We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
Published online by Cambridge University Press: 03 May 2010
Differential homological algebra is used in algebraic topology to study the homology of fibrations. The Serre spectral sequence determines the homology of the total space given the homologies of the base and fibre. In general, given the homologies of two out of the three, base, fibre, and total space, various techniques have been developed to determine the homology of the third.
Two cases of historical interest are the following: (1) Given the homology of a space, determine the homology of the loop space. (2) Given the homology of a topological group, determine the homology of the classifying space. In a real sense, these problems are dual to one another and have specific solutions.
The problem of determining the homology of the loop space is solved with the help of the cobar construction introduced by Frank Adams. This differential algebra, when applied to the chains on a space, is homologically equivalent to the chains on the loop space together with the multiplication induced by loop multiplication.
The problem of determining the homology of the classifying space is solved with the help of the bar construction introduced by Eilenberg and MacLane and followed up in a geometric form by John Milnor. This differential coalgebra, when applied to the chains on a topological group, is homologically equivalent to the chains on the classifying space together with the coalgebra structure given by the diagonal.
To save this book to your Kindle, first ensure no-reply@cambridge.org 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.
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.
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.
