lecture29

# lecture29 - Lecture 29 We have been studying the important...

This preview shows pages 1–2. Sign up to view the full content.

Lecture 29 We have been studying the important invariant called the degree of f . Today we show that the degree is a “topological invariant.” 5.3 Topological Invariance of Degree Recall that given a subset A of R m and a function F : A R , we say that F is C if it extends to a C map on a neighborhood of A . Let U be open in R n , let V be open in R k , and let A = U × [0 , 1]. Definition 5.22. Let f 0 , f 1 : U V be C maps. The maps f 0 and f 1 are homotopic if there is a C map F : U × [0 , 1] V such that F ( p, 0) = f 0 ( p ) and F ( p, 1) = f 1 ( p ) for all p U . Let f t : U V be the map defined by f t ( p ) = F ( p, t ) . (5.144) Note that F C = So, f t : U V , where 0 t 1, gives a family f t C . of maps parameterized by t . The family of maps f t is called a C deformation of f 0 into f 1 . Definition 5.23. The map F is a proper homotopy if for all compact sets A V , the pre-image F 1 ( A ) is compact. Denote by π the map π : U × [0 , 1] U that sends ( p, t ) t . Let A V be compact. Then B = π ( F 1 ( A )) is compact, and for all t , f 1 ( A ) B . As a t consequence, each f t is proper.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern