A contradiction in Spivak's Calculus on manifold?

[quasar987]

Homework Statement

I don't have great expectation that this will get a reply but here goes, because this is bugging me.

I will assume that you are familiar with the notation used by Spivak.

In the last section of chapter 4, he shows how to integrate a k-form on R^m over a singular k-cube in R^m, namely,

$$\int_c \omega = \int_{[0,1]^k}c^{*}\omega$$

He then goes on to define the integral of a k-form over a k-chain $c=\sum a_i c_i$ in R^m by

$$\int_c \omega = \sum a_i\int_{c_i} \omega=\sum a_i\int_{[0,1]^k}c_{i}^{*}\omega$$

But actually, a k-chain in R^m is a k-cube in R^m, so in principle, the integral of a k-chain is already defined, namely by

$$\int_c \omega =\int_{[0,1]^k}(\sum a_i c_i)^{*}\omega$$

and this will be consistent with the above definition if $(\sum a_i c_i)^{*}=\sum a_ic_i^{*}$. But is this so? Taking the case where w is a 1-form for simplicity,

$$(\sum a_i c_i)^{*}\omega (p)(v_p)=\omega (\sum a_i c_i(p))((\sum a_i c_i)_{*}(v_p))=\omega (\sum a_i c_i(p))(\sum a_i (Dc_i(p)(v))_{c(p)})=\sum a_i \omega (\sum a_i c_i(p))((Dc_i(p)(v))_{c(p)})$$

On the other hand, supposing we define "+" and multiplication by a scalar in the natural way on the f* operators,

$$(\sum a_ic_i^{*})\omega(p)(v_p)=\sum a_ic_i^{*}\omega(p)(v_p) = \sum a_i \omega(c_i(p))(c_{i*}(v_p))=\sum a_i \omega(c_i(p))(Dc_{i}(v))_{c_i(p)})$$

They are not at all the same.

 

[olgranpappy]

This will not do. Math is obviously far too easy to be left to the mathematicians.

 

[quasar987]

Better than nothing. :p

(Where have the smiley faces gone?)

 

[olgranpappy]

Better than nothing. :p

(Where have the smiley faces gone?)

up in the toolbar-thing (look for the smileyface with a down-arrow next to it).

 

[quasar987]

Could it be that chains are not really functions, and when we write $c=\sum a_i c_i$, it is purely notational?

Because I happen to be reading at the moment that this whole cube & chains business can be extended to the general setting of manifold. Namely, if M is a manifold, a singular k-cube in M is a map c:[0,1]^k --> M. But since there is a priori no algebra on manifolds, it does not make sense to add and multiply cubes by constants, such that $c=\sum a_i c_i$ is only a formal sum.

 

[Dick]

I think you have it. A chain is a piece of geometry. It's not the coordinate map, since the integration is independent of the coordinates. So, yes, it's a formal sum. Probably best to think of it as a 'weight' on the chain.