Eric Forgy Smooth Spaces




Let UU denote an open subset of n\mathbb{R}^n and

C(U)= r0C r(U)C(U) = \bigoplus_{r \ge 0} C_r(U)

be the space of chains on UU.

Let XX be a smooth space. For any plot ϕ:UX\phi:U\to X, and any chain cC(U)c\in C(U), the push forward

ϕ *c\phi_* c

is a chain in XX. The collection of all plots allow us to push forward all such spaces of chains to obtain the space of chains on XX, i.e.

C(X)= r0C r(X).C(X) = \bigoplus_{r\ge 0} C_r(X).


Using plots ϕ:UX\phi:U\to X on chains on the domains of plots C(U)C(U) gives chains C(X)C(X) on XX. Once we have chains on XX, we can define cochains on XX to be the formal duals of chains, i.e.

C *(X)= r0C r *(X).C^*(X) = \bigoplus_{r\ge 0} C^*_r(X).

For example, if αC *(X)\alpha\in C^*(X) we can define its value on ϕ *cC(X)\phi_* c\in C(X) via

α,ϕ *c=ϕ *α,c.\langle \alpha,\phi_* c\rangle = \langle \phi^*\alpha, c\rangle.

Differential Forms

A differential form αΩ p(X)\alpha\in\Omega^p(X) on XX is a cochain such that its pull back via any plot is a differential form on the domain of that plot, i.e.

ϕ *αΩ r(U)ϕ:UXαΩ r(X).\phi^*\alpha\in\Omega^r(U)\forall\phi:U\to X\implies\alpha\in\Omega^r(X).


Let ϕ i:U iX\phi_i:U_i\to X denumerate a collection of plots such that any chain cC(X)c\in C(X) can be expressed as

c= i(ϕ i) *c ic = \sum_i (\phi_i)_* c_i

for some chain c iC(U i)c_i\in C(U_i). Then we can define integration on XX via

cα= i (ϕ i) *c iα= i c iϕ i *α.\int_c \alpha = \sum_i \int_{(\phi_i)_* c_i} \alpha = \sum_i \int_{c_i} \phi_i^* \alpha.


ϕ *=ϕ *\partial\circ \phi_* = \phi_*\circ\partial


dϕ *=ϕ *dd\circ \phi^* = \phi^*\circ d

Stokes Theorem

cdα= i c i(ϕ i) *(dα)= i c id(ϕ i *α)= i (ϕ i) *(c i)α= i (ϕ i) *c iα= cα.\int_c d\alpha = \sum_i \int_{c_i} (\phi_i)^* (d\alpha) = \sum_i \int_{c_i} d(\phi_i^*\alpha) = \sum_i \int_{(\phi_i)_*(\partial c_i)} \alpha = \sum_i \int_{\partial (\phi_i)_* c_i} \alpha = \int_{\partial c}\alpha.


Eventually the following will be a commented list – promised.

There are also Hofer’s polyfolds.

Concerning smooth ∞-stacks there is useful material in


Dual to generalized smooth spaces are generalized smooth algebras of functions on them, according to the general lore of space and quantity.

Further discussion

We had extensive discussion of generalized smooth spaces at the nn-Café:

Revised on May 2, 2010 at 09:56:02 by Eric Forgy