

Cohomology Operations

Theory of cohomology operations was developed by
Steenrod for
the singular cohomology theory
.
There are several ways to construct cohomology
operations in the singular cohomology theory. There is
a purely algebraic treatment. See
[
Woo97
]
by Wood and
[
Smi
]
by Larry Smith. It seems their
q
deformations are defined
[
HT04
]
and studied from algebraic viewpoints
[
BGW
]
.
We can also develope theory of cohomology operations
in
generalized cohomology theories
, although they are much more complicated in general.
We also have notions of secondary and even higher
cohomology operatioins, which can be used to detect
topological informations that are invisible by primary
operations.
The Adams spectral sequence
can be regarded as a systematic way of handling
higher order cohomology operations.
Cohomology operations are also important for
cohomology theories defined on categories other than
the category of spaces. For example, Batanin, Berger,
and Markl studied
operads
acting on
the Hochschild cochains
in
[
BBM
]
.
[BBM]
Michael Batanin, Clemens Berger, and Martin Markl.
Operads of natural operations I: Lattice paths,
braces and Hochschild cochains,
arXiv:0906.4097
.
[BGW]
Francois Bergeron, Adriano Garsia, and Nolan
Wallach. Harmonics for Deformed Steenrod Operators,
arXiv:0812.3566
.
[HT04]
F. Hivert and N. M. Thié§»y.
Deformation of symmetric functions and the rational
Steenrod algebra. In
Invariant theory in all
characteristics
, volume 35 of
CRM Proc. Lecture Notes
, pages 91–125. Amer. Math. Soc., Providence,
RI, 2004,
arXiv:0812.3056
.
[Smi]
Larry Smith. An algebraic introduction to the
Steenrod algebra,
arXiv:0903.4997
.
[Woo97]
R. M. W. Wood. Differential operators
and the Steenrod algebra.
Proc. London Math. Soc. (3)
, 75(1):194–220, 1997.


