- #1
Bacle
- 656
- 1
Hi, All:
I am trying to understand the formal machinery leading to a proof that the homology of the disjoint union of spaces is the disjoint (group) sum of the homologies of the respective spaces; the idea seems clear: if a cycle bounds in a given space Xi, then it will bound in the disjoint sum ( but it will bound only in Xi itself), and, conversely, a trivial, bounding cycle will also be trivial in the disjoint union. Still, I have been told--very non-specifically--that a formal proof needs a lot of machinery.
I guess part of the problem is that we may have more than countably-many possible spaces, so standard induction may not work; we may have to somehow use transfinite induction ( so it starts getting ugly here ), and maybe inverse limits (uglier), etc.
Anyone know what a more formal proof would entail?
Thanks.
I am trying to understand the formal machinery leading to a proof that the homology of the disjoint union of spaces is the disjoint (group) sum of the homologies of the respective spaces; the idea seems clear: if a cycle bounds in a given space Xi, then it will bound in the disjoint sum ( but it will bound only in Xi itself), and, conversely, a trivial, bounding cycle will also be trivial in the disjoint union. Still, I have been told--very non-specifically--that a formal proof needs a lot of machinery.
I guess part of the problem is that we may have more than countably-many possible spaces, so standard induction may not work; we may have to somehow use transfinite induction ( so it starts getting ugly here ), and maybe inverse limits (uglier), etc.
Anyone know what a more formal proof would entail?
Thanks.