Why is the iterated integral of two 1-forms homotopy invariant?

[lavinia]

I am trying to understand why the following expression is a homotopy invariant.

Start with 2 closed 1 forms whose wedge product is null homologous.

w1^w2 + dw12 = 0

I want to understand why

$$\int w_ {1}w_ {2} + \int w_ {12}$$

is a homotopy invariant of smooth paths where

$$\int w_{1}w_{2}$$

is the iterated integral of the two 1 forms along the path

 

[lavinia]

Never mind - I figured it out