
$TM|_{\Sigma}$. Since the normal bundle $\mathfrak{N}$ of $\Sigma$ is trivial,
we thus have

$\displaystyle\big{\langle}c_{1}(\mathscr{H}),[\hat{\Sigma}]\big{\rangle}=\big{\langle}c_{1}(\mathbb{C}\oplus
T^{1,0}\Sigma),[{\Sigma}]\big{\rangle}=\chi(\Sigma)=2(1-\mathfrak{g}),$

where $\mathfrak{g}$ denotes the genus of $\Sigma$ and where, once again,
$\mathscr{H}=\mathsf{H}^{1,0}$.

Now consider the homology class $A=[\hat{\Sigma}]+(\mathfrak{g}-1)F\in
H_{2}(Z,\mathbb{Z})$, where $F$ once again denotes the homology class of a
fiber of the twistor projection $\wp\colon Z\to M$. Since $\wp$ induces an
oriented diffeomorphism $\hat{\Sigma}\to\Sigma$, and since $\wp$ collapses $F$
to a point, $\wp_{*}\colon H_{2}(Z,\mathbb{Z})\to H_{2}(M,\mathbb{Z})$
therefore sends $A\mapsto[\Sigma]$. We will now prove that $A$ is a torsion
class. To show this, it suffices to check that $\langle\mathbf{a},A\rangle=0$
for every $\mathbf{a}\in H^{2}(Z,\mathbb{R})$. However, the
$\mathbb{R}$-coefficient version of the Gysin sequence (2.1) implies that
$H^{2}(Z,\mathbb{R})=\wp^{*}H^{2}(M,\mathbb{R})\oplus\mathbb{R}c_{1}(\mathscr{H})$,
since $\langle c_{1}(\mathscr{H}),F\rangle=2\neq 0$. But pairing $A$ with an
element of $\wp^{*}H^{2}(M,\mathbb{R})$ amounts to pairing
$\wp_{*}(A)=[\Sigma]$ with an element of $H^{2}(M,\mathbb{R})$, which yields
zero because $[\Sigma]$ is a torsion class by hypothesis. But because the
restriction of $\mathscr{H}$ to an $S^{2}$ fiber has degree $+2$, we also have

$\displaystyle\langle
c_{1}(\mathscr{H}),A\rangle=\big{\langle}c_{1}(\mathscr{H}),[\hat{\Sigma}]\big{\rangle}+(\mathfrak{g}-1)\langle
c_{1}(\mathscr{H}),F\rangle=2(1-\mathfrak{g})+(\mathfrak{g}-1)2=0.$

This shows that $A$ is a torsion class in $H_{2}(Z,\mathbb{Z})$ with
$\wp_{*}(A)=[\Sigma]$, and therefore proves that
$\wp_{*}\colon\mathfrak{T}_{2}(Z)\to\mathfrak{T}_{2}(M)$ is surjective, as
claimed. ∎

Applying Proposition 2.1 to the Gysin sequence, we thus have:

######  Corollary 3.2.

Any compact oriented Riemannian $4$-manifold $(M,g)$ satisfies
$\mathsf{e}(\Lambda^{+})=0$. Consequently, with either $\mathbb{Z}$ or
$\mathbb{Z}_{2}$ coefficients,

$\displaystyle 0\to
H^{k}(M)\stackrel{{\scriptstyle\wp^{*}}}{{\to}}H^{k}(Z)\stackrel{{\scriptstyle\wp_{*}}}{{\to}}H^{k-2}(M)\to
0$

is exact for every $k$, where $\wp\colon Z\to M$ is the twistor projection.

As an application, we therefore have the following:

######  Proposition 3.3.

One has $w_{2}(TM)=w_{2}(\Lambda^{+})$ for every compact oriented Riemannian
four-manifold $(M,g)$. Moreover, $w_{2}(TM)\in H^{2}(M,\mathbb{Z}_{2})$ is the
mod-$2$ reduction of some integer cohomology class $\mathbf{c}\in
H^{2}(M,\mathbb{Z})$.

###### Proof.

We have
$\wp^{*}w_{2}(TM)=w_{2}(\wp^{*}TM)=w_{2}(\mathscr{H})=\rho[c_{1}(\mathscr{H})]=\rho\big{[}c_{1}\big{(}{\wedge}^{2}\mathscr{H}\big{)}\big{]}=w_{2}\big{(}{\wedge}^{2}\mathscr{H}\big{)}\allowbreak=w_{2}\big{(}\mathbb{R}\oplus\wedge^{2}\mathscr{H}^{*}\big{)}=w_{2}(\wp^{*}\Lambda^{+})=\wp^{*}w_{2}(\Lambda^{+})$.
Since $\wp^{*}\colon H^{2}(M,\mathbb{Z}_{2})\to H^{2}(Z,\mathbb{Z}_{2})$ is
injective by Corollary 3.2, this implies that $w_{2}(TM)=w_{2}(\Lambda^{+})$.

Now consider the third integer Stiefel–Whitney class defined by
$W_{3}:=\beta(w_{2})$, where $\beta$ is the Bockstein homomorphism of the long
exact sequence

$\displaystyle\cdots\to
H^{2}(\underline{\hphantom{M}},\mathbb{Z})\stackrel{{\scriptstyle
2\cdot}}{{\to}}H^{2}(\underline{\hphantom{M}},\mathbb{Z})\stackrel{{\scriptstyle\rho}}{{\to}}H^{2}(\underline{\hphantom{M}},\mathbb{Z}_{2})\stackrel{{\scriptstyle\beta}}{{\to}}H^{3}(\underline{\hphantom{M}},\mathbb{Z})\to\cdots$
(3.1)

induced by

$\displaystyle 0\to\mathbb{Z}\stackrel{{\scriptstyle
2\cdot}}{{\to}}\mathbb{Z}\stackrel{{\scriptstyle\rho}}{{\to}}\mathbb{Z}_{2}\to
0,$

with $\rho$ denoting reduction mod $2$. Since
$w_{2}(\wp^{*}TM)=\rho[c_{1}(\mathscr{H})]$, it follows that
$\wp^{*}W_{3}(TM)=W_{3}(\wp^{*}TM)=\beta[w_{2}(\wp^{*}TM)]=0$. The injectivity
of $\wp^{*}\colon H^{3}(M,\mathbb{Z})\to H^{3}(Z,\mathbb{Z})$ guaranteed by
Corollary 3.2 thus implies that $W_{3}(TM)=\beta[w_{2}(TM)]$ vanishes, and the
exactness of (3.1) then tells us that $w_{2}(TM)$ belongs to the image of
$\rho\colon H^{2}(M,\mathbb{Z}){\to}H^{2}(M,\mathbb{Z}_{2})$. ∎

It is now easy to deduce an analogous result for the simplest non-compact
$4$-manifolds:

######  Corollary 3.4.

Let $M$ be the interior of a smooth compact oriented $4$-manifold-with-
boundary. Then $M$ admits spinc structures.

###### Proof.

If $M$ is displayed as the interior of a compact oriented $4$-manifold-with-
boundary $X$, we first construct the double $N=X\cup_{\partial X}\overline{X}$
of $X$, where $\overline{X}$ denotes $X$ equipped with the opposite
orientation. Each component of $N$ then admits a spinc structure by Theorem
3.1, so taking a union over components gives us a complex line bundle
$\mathscr{L}\to S(\Lambda^{+})$, where $\Lambda^{+}\to N$ is defined with
respect to some Riemannian metric on $N$. Since $M$ is an open subset of $N$,
the twistor space of $M$ is an open subset of the twistor space of $N$, and
restricting $\mathscr{L}\to S(\Lambda^{+})$ to this subset now gives us a
spinc structure on $M$. Of course, this construction is carried out with
respect to a Riemannian metric on $M$ that happens to arises by restriction
from $N$, but the conclusion does not depend on a choice of metric, since the
bundle-isomorphism class of $\Lambda^{+}\to M$ is actually metric-independent.
∎

There seems to be a widespread consensus [2, 5] that this result should also
hold for general non-compact $4$-manifolds, including those that are not
homotopy-equivalent to finite cell complexes. Unfortunately, however, a
watertight proof of this assertion is currently lacking. For example, it does
not suffice to exhaust $M$ by precompact regions $M_{1}\subset
M_{2}\subset\cdots\subset M_{j}\subset\cdots$ with smooth boundary, and then
observe that $\mathsf{e}(\Lambda^{+})=0$ vanishes on each $M_{j}$ by Corollary
3.4, because $H^{*}(M,\mathbb{Z})\neq\varprojlim H^{*}(M_{j},\mathbb{Z})$ in
general; cf. [6, p. 109].

##  4 Spin and spinc geometry

While every compact oriented $4$-manifold admits spinc structures, such
structures are typically far from unique:

######  Theorem 4.1.

On any smooth compact oriented $4$-manifold $M$, the cohomology group
$H^{2}(M,\mathbb{Z})$ acts freely and transitively on the set of spinc
structures on $M$.

###### Proof.

Assuming, for simplicity, that $M$ is connected, Corollary 3.2 tells us that
there is an exact sequence

$\displaystyle 0\to
H^{2}(M,\mathbb{Z})\stackrel{{\scriptstyle\wp^{*}}}{{\to}}H^{2}(Z,\mathbb{Z})\stackrel{{\scriptstyle\wp_{*}}}{{\to}}\mathbb{Z}\to
0,$

while our geometric definition tells us that $\\{\mbox{spin${}^{c}$ structures
on }M\\}$ is exactly $\wp_{*}^{-1}(1)\\!\subset\\!H^{2}(Z,\mathbb{Z})$. This
corresponds to tensoring $\mathscr{L}\to Z$ with pull-backs of line bundles on
$M$. ∎

By contrast, many $4$-manifolds do not admit spin structures:

######  Theorem 4.2.

A smooth oriented $4$-manifold $M$ admits a spin structure iff $w_{2}(TM)=0$.
When this happens, $H^{1}(M,\mathbb{Z}_{2})$ acts freely and transitively on
the spin structures of $M$.

###### Proof.

By our geometric definition, a spin structure exists iff the vertical line
bundle $\mathscr{V}\to Z$ has a square root. However, as pointed out by
Hitchin [4], there is a canonical isomorphism
$\mathscr{V}\cong\wedge^{2}\mathscr{H}$, because these two bundles have
tautological identifications with the very same line sub-bundle of
$\mathbb{C}\otimes\wp^{*}(\Lambda^{+})$. Thus, $M$ admits a spin structure iff
$c_{1}(\mathscr{H})=c_{1}\big{(}{\wedge}^{2}\mathscr{H}\big{)}=c_{1}(\mathscr{V})$
is divisible by $2$ in $H^{2}(Z,\mathbb{Z})$. But because

$\displaystyle\cdots\to H^{2}(Z,\mathbb{Z})\stackrel{{\scriptstyle
2\cdot}}{{\to}}H^{2}(Z,\mathbb{Z})\stackrel{{\scriptstyle\rho}}{{\to}}H^{2}(Z,\mathbb{Z}_{2})\to\cdots$

is exact, this happens iff
$\rho[c_{1}(\mathscr{H})]=w_{2}(\mathscr{H})=\wp^{*}w_{2}(TM)$ vanishes. Since
$\wp^{*}\colon H^{2}(M,\mathbb{Z}_{2})\to H^{2}(Z,\mathbb{Z}_{2})$ is
injective by Corollary 3.2, it therefore follows that $M$ admits a spin
structure iff $w_{2}(TM)=0$.

When $w_{2}(TM)=0$, the spin structures are exactly those double covers of the
principal $\mathbb{C}^{\times}$-bundle
$\mathscr{V}^{\times}=\mathscr{V}-\mathsf{0}_{Z}$ that also doubly cover the
fiber; equivalently, they correspond to elements of
$H^{1}(S(\mathscr{V}),\mathbb{Z}_{2})$ that are non-zero on the fiber circle.
Since $w_{2}(TM)=0$ implies that $w_{2}(\mathscr{V})=0$, the Gysin sequence of
$\mathscr{V}\to Z$ then simplifies to yield

$\displaystyle 0\to H^{1}(Z,\mathbb{Z}_{2})\to
H^{1}(S(\mathscr{V}),\mathbb{Z}_{2})\to H^{0}(Z,\mathbb{Z}_{2})\to 0$

and it follows that $H^{1}(Z,\mathbb{Z}_{2})$ acts freely and transitively on
such elements of $H^{1}(S(\mathscr{V}),\mathbb{Z}_{2})$. Since we also have
$H^{1}(M,\mathbb{Z}_{2})=H^{1}(Z,\mathbb{Z}_{2})$ by Corollary 3.2,
$H^{1}(M,\mathbb{Z}_{2})$ therefore acts freely and transitively on spin
structures by tensoring $\mathscr{V}^{1/2}$ with pull-backs of real line
bundles. ∎

We now give a direct twistorial construction of the spinor bundles of a spin
structure, and twisted-spinor bundles of a spinc structure. This provides yet
another way of seeing that our geometric definitions of such structures are
equivalent to the standard definitions.

Given a square-root $\mathscr{V}^{1/2}$ of the vertical line bundle
$\mathscr{V}\to Z$ on the twistor space, we begin by noticing that
$\mathscr{V}$ is canonically isomorphic to the tangent bundle
$T^{1,0}\mathbb{CP}_{1}(x)$ on any twistor fiber
$\mathbb{CP}_{1}(x):=\wp^{-1}(x)$, $x$. This gives $\mathscr{V}^{1/2}$ a
natural fiber-wise holomorphic structure, and we may therefore define two
$2$-dimensional vector spaces at each $x\in M$ by

$\displaystyle{\mathbb{S}}_{+{x}}=H^{0}\big{(}{\mathbb{CP}_{1}}({x}),\mathcal{O}\big{(}\mathscr{V}^{1/2}\big{)}\big{)},\qquad{\mathbb{S}}_{-{x}}=H^{0}\big{(}{\mathbb{CP}_{1}}({x}),\mathcal{O}\big{(}{\nu}\otimes\mathscr{V}^{-1/2}\big{)}\big{)},$

where the normal bundle $\nu=\mathscr{H}|_{\mathbb{CP}_{1}(x)}$ of
$\mathbb{CP}_{1}(x)\subset Z$ is thought of as a holomorphic bundle
$\cong\mathcal{O}(1)\oplus\mathcal{O}(1)$. These naturally define smooth
vector bundles $\mathbb{S}_{\pm}$, because we may define the smooth sections
of $\mathbb{S}_{+}\to M$ (respectively, $\mathbb{S}_{-}\to M$) to be the
smooth sections of $\mathscr{V}^{1/2}\to Z$ (respectively,
${\nu}\otimes\mathscr{V}^{-1/2}\to Z$) that are holomorphic on each fiber of
$\wp$. These bundles naturally reduce to the structure group ${\rm SU}(2)={\rm
Sp}(1)$, and the principal ${\rm Spin}(4)$-bundle $\widetilde{\mathfrak{F}}\to
M$ then arises as a bundle of adapted frames for
$\mathbb{S}_{+}\oplus\mathbb{S}_{-}$.

If we instead start with a degree-$1$ complex line-bundle $\mathscr{L}\to Z$,
the Gysin sequence allows us to choose an isomorphism
$\mathscr{L}^{2}=\mathscr{V}\otimes\wp^{*}\mathsf{L}$ for a unique complex
line-bundle $\mathsf{L}\to M$. Since $\wp^{*}\mathsf{L}$ has a natural flat
connection on each twistor fiber $\mathbb{CP}_{1}(x)$, this gives
${\mathscr{L}}=(\mathscr{V}\otimes\wp^{*}\mathsf{L})^{1/2}$ a natural fiber-
wise holomorphic structure. We thus obtain a pair of vector bundles
$\mathbb{V}_{\pm}\to M$ by setting

$\displaystyle{\mathbb{V}}_{+{x}}=H^{0}\big{(}{\mathbb{CP}_{1}}({x}),\mathcal{O}({\mathscr{L}})\big{)},\qquad{\mathbb{V}}_{-{x}}=H^{0}\big{(}{\mathbb{CP}_{1}}({x}),\mathcal{O}({\nu}\otimes{\mathscr{L}}\otimes{\mathscr{V}^{*}})\big{)}.$

On any spin subset of $M$, these bundles can then be re-expressed as
$\mathbb{V}_{\pm}=\mathbb{S}_{\pm}\otimes\mathsf{L}^{1/2}$. Moreover,
$\wedge^{2}\mathbb{V}_{+}=\wedge^{2}\mathbb{V}_{-}=\mathsf{L}$, and the
principal ${\rm Spin}^{c}(4)$-bundle $\widehat{\mathfrak{F}}\to M$ arises as a
bundle of adapted frames for $\mathbb{V}_{+}\oplus\mathbb{V}_{-}$.

In particular, any spinc structure gives us a rank-$2$ complex vector bundle
$\mathbb{V}_{+}\to M$ such that $\mathbb{P}(\mathbb{V}_{+})=Z=S(\Lambda^{+})$.
However, because $\operatorname{rank}_{\mathbb{R}}\mathbb{V}_{+}=4=\dim M$,
the Poincaré–Hopf paradigm applies, and can be used to predict the existence
of non-zero sections of $\mathbb{bV}_{+}$. If $M$ is compact, we can always
choose a generic section of $\mathbb{V}_{+}$ that vanishes at only a finite
number of points; by following a suitable self-isotopy of $\mathbb{V}_{+}\to
M$, we can then arrange for all of these zeroes to be contained in an
arbitrarily small ball $B_{\varepsilon}(p)\subset M$, and then use a local
trivialization over this ball to alter this section so that it only vanishes
at the center $p$ of the ball (albeit typically with high multiplicity).
Applying the projection
$\mathbb{V}_{+}-\mathsf{0}_{M}\to\mathbb{P}(\mathbb{V}_{+})=Z$, we thus obtain
a section of $\wp\colon Z-\wp^{-1}(p)\to M-\\{p\\}$, and we may then interpret
this section as an almost-complex structure $J$ on $M-\\{p\\}$. Moreover, the
image of this $J$ is a closed codimension-$2$ submanifold that is Poincaré
dual to $c_{1}(\mathscr{L})$ for the given spinc structure, and so completely
determines the spinc structure on $M-\\{p\\}$. On the other hand, removing a
point from $M$ does not change $H^{2}(\mathbb{Z})$, and Corollary 3.2
therefore tells us that spinc structures on $M$ are completely determined by
their restrictions to $M-\\{p\\}$. In summary, we have proved:

######  Theorem 4.3.

Let $(M,g)$ be a compact connected oriented Riemannian $4$-manifold, and let
${p}\in M$ be an arbitrary base-point. Then

  * •

$M-\\{p\\}$ admits almost-complex structures $J$ compatible with the given
metric and orientation;

  * •

any such $J$ determines a spinc structure on $M$; and

  * •

every spinc structure on $M$ arises this way.

Similarly, on the interior of any compact oriented $4$-manifold-with-nonempty-
boundary, there always exist almost-complex structures compatible with the
given orientation. Every such almost-complex structure moreover determines a
spinc structure, and every spinc structure arises in this way.

However, in the compact case, the count of zeroes with multiplicity for a
section of $\mathbb{V}_{+}$ is given by the Euler number

$\displaystyle\int_{M}c_{2}(\mathbb{V}_{+})=\frac{c_{1}^{2}(\mathsf{L})-(2\chi+3\tau)(M)}{4},$
(4.1)

where $\chi(M)$ and $\tau(M)$ respectively denote Euler characteristic and
signature of $M$. One can therefore find a global almost-complex structure on
$M$ if and only if $c_{1}(\mathsf{L})\in H^{2}(M,\mathbb{Z})$ can be chosen to
make the right-hand side of (4.1) vanish. This therefore happens [3] if and
only if there is some $\mathbf{c}\in H^{2}(M,\mathbb{Z})$ with
$\rho(\mathbf{c})=w_{2}(TM)$ and $\mathbf{c}^{2}=(2\chi+3\tau)(M)$.

##  5 Other dimensions

Our discussion has shown that a twistorial perspective can shed new light on
spinc geometry in dimension four. We now conclude by pointing out some partial
generalizations of these ideas to other dimension.

If $(M,g)$ is any oriented Riemannian $2m$-manifold, its twistor space $Z$ is
the fiber bundle $Z=\mathfrak{F}/{\rm U}(m)$, where $\mathfrak{F}\to M$ is the
principal ${\rm SO}(2m)$-bundle of oriented orthonormal frames for $TM$. Each
fiber ${\digamma}$ of $Z$ is thus a copy of the homogeneous space ${\rm
SO}(2m)/{\rm U}(m)$ of real dimension $m(m-1)$. However, ${\digamma}$ is
actually a compact Hermitian symmetric space, and so is naturally a compact
complex Fano manifold of complex dimension $\mathsf{d}=\binom{m}{2}$. The
twistor fiber $\digamma_{x}$ over $x\in M$ thus parameterizes the complex
structures on $T_{x}M\cong\mathbb{R}^{2m}$ that are compatible with the given
metric and orientation; equivalently, $\digamma$ is just the space of
$\alpha$-planes [8, Appendix] in $\mathbb{C}\otimes
T_{x}M\cong\mathbb{C}^{2m}$. Consequently, the twistor space $Z$ again admits
a tautological almost-complex structure $J=J_{\mathsf{H}}\oplus
J_{\mathsf{V}}$, allowing us to view the horizontal and vertical subspaces of
$TZ$ as complex vector bundles $\mathscr{H}=\mathsf{H}^{1,0}$ and
$\mathscr{V}=\mathsf{V}^{1,0}$. Because there is a natural isomorphism
$\mathscr{V}=\wedge^{2}\mathscr{H}$, the vertical anti-canonical line-bundle
$K^{-1}_{\wp}=\wedge^{\mathsf{d}}\mathscr{V}$ has a natural $(m-1)^{\rm st}$
root $K^{-1/(m-1)}_{\wp}$ given by $\wedge^{m}\mathscr{H}$. On the other hand,
$c_{1}\big{(}K^{-1/(m-1)}_{\wp}\big{)}$ restricts to any fiber as twice the
generator of $H^{2}({{\digamma}},\mathbb{Z})=\mathbb{Z}$. A spin structure on
$M$ is just a square root $K^{-1/2(m-1)}_{\wp}$ of $K^{-1/(m-1)}_{\wp}$, while
a spinc structure on $M$ is simply a line bundle on $Z$ whose Chern class
restricts to a fiber ${{\digamma}}$ as the generator of
$H^{2}({{\digamma}},\mathbb{Z})=\mathbb{Z}$. The corresponding spinor and
twisted-spinor vector bundles can then be manufactured by a straightforward
generalization of the constructions described in Section 4.

The odd-dimensional case is similar. If $(M,g)$ is an oriented Riemannian
$(2m-1)$-manifold, with oriented orthonormal frame bundle $\mathfrak{F}\to M$,
its twistor space is defined to be $\mathfrak{F}/{\rm U}(m-1)$, and its fibers
${\digamma}={\rm SO}(2m-1)/{\rm U}(m-1)\cong{\rm SO}(2m)/{\rm U}(m)$ are thus
identical to the fibers discussed above. The vertical anti-canonical line
bundle still has a natural $(m-1)^{\rm st}$ root $K^{-1/(m-1)}_{\wp}$, and a
spin structure on $M$ is again just a square-root $K^{-1/2(m-1)}_{\wp}$ of
this line-bundle. A spinc structure is once again just a line-bundle on $Z$ of
fiber-degree $1$.

However, our Gysin-sequence approach to the $4$-dimensional case does not
generalize to higher dimensions. This makes the use of the Leray–Serre
spectral sequence absolutely essential for a full understanding of the
topological issues that arise in this broader context.

### Acknowledgements

This article is dedicated to my friend and teacher Sir Roger Penrose, in
celebration of his ninetieth birthday and recent Nobel Prize in Physics. It is
a pleasure to thank Dennis Sullivan for his advice and encouragement, and
Jiahao Hu for some very illuminating conversations. This research was
supported in part by NSF grant DMS-1906267.

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