differential geometry w notes from teacher_Part_90

# differential geometry w notes from teacher_Part_90 - 179...

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

7.3. SINGULAR HOMOLOGY GROUPS 179 Let M and V be manifolds, G be a n Abelain group, F : M V be a map and F * : C p ( M ; G ) C p ( V ; G ) be the induced homomorphism of chain groups. Since the induced homomorphism F * commutes with the boundary homo- morphism , the groups Z p ( M ; G ) and B p ( M ; G ) are closed under F * . Therefore, the homomorphism F * naturally acts on the homology groups F * : H p ( M ; G ) H p ( V ; G ) . If F : M V is a homeomorphism, then there is the inverse homeomor- phism F - 1 : V M and the inverse induced homomorphism F - 1 * : H p ( V ; G ) H p ( M ; G ) . In this case, the induced homomorphism F * is an isomorphism. Theorem 7.3.2 Let M and V be compact homeomorphic manifolds and G be an Abelian group. Then their homology groups are isomor- phic, that is, for any p H p ( M ; K ) H p ( V ; G ) . Proof : Follows from above. Thus, homology groups are topological invariants . Corollary 7.3.2 Let M and V be compact manifolds. If there is an Abelian group G and an integer p such that their homology groups are

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