Bachman's Geometric Approach to Differential Forms: Chains & Cells

[Rasalhague]

Bachman: A Geometric Approach to Differential Forms, p. 65:

A k-chain is a formal linear combination of k-cells. As one would expect, we assume the following relations hold:

S - S = {}

nS + mS = (n+m)S

S + T = T + S

Questions. (1) I know the term linear combination from linear algebra, but what is a formal linear combination?

(2) What are n and m: integers, rational numbers, real numbers?

(3) Is it also assumed that m(S + T) = mS + mT, and does 1S = S? The numbers 1 and -1 distribute over cells in his definition of the boundary of a cell, and I think he assumes that the latter relation holds too. This would imply that k-chains make a vector space or a module, if m and n represent elements of a field or ring respectively.

(NOTE: Bachman writes n-chain and n-cells; I changed this to k, as I'm guessing the n in the name has no connection to the n in his second relation.)

 

[Rasalhague]

http://www.math.upenn.edu/~pemantle/581-html/chapter04.pdf source also uses the expression "formal linear combination" in defining a chain (also without definition), although here the elements combined are maps from "p-simplices" that are subsets of R^p to a manifold M, rather than the images of unit cubes in R^p under a map to M. He also says, "Boundaries are a subset (in fact, a sub-vector space) of the cycles" (p. 64), and the cycles are a subset of the chains. So, reading between the lines, perhaps Bachman's chains and this guy (Pemantle)'s chains are vectors, in which case the coefficients of Bachman's cells would belong to a field, such as the rationals or the reals.

 

[lavinia]

Bachman: A Geometric Approach to Differential Forms, p. 65:



Questions. (1) I know the term linear combination from linear algebra, but what is a formal linear combination?

The simplices are geometric objects not elements of a Z-module. But on can formally write down Z-linear combinations of them as though they were. These linear combinations are really just symbols but they produce a Z-module just the same.

(2) What are n and m: integers, rational numbers, real numbers?

chains are usually first presented over the integers but one could just as easily use any commutative ring with identity. In this case I think n and m are integers because these letters typically denote integers.

(3) Is it also assumed that m(S + T) = mS + mT, and does 1S = S? The numbers 1 and -1 distribute over cells in his definition of the boundary of a cell, and I think he assumes that the latter relation holds too. This would imply that k-chains make a vector space or a module, if m and n represent elements of a field or ring respectively.

(NOTE: Bachman writes n-chain and n-cells; I changed this to k, as I'm guessing the n in the name has no connection to the n in his second relation.)

yes, 1S = S, m(S + T) = mS + mT is just scalar multiplication in the formal module.

To be precise the formal module is all formal linear combinations of the simplices modulo the equivalence relations, m(S + T) = mS + mT and so forth.

 

[Rasalhague]

Thanks, lavinia. To summarise, the structure being defined by Bachman is a Z-module, meaning a module over the integers. Its elements are chains of cells with integer coefficients. "Formal" is (operationally) superfluous in formal linear combination, like the "linear" in "linear vector space"? But perhaps, although it doesn't denote any difference, it carries a connotation of being "merely symbolic". Pemantle defines his chains slightly differently with maps called simplices in place of Bachman's sets called cells. He probably allows the coefficients to be at least rationals, possibly reals, since he describes chains as vectors.