Hatcher Exercise 3.1.5

Let be a 1-cochain in with coefficients in -- i.e. .

\begin{align} & (\delta1 \varphi)(F) = \varphi(F \vert{[bc]}) - \varphi(F \vert{[ac]}) + \varphi(F{[ab]}) = 0 \ \Rightarrow \enspace & \varphi(g) - \varphi(f \cdot g) + \varphi(f) = 0 \ \Rightarrow \enspace & \varphi(f \cdot g) = \varphi(f) + \varphi(g) \end{align}

Corollary:


Theorem: If is homotopic to (fixing endpoints), then


Proof: Let in explicitly describe the vertices of a 2-simplex. We will explicitly describe two singular 2-chains . Let be the homotopy from to , so that and .

Let . Observe that and (the constant path). Then , where is the path . Let . Then, (the constant path) and . In particular, again. Then, we have:

But since is zero on constant paths, the last term is zero and we have . Applying the same argument to gives , and we have by transitivity.


Theorem:


Proof: Let be an arbitrary singular 1-chain. If is a coboundary, then for a 0-cochain . But then , so only depends on the endpoints.

Suppose that depends only on the endpoints of for any . Pick a basepoint and define a 0-cochain by where is any path from to . This is well-defined due to the supposition about . Then, for any path from to , pick some path from to , and notice that is some path from to . Then, we have:

using the first theorem. Thus, .