**Lemma**. Let be a commutative ring and let be a group. We define the map by

Then is an onto ring homomorphism and is a basis for the free -module… 217 more words

**Lemma**. Let be a commutative ring and let be a group. We define the map by

Then is an onto ring homomorphism and is a basis for the free -module… 217 more words

In the previous lecture, we saw that Ricci flow with surgery ensures that the second homotopy group became extinct in finite time (assuming, as stated in the above erratum, that there is no embedded with trivial normal bundle). 2,100 more words