Bordism
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1 Introduction
The theory of bordism is is one of the most deep and influential parts of algebraic topology. The foundations of bordism were laid in the pioneering works of Pontrjagin [Pontryagin1959] and Thom [Thom1954], and the theory experienced a spectacular development in the 1960s. The main introductory reference is the monograph [Stong1968].
Basic geometric constructions of bordism and cobordism, as well as homotopical definitions are summarised here. For more information, see B-Bordism and the pages on specific bordism theories, such as unoriented, oriented and complex bordism.
2 The bordism relation
All manifolds here are assumed to be smooth, compact and closed
(without boundary), unless otherwise specified. Given two
-dimensional manifolds
and
, a bordism
between them is an
-dimensional manifold
with boundary,
whose boundary is the disjoint union of
and
, that is,
. If such
exists,
and
are called bordant. The bordism relation splits manifolds
into equivalence classes (see Figure), which are called
bordism classes.
3 Unoriented bordism
We denote the bordism class of by
, and denote by
the set of bordism classes of
-dimensional
manifolds. Then
is an abelian group with respect
to the disjoint union operation:
.
Zero is represented by the bordism class of an empty set (which is
counted as a manifold in any dimension), or by the bordism class
of any manifold which bounds. We also have
, so that
is a 2-torsion group.
Set . The
product of bordism classes, namely
, makes
a graded commutative ring known as
the unoriented bordism ring.
For any (good) space the bordism relation can be extended to
maps of
-dimensional manifolds to
: two maps
and
are bordant if there is a bordism
between
and
and the map
extends to a map
. The set of bordism classes of maps
forms an
abelian group called the
-dimensional unoriented bordism group of
and denoted
(other notations:
,
).
The assignment defines a
generalised homology theory, that is, satisfies the homotopy invariance, has
the excision property and exact sequences of pairs. For this
theory we have
, and
is an
-module.
The Pontrjagin-Thom construction reduces the calculation of the bordism groups to a homotopical problem:
![\displaystyle O_n(X)=\lim_{k\to\infty}\pi_{k+n}\bigl((X_+)\wedge MO(k)\bigr)](/images/math/7/2/9/729ef1affd3c041b889a5b73198cd51e.png)
where , and
is the Thom space of
the universal vector
-plane bundle
. The
cobordism groups are defined dually:
![\displaystyle O^n(X)=\lim_{k\to\infty}[\Sigma^{k-n}(X_+),MO(k)]](/images/math/b/b/0/bb05a687d97e200b92528c433a6d0273.png)
where denotes the set of based homotopy classes of maps from
to
. The resulting generalised cohomology theory is multiplicative, which implies that
is a graded commutative ring. It follows from the definitions that
. The graded ring
with
is called the unoriented cobordism ring. It has nonzero elements only in nonpositively graded components. The bordism ring
and the cobordism ring
differ only by their gradings, so the notions of the "bordism class" and "cobordism class" of a manifold
are interchangeable. The difference between bordism and cobordism appears only when one considers the generalised cohomology theory.
4 Oriented and complex bordism
The most important examples of bordism theories arise from extending the bordism relation to manifolds endowed with some additional structure. To take account of this structure in the definition of bordism one requires that , where the structure on
is induced from that on
, and
denotes the manifold with the opposite structure. The universal homotopical framework for geometric bordism with additional structure is provided by the theory of B-bordism.
The simplest additional
structure is an orientation. The oriented bordism relation
arises accordingly. The oriented bordism ring
is defined similarly to
, with
the only difference that
. Elements of
generally do not have order 2.
Complex structure gives another important example of an additional
structure on manifolds. However, a direct attempt to define the
bordism relation on complex manifolds fails because the manifold
is odd-dimensional and therefore cannot be complex. This can
be remedied by considering stably complex (also known as
weakly almost complex, stably almost complex or quasicomplex) structures.
Let denote the tangent bundle of
, and
the product vector bundle
over
. A tangential stably complex structure on
is
determined by a choice of an isomorphism
![\displaystyle c_{\mathcal T}\colon {\mathcal T}\!M\oplus \underline{\mathbb R}^k\to \xi](/images/math/2/f/1/2f19edd9c6cb0b41b14d224c85c7312c.png)
between the "stable" tangent bundle and a complex vector
bundle over
. Some of the choices of such isomorphisms
are deemed to be equivalent, i.e. determining the same stably
complex structures (see details in Chapters II and VII of [Stong1968]). In
particular, two stably complex structures are equivalent if they
differ by a trivial complex summand. A normal stably complex structure on
is determined by a choice of a complex bundle
structure in the normal bundle
of an embedding
. Tangential and normal stably
complex structures on
determine each other by means of the
canonical isomorphism
. We therefore may restrict
our attention to tangential structures only.
A stably complex manifold is a pair consisting of a manifold
and a stably complex structure
on it. This is a generalisation of a complex and
almost complex manifold (where the latter means a manifold
with a choice of a complex structure on
, i.e. a
stably complex structure
with
).
Example 4.1.
Let . The standard complex structure on
is
equivalent to a stably complex structure determined by the
isomorphism
![\displaystyle {\mathcal T}(\mathbb{C}P^1)\oplus\underline{\mathbb R}^2\stackrel{\cong}{\longrightarrow} \overline{\eta}\oplus \overline{\eta}](/images/math/4/b/b/4bbdec9b83c90eda22e6035e1b946dc2.png)
where is the Hopf line bundle. On the other hand, the
isomorphism
![\displaystyle {\mathcal T}(\mathbb{C}P^1)\oplus\underline{\mathbb R}^2\stackrel{\cong}{\longrightarrow} \eta\oplus \overline{\eta}\cong \underline{\mathbb C}^2](/images/math/d/e/7/de72e48450e5f811dd4ad211c544427b.png)
determines a trivial stably complex structure on .
The bordism relation can be defined between stably complex
manifolds. Like the case of unoriented bordism, the set of
bordism classes of
-dimensional stably complex manifolds is an Abelian group with respect to the disjoint union. This group is called the
-dimensional complex bordism group and
denoted by
. The zero is represented by the bordism
class of any manifold
which bounds and whose stable tangent
bundle is trivial (and therefore isomorphic to a product complex
vector bundle
). The sphere
provides an example
of such a manifold. The opposite element to the bordism class
in the group
may be
represented by the same manifold
with the stably complex
structure determined by the isomorphism
![\displaystyle {\mathcal T}\!M\oplus\underline{\mathbb R}^k\oplus\underline{\mathbb R}^2\stackrel{c_{\mathcal T}\oplus e}{\relbar\joinrel\hspace{-1pt}\relbar\joinrel\hspace{-1pt}\longrightarrow}\xi\oplus\underline{\mathbb C}](/images/math/3/0/0/30016dd60828ef8d72d3e84bf11c43ff.png)
where is given by
.
An abbreviated notation for the complex
bordism class will be used whenever the stably complex structure
is clear from the context.
The complex bordism and cobordism groups of a space
are defined similarly to the unoriented case:
![\displaystyle \begin{aligned} U_n(X)&=\lim_{k\to\infty}\pi_{2k+n}((X_+)\wedge MU(k)),\\ U^n(X)&=\lim_{k\to\infty}[\Sigma^{2k-n}(X_+),MU(k)] \end{aligned}](/images/math/3/8/8/388203cfec203d491a8ba0f9062a963f.png)
where is the Thom space of the universal complex
-plane
bundle
. These groups are
-modules
and give rise to a multiplicative (co)homology theory. In
particular,
is a graded ring.
The graded ring with
is
called the complex cobordism ring; it has nontrivial
elements only in nonpositively graded components.
5 Connected sum and bordism
For manifolds of positive dimension the disjoint union representing the sum of bordism classes
may be
replaced by their "connected sum", which represents the same
bordism class.
The connected sum of manifolds
and
of the same dimension
is constructed as follows. Choose points
and
, and take closed
-balls
and
around them (both
manifolds may be assumed to be endowed with a Riemannian metric).
Fix an isometric embedding
of a pair of standard
-balls
(here
) into
which maps
onto
and
onto
. If both
and
are oriented
we additionally require the embedding
to preserve the
orientation on the first ball and reverse in on the second. Now, using this
embedding, replace in
the pair of balls
by a "pipe"
. After smoothening the
angles in the standard way we obtain a smooth manifold
.
If both and
are connected the smooth structure on
does not depend on a choice
of points
,
and embedding
. It does however depend on the orientations;
and
are not diffeomorphic in general.
There are smooth contraction maps and
. In the oriented case the manifold
can be oriented in such a way that both contraction
maps preserve the orientations.
A bordism between and
may be
constructed as follows. Consider a cylinder
, from
which we remove an
-neighbourhood
of the point
. Similarly,
remove the neighbourhood
from
(each of these two neighbourhoods can be identified
with the half of a standard open
-ball). Now connect the
two remnants of cylinders by a "half pipe"
in
such a way that the half-sphere
is identified
with the half-sphere on the boundary of
, and
is identified
with the half-sphere on the boundary of
. Smoothening the angles we obtain a manifold with boundary
(or
in the
oriented case), see Figure.
If and
are stably complex manifolds, then
there is a canonical stably complex structure on
,
which is constructed as follows. Assume the stably complex
structures on
and
are determined by isomorphisms
![\displaystyle c_{\mathcal T\!,1}\colon{\mathcal T}\!M_1\oplus\underline{\mathbb R}^{k_1}\to\xi_1 \quad\text{and}\quad c_{\mathcal T\!,2}\colon{\mathcal T}\!M_2\oplus\underline{\mathbb R}^{k_2}\to\xi_2.](/images/math/a/d/8/ad8c69fb29435893bf4a92dd2593bcfe.png)
Using the isomorphism , we define a stably complex structure on
by the isomorphism
![\displaystyle {\mathcal T}(M_1\mathbin\# M_2)\oplus\underline{\mathbb R}^{n+k_1+k_2} \cong p_1^*{\mathcal T}\!M_1\oplus\underline{\mathbb R}^{k_1}\oplus p_2^*{\mathcal T}\!M_2\oplus\underline{\mathbb R}^{k_2} \xrightarrow{c_{{\mathcal T},1}\oplus c_{{\mathcal T},2}} p_1^*\xi_1\oplus p_2^*\xi_2.](/images/math/7/a/8/7a8eebea1b6d7ffc84573feb26a65b37.png)
This stably complex structure is called the
connected sum of stably complex structures on and
. The corresponding complex bordism class is
.
6 Structure results
The theory of unoriented (co)bordism was first to be completed: its coefficient ring was calculated by Thom, and the bordism groups
of cell complexes
were reduced to
homology groups of
with coefficients in
. The
corresponding results are summarised as follows:
Theorem 6.1.
- Two manifolds are unorientedly bordant if and only if they have identical sets of Stiefel-Whitney characteristic numbers.
is a polynomial ring over
with one generator
in every positive dimension
.
- For every cell complex
the module
is a free graded
-module isomorphic to
.
Parts 1 and 2 were done by [Thom1954]. Part 3 is due to [Conner&Floyd1964].
Calculating the complex bordism ring turned out to
be a much more difficult problem:
Theorem 6.2.
is a polynomial ring over
generated by the bordism classes of complex projective spaces
,
.
- Two stably complex manifolds are bordant if and only if they have identical sets of Chern characteristic numbers.
is a polynomial ring over
with one generator
in every even dimension
, where
.
Part 1 can be proved by the methods of Thom. Part 2 follows from the results of [Milnor1960] and [Novikov1960]. Part 3 is the most difficult one; it was done in 1960 by [Novikov1960] (see also [Novikov1962] for a more detailed account) and Milnor (unpublished, but see [Thom1995]).
Note that part 3 of Theorem 6.1 does not extend to
complex bordism; is not a free
-module in
general. Unlike the case of unoriented bordism, the calculation of complex bordism of a space
does not reduce to
calculating the coefficient ring
and homology
groups
.
The calculation of the oriented bordism ring was completed by
[Novikov1960] (ring structure modulo torsion) and [Wall1960] (additive torsion), with important contributions made by
Rokhlin, Averbuch, and Milnor. Unlike complex bordism, the
ring has additive torsion. We give only a partial
result here, which does not fully describe the torsion elements. For the complete description of the ring
see the Oriented bordism page.
Theorem 6.3.
is a polynomial ring over
generated by the bordism classes of complex projective spaces
,
.
- The subring
of torsion elements contains only elements of order 2. The quotient
is a polynomial ring over
with one generator
in every dimension
where
.
- Two oriented manifolds are bordant if and only if they have identical sets of Pontrjagin and Stiefel-Whitney characteristic numbers.
For more specific information about the three bordism theories, including constructions of manifolds representing polynomial generators in the bordism rings and applications, see unoriented bordism, oriented bordism, and complex bordism pages.
7 References
- [Conner&Floyd1964] P. E. Conner and E. E. Floyd, Differentiable periodic maps, Academic Press Inc., Publishers, New York, 1964. MR0176478 (31 #750) Zbl 0417.57019
- [Milnor1960] J. Milnor, On the cobordism ring
and a complex analogue. I, Amer. J. Math. 82 (1960), 505–521. MR0119209 (22 #9975) Zbl 0095.16702
- [Novikov1960] S. P. Novikov, Some problems in the topology of manifolds connected with the theory of Thom spaces, Soviet Math. Dokl. 1 (1960), 717–720. MR0121815 (22 #12545) Zbl 0094.35902
- [Novikov1962] S. P. Novikov, Homotopy properties of Thom complexes, Mat. Sb. (N.S.) 57 (99) (1962), 407–442. MR0157381 (28 #615) Zbl 0193.51801
- [Pontryagin1959] L. S. Pontryagin, Smooth manifolds and their applications in homotopy theory, Amer. Math. Soc. Translations, Ser. 2, Vol. 11, Providence, R.I. (1959), 1–114. MR0115178 (22 #5980) Zbl 0084.19002
- [Stong1968] R. E. Stong, Notes on cobordism theory, Princeton University Press, Princeton, N.J., 1968. MR0248858 (40 #2108) Zbl 0277.57010
- [Thom1954] R. Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28 (1954), 17–86. MR0061823 (15,890a) Zbl 0057.15502
- [Thom1995] R. Thom, Travaux de Milnor sur le cobordisme, Séminaire Bourbaki, Vol. 5, Exp. No. 180, Soc. Math. France, Paris, (1995), 169–177. MR1603465 Zbl 0116.40402
- [Wall1960] C. T. C. Wall, Determination of the cobordism ring, Ann. of Math. (2) 72 (1960), 292–311. MR0120654 (22 #11403) Zbl 0097.38801