Published online by Cambridge University Press: 05 June 2014
All differences in this world are of degree, and not of kind, because oneness is the secret of everything.
Swami VivekanandaSuppose you are walking your dog Spot. You put the leash on Spot and take him to his favorite tree to do his business, but he sees a squirrel on the tree and takes off after it. The squirrel is really tired of being chased, and decides to teach Spot a lesson. So, instead of climbing back up the tree, he runs in a counterclockwise direction around the trunk with Spot not far behind. As Spot follows him around the tree, the leash gets wound around the tree k times (assuming you stay in place while he is running). At this point Spot gives up and sits down near your feet to bark sullenly at the squirrel. We say that the winding number of the leash is k. No matter how you try to move the leash, unless you cut it, it will remain wound around the tree k times. That is, its winding number is a homotopy invariant. This prosaic example generalizes to higher dimensions and has interesting mathematical and physical applications.
We start with the stack of records theorem, so-called because it reveals that all smooth maps from a compact manifold to another manifold of the same dimension look like a smooth covering by a stack of records. (See Figure 9.1.)
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.