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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) sites Poincare's formula for describing the cell structure of a manifold in terms of an incidence matrix.

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

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) sites Poincare:

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) sites Poincare's formula for describing the cell structure of a manifold in terms of an incidence matrix.

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).]

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) sites Poincare: