Re: Topology on cohomology groups

[Andrew Salch <asalch@math.jhu.edu>]

This certain does happen--if E^* is a generalized cohomology theory, and X
is an infinite complex, we often want to topologize E^*(X) by the inverse
limit of the E-cohomology of the finite skeleta of X. This is why, for
example, if E is a complex-oriented cohomology theory, the E-cohomology of
infinite-dimensional complex projective space is a power series ring (and
not merely a polynomial ring), in one variable, over the coefficient ring
E^*.

This is sometimes important and useful in topology (for example, in the
situation above, where one uses the above description of E^*(CP^{\infty})
to associate a 1-dimensional formal group law to E), and sometimes it's
more just a hassle: for example, the early papers on the Adams-Novikov
spectral sequence used MU-cohomology (i.e., complex cobordism), and this
necessarily meant keeping track of the topology on MU^*(X) of various
spectra E, since for example MU^*(MU), the ring of stable natural
transformations of MU^*, has infinite homogeneous sums, and one had to
handle completed tensor products of MU^*(MU)-modules; the modern way is to
use generalized homology instead of generalized cohomology for these
generalized Adams spectral sequences, which does away with the topologies
and the need for completed tensor products (of course, the price one pays
is that one is then, in the case of MU, dealing with MU_*(MU)-comodules
rather than (topological) MU^*(MU)-modules, and computing Cotor rather
than Ext; but this seems to be worth it).

There's some discussion of this in Ravenel's green book. The paper on
unstable operations by Boardman, Johnson, and Wilson in the Handbook of
Algebraic Topology also includes some discussion and some nice
manipulations of topologies (again, coming from the finite skeleta of an
infinite complex) on some generalized cohomology rings and modules.

There are also generalized homology theories, like the Morava E-theories,
which occur as completions of other generalized homology theories, and so
E_*(X) naturally has a topology (coming from the completion) when E is one
of these theories; recent developments in stable homotopy theory make it
seem likely that there will be more such theories in our future.

Hope this is useful to you,
Andrew S.


On Fri, 19 Jun 2009, Steve Vickers wrote:

> A cohomology group can easily be an infinite power of the coefficient
> group. But such a group has a natural non-discrete topology, namely the
> compact-open (which in this case is also the product topology).
>
> Are there approaches to cohomology that, as part of the process, also
> supply topologies on the cohomology groups?
>
> [I'm trying to understand the topos-theoretic account of cohomology as
> in Johnstone's "Topos Theory". But it looks heavily dependent on having
> a classical base topos, since it uses the classical proof of sufficiency
> of injectives (together with the existence of Barr covers) to deduce the
> same property internally in any Grothendieck topos. For a more fully
> constructive theory I wonder if one needs to take better care of the
> topologies.]
>
> Steve Vickers.
>

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