Skip to main content Accessibility help
×
Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-23T06:53:04.812Z Has data issue: false hasContentIssue false

1 - Operations of the nth kind in K-theory, and what we don't know about RP

Published online by Cambridge University Press:  05 April 2013

Get access

Summary

Operations of the nth kind in K-theory

In the old days, if you wanted to solve some concrete problem in homotopy theory, you began by calculating the ordinaty cohomology groups of all the spaces involved. Then you used primary cohomology operations, such as cup-products and the Steenrod operations. If, or when, those didn't yield enough information you tried secondary ones, and then tertiary and higher ones. Of course, if the problem needed tertiary operations you didn't publish the argument in that form because it was too nasty. However, it was sometimes possible to avoid some of the nastiness by using suitable formal machinery like the Adams spectral sequence.

A little later we realised, with great pleasure, that sometimes by using a generalised cohomology theory – perhaps with primary operations – you could successfully tackle a geometrical problem which if done by ordinary cohomology would have needed operations of arbitrarily high kind. It was always conceded that the choice of the cohomology theory most useful for a particular problem might take hard work, or luck, or both. But there was a sort of democratic movement, which proclaimed that every generalised cohomology theory deserved equal rights. For example, Atiyah showed that it is technically possible to teach K-theory before ordinary cohomology. Of course all normal people still did their calculations in ordinary cohomology first, but they were made to feel that they were mere slaves of habit. The philosophy prevailed that all the apparatus of calculation, with which we are so familiar in the ordinary case, should be set up for generalised cohomology. It was conceded that some results like the Kunneth theorem might require restrictive hypotheses.

Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 1974

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×