In mathematics, complex cobordism is a
generalized cohomology theory related to
cobordism of
manifolds. Its
spectrum
A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors ...
is denoted by MU. It is an exceptionally powerful
cohomology
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
theory, but can be quite hard to compute, so often instead of using it directly one uses some slightly weaker theories derived from it, such as
Brown–Peterson cohomology In mathematics, Brown–Peterson cohomology is a generalized cohomology theory introduced by
, depending on a choice of prime ''p''. It is described in detail by .
Its representing spectrum is denoted by BP.
Complex cobordism and Quillen's idempo ...
or
Morava K-theory In stable homotopy theory, a branch of mathematics, Morava K-theory is one of a collection of cohomology theories introduced in algebraic topology by Jack Morava in unpublished preprints in the early 1970s. For every prime number ''p'' (which is sup ...
, that are easier to compute.
The generalized homology and cohomology complex cobordism theories were introduced by using the
Thom spectrum In mathematics, the Thom space, Thom complex, or Pontryagin–Thom construction (named after René Thom and Lev Pontryagin) of algebraic topology and differential topology is a topological space associated to a vector bundle, over any paracompact ...
.
Spectrum of complex cobordism
The complex bordism
of a space
is roughly the group of bordism classes of manifolds over
with a complex linear structure on the stable
normal bundle
In differential geometry, a field of mathematics, a normal bundle is a particular kind of vector bundle, complementary to the tangent bundle, and coming from an embedding (or immersion).
Definition
Riemannian manifold
Let (M,g) be a Riemannian m ...
. Complex bordism is a generalized
homology theory, corresponding to a spectrum MU that can be described explicitly in terms of
Thom space In mathematics, the Thom space, Thom complex, or Pontryagin–Thom construction (named after René Thom and Lev Pontryagin) of algebraic topology and differential topology is a topological space associated to a vector bundle, over any paracompact s ...
s as follows.
The space
is the
Thom space In mathematics, the Thom space, Thom complex, or Pontryagin–Thom construction (named after René Thom and Lev Pontryagin) of algebraic topology and differential topology is a topological space associated to a vector bundle, over any paracompact s ...
of the universal
-plane bundle over the
classifying space
In mathematics, specifically in homotopy theory, a classifying space ''BG'' of a topological group ''G'' is the quotient of a weakly contractible space ''EG'' (i.e. a topological space all of whose homotopy groups are trivial) by a proper free ac ...
of the
unitary group
In mathematics, the unitary group of degree ''n'', denoted U(''n''), is the group of unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group . Hyperorthogonal group is ...
. The natural inclusion from
into
induces a map from the double
suspension
Suspension or suspended may refer to:
Science and engineering
* Suspension (topology), in mathematics
* Suspension (dynamical systems), in mathematics
* Suspension of a ring, in mathematics
* Suspension (chemistry), small solid particles suspende ...
to
. Together these maps give the spectrum
; namely, it is the
homotopy colimit
In mathematics, especially in algebraic topology, the homotopy limit and colimitpg 52 are variants of the notions of limit and colimit extended to the homotopy category \text(\textbf). The main idea is this: if we have a diagramF: I \to \textbfc ...
of
.
Examples:
is the sphere spectrum.
is the
desuspension In topology, a field within mathematics, desuspension is an operation inverse to suspension.
Definition
In general, given an ''n''-dimensional space X, the suspension \Sigma has dimension ''n'' + 1. Thus, the operation of suspension creat ...
of
.
The
nilpotence theorem
In algebraic topology, the nilpotence theorem gives a condition for an element in the homotopy groups of a ring spectrum to be nilpotent, in terms of the complex cobordism spectrum \mathrm. More precisely, it states that for any ring spectrum R, t ...
states that, for any
ring spectrum
In stable homotopy theory, a ring spectrum is a spectrum ''E'' together with a multiplication map
:''μ'': ''E'' ∧ ''E'' → ''E''
and a unit map
: ''η'': ''S'' → ''E'',
where ''S'' is the sphere spectrum. These maps have to satisfy a ...
, the kernel of
consists of nilpotent elements.
[http://www.math.harvard.edu/~lurie/252xnotes/Lecture25.pdf ] The theorem implies in particular that, if
is the sphere spectrum, then for any
, every element of
is nilpotent (a theorem of
Goro Nishida
was a Japanese mathematician. He was a leading member of the Japanese school of homotopy theory, following in the tradition of Hiroshi Toda.
Nishida received his Ph.D. from Kyoto University in 1973, after spending the 1971–72 academic year a ...
). (Proof: if
is in
, then
is a torsion but its image in
, the
Lazard ring In mathematics, Lazard's universal ring is a ring introduced by Michel Lazard in over which the universal commutative one-dimensional formal group law is defined.
There is a universal commutative one-dimensional formal group law over a universal ...
, cannot be torsion since
is a polynomial ring. Thus,
must be in the kernel.)
Formal group laws
and showed that the coefficient ring
(equal to the complex cobordism of a point, or equivalently the ring of cobordism classes of stably complex manifolds) is a polynomial ring