# On the notion of a chain (as for example in chain complex)

The thing with mathematics is that on one if you define something, you are completely free in choosing any name you want, and on the other hand you should find a meaningful name that evokes some intuitive ideas that coincide with what you actually wanted to do. So long story short, one thing that always bothered me is the following:

What is the origin of the term chain? When was it first used and what explanation (if any) was given?

To be a bit more precise, by chain I mean a finite linear combination of (oriented) simplices.

I looked into Dieudonné's "A History of Algebraic and Differential Topology" but there he only points out in a footnote that the term was coined by J.W. Alexander without giving any source.

Maybe related to this is the notion of a chain complex, which I think at least once someone explained to me as being a chain of arrows, eventhough these arrows are actually the things mapping chains to chains. This is of course not helped by the fact that some useful abstractions of a chain are called complexes (as in CW or simplicial). I guess that the term chain complex was coined some time afterwards, but in some cases concepts are found and given a name much later, so maybe there is a chance they are related.

Finally I may need to add that my background is in mathematical analysis, my primary interest in chains is integrating something along them. I am not very well versed in abstract algebra, so there may be some simple source that I missed. However I am more interested in the basic geometrical objects not in the abstract generalisations. (only there is no algebraic geometry tag...)

• I believe the term chain was introduced by Poincare as part of his second definition of the term manifold; the definition I am referring to is on pg. 24-25 of Stillwell's wonderful translation of Analysis Situs. "We then say that the two manifolds V and V ′ are analytic continuations of each other. In this way we can form a chain of manifolds V1, V2, ..., Vn such that each is an analytic continuation of its predecessor, and there is a common part between any two consecutive manifolds of the chain. I shall call this a connected chain." Commented Aug 25, 2015 at 1:27
• Poincare goes on and mentions something akin to a combination of oriented simplices: "...There are, in fact, manifolds (and we shall see examples later) which can be decomposed into a certain number of partial manifolds forming a connected chain or network and such that each of them can be defined by equations of the form (8)..." [(8) references that a collection of equations of the form $x_1 = \theta(y_1, ... y_m); x_2 = \theta(y_1, ... y_m); ... x_n = \theta(y_1, ... y_m)$ represents a manifold of m dimensions (if the $y$ are considered as independent variables).] Commented Aug 25, 2015 at 1:38
• @Catherine Ray You should put your comments into an answer, it looks like this is the answer. Commented Aug 25, 2015 at 17:30

One learns quite a bit by searching I. M. James' History of Topology (1999):

The name: Lefschetz (p. 546) confirms that

The felicitous term "chain" is due to Alexander

while Katz (p. 119) specifies that it was coined in the paper Combinatorial analysis situs (1926). Weibel (p. 801) adds that "chain complex" was introduced by Mayer in Über abstrakte Topologie (1929). Finally Vanden Eynde (p. 97) points to the earlier use of chains ("Ketten") with a different meaning by Brouwer (following Cantor) in Beweis der Invarianz der geschlossenen Kurve (1912).

The thing: Sarkaria (p. 141) traces it to Poincaré's Analysis Situs (1895) where it is among the four(!) different things called "variety":

In § 4 "varieties" are oriented, and then in §§ 5 and 6 Poincaré considers integral or rational linear combinations of the oriented varieties of an $M$ to define its homologies and Betti numbers: in this context it is best to think of his "varieties" as smooth chains of $M$.

(As @CatherineRay notes, Poincaré's paper ironically also used the term "(connected) chain" to mean something altogether different.)

• Perhaps this is incorrect, but Alexander, in Combinatorial analysis situs, uses the word chains as if it is an established term and he is modifying it to work with a different sort of chain. "We then give a short account of the classical theory of connectivity and of a modified form of the theory where we operate with chains reduced modulo $\pi$." Commented Aug 25, 2015 at 17:43
• @CatherineRay You're right he sounds that way, but it can also mean the word first got "established" in Princeton conversations before the paper -- that would be how Lefschetz, who was there, can write what he does. Commented Aug 25, 2015 at 17:53
• I agree about the ambiguity there. I was unable to post a picture in this comment thread, so please see the picture I have appended to my answer below. Commented Aug 25, 2015 at 17:58

I believe the term chain was introduced by Poincare as part of his second definition of the term manifold; the definition I am referring to is on pg. 24-25 of Stillwell's wonderful translation of Analysis Situs.

"We then say that the two manifolds V and V ′ are analytic continuations of each other. In this way we can form a chain of manifolds V1, V2, ..., Vn such that each is an analytic continuation of its predecessor, and there is a common part between any two consecutive manifolds of the chain. I shall call this a connected chain."

Poincare goes on and mentions something akin to a combination of oriented simplices:

"...There are, in fact, manifolds (and we shall see examples later) which can be decomposed into a certain number of partial manifolds forming a connected chain or network and such that each of them can be defined by equations of the form (8)..."

[(8) references that a collection of equations of the form $x_1= \theta_1(y_1,...y_m) ; x_2=\theta_2(y_1,...y_m) ;...; x_n=\theta_n(y_1,...y_m)$ represents a manifold of m dimensions (if the $y$ are considered as independent variables).]

Also note, Alexander's definition of the term chain (page 301 of Combinatorial Analysis Situs) cites Poincare's formula for describing the cell structure of a manifold in terms of an incidence matrix.

"In his first long paper on topology Poincare (1895) continued to explore the Betti numbers. He set up machinery for computing them by assuming that each manifold has a decomposition into cells homeomorphic to simplices, reading off linear equations he called homologies, and computing the Betti numbers by linear algebra."

-- Mathematical perspectives, Stillwell

• This is also an interesting answer, which may be an earlier usage of the word chain. However as far as I see it is not really used as a definition but rather as a description on how to define a manifold. But you are probably right in that it will have inspired the modern usage.
– mlk
Commented Aug 25, 2015 at 19:33
• I agree that the introduction if the word chain to denote a linear combination of simplicies is motivated in order to both define manifolds and to calculate their betti numbers. I'd appreciate it if you would explain how Alexander is defining a concept that is different from Poincare's definition of chain. Commented Aug 25, 2015 at 19:38
• I agree that from the point of algebraic topology those two concepts are essentially identical. But must of the stuff I do is geometric measure theory, where both also occur and where a chain and a manifold are different objects, since one is flat and has only piecewise differentiable structure, while the other is curved and smooth. Of course even there, they are closely related, both being a special case of integer rectifiable currents. Also the definition of boundary for a chain as the sum of boundaries of the parts does not work for Poincaré's overlapping pieces.
– mlk
Commented Aug 25, 2015 at 19:57