The terminology in topology seems to go back a long way: Über die Torsionszahlen von Produktmannigfaltigkeiten (Mathematische Annalen 1924) is the paper in which Künneth sets out to show how the torsion in homology behaves in a product, and there he footnotes earlier work of Poincaré and Tietze (his advisor) on the "Torsionszahlen". So it is a fair bet the terminology goes back at least to Tietze. In terms of the question "algebra or topology first?", I'd therefore lay bets on topology, bearing in mind that homology wasn't clearly formulated as taking values in abelian groups until a little later. The more general idea of "twisting" probably awaited the theory of bundles, a little later again.

There is some quite detailed discussion of the history and Emmy Noether's intervention in http://math.unice.fr/~nbasbois/Articles/groupe_homologie2.pdf. A very interesting point, for algebraists, is that the issue is tied up with the structure theory of finitely-generated abelian groups and the chicken-and-egg question of whether you derive it from the theory of elementary divisors of integral matrices, or vice versa (which would be the modern approach in abstract algebra). Psychologically this makes quite a difference to what one is looking for as a canonical form. Poincaré definitely approached it from elementary divisors, and as the paper explains, the introduction of abelian groups as adjuncts to the incidence matrix had to be "justified" to topologists.

The terminology in topology seems to go back a long way: Über die Torsionszahlen von Produktmannigfaltigkeiten (Mathematische Annalen 1924) is the paper in which Künneth sets out to show how the torsion in homology behaves in a product, and there he footnotes earlier work of Poincaré and Tietze (his advisor) on the "Torsionszahlen". So it is a fair bet the terminology goes back at least to Tietze. In terms of the question "algebra or topology first?", I'd therefore lay bets on topology, bearing in mind that homology wasn't clearly formulated as taking values in abelian groups until a little later. The more general idea of "twisting" probably awaited the theory of bundles, a little later again.

There is some quite detailed discussion of the history and Emmy Noether's intervention in http://math.unice.fr/~nbasbois/Articles/groupe_homologie2.pdf. A very interesting point, for algebraists, is that the issue is tied up with the structure theory of finitely-generated abelian groups and the chicken-and-egg question of whether you derive it from the theory of elementary divisors of integral matrices, or vice versa (which would be the modern approach in abstract algebra, despite what is said on "Is there a slick proof of the classification of finitely generated abelian groups?"). Psychologically this makes quite a difference to what one is looking for as a canonical form. Poincaré definitely approached it from elementary divisors, and as the paper explains, the introduction of abelian groups as adjuncts to the incidence matrix had to be "justified" to topologists.

The terminology in topology seems to go back a long way: Über die Torsionszahlen von Produktmannigfaltigkeiten (Mathematische Annalen 1924) is the paper in which Künneth sets out to show how the torsion in homology behaves in a product, and there he footnotes earlier work of Poincaré and Tietze (his advisor) on the "Torsionszahlen". So it is a fair bet the terminology goes back at least to Tietze. In terms of the question "algebra or topology first?", I'd therefore lay bets on topology, bearing in mind that homology wasn't clearly formulated as taking values in abelian groups until a little later. The more general idea of "twisting" probably awaited the theory of bundles, a little later again.

The terminology in topology seems to go back a long way: Über die Torsionszahlen von Produktmannigfaltigkeiten (Mathematische Annalen 1924) is the paper in which Künneth sets out to show how the torsion in homology behaves in a product, and there he footnotes earlier work of Poincaré and Tietze (his advisor) on the "Torsionszahlen". So it is a fair bet the terminology goes back at least to Tietze. In terms of the question "algebra or topology first?", I'd therefore lay bets on topology, bearing in mind that homology wasn't clearly formulated as taking values in abelian groups until a little later. The more general idea of "twisting" probably awaited the theory of bundles, a little later again.

There is some quite detailed discussion of the history and Emmy Noether's intervention in http://math.unice.fr/~nbasbois/Articles/groupe_homologie2.pdf. A very interesting point, for algebraists, is that the issue is tied up with the structure theory of finitely-generated abelian groups and the chicken-and-egg question of whether you derive it from the theory of elementary divisors of integral matrices, or vice versa (which would be the modern approach in abstract algebra). Psychologically this makes quite a difference to what one is looking for as a canonical form. Poincaré definitely approached it from elementary divisors, and as the paper explains, the introduction of abelian groups as adjuncts to the incidence matrix had to be "justified" to topologists.