E -spectra and Dyer-Lashof operations

Authored by: Tyler Lawson

Handbook of Homotopy Theory

Print publication date:  December  2019
Online publication date:  December  2019

Print ISBN: 9780815369707
eBook ISBN: 9781351251624
Adobe ISBN:


 Download Chapter



Cohomology operations are absolutely essential in making cohomology an effective tool for studying spaces. In particular, the mod-p cohomology groups of a space X are enhanced with a binary cup product, a Bockstein derivation, and Steenrod’s reduced power operations; these satisfy relations such as graded-commutativity, the Cartan formula, the Adem relations, and the instability relations [90]. The combined structure of these cohomology operations is very effective in homotopy theory because of three critical properties.

These operations are natural. We can exclude the possibility of certain maps between spaces because they would not respect these operations.

These operations are constrained. We can exclude the existence of certain spaces because the cup product and power operations would be incompatible with the relations that must hold.

These operations are complete. Because cohomology is representable, we can determine all possible natural operations which take an n-tuple of cohomology elements and produce a new one. All operations are built, via composition, from these basic operations. All relations between these operations are similarly built from these basic relations.

Search for more...
Back to top

Use of cookies on this website

We are using cookies to provide statistics that help us give you the best experience of our site. You can find out more in our Privacy Policy. By continuing to use the site you are agreeing to our use of cookies.