Whitehead group
Topologists want to know when two spaces that look homotopy-equivalent are actually the same up to a controlled deformation. The Whitehead group is the home of the obstruction: it is a quotient of K_1 of an integral group ring, cleaned of the trivial units that carry no geometric content. A homotopy equivalence between finite complexes carries an invariant, the Whitehead torsion, living in this group, and it vanishes exactly when the equivalence is a simple homotopy equivalence.
Let G be a group and Z[G] its integral group ring. The Whitehead group is Wh(G) = K_1(Z[G]) / (±G), where (±G) denotes the image in K_1 of the trivial units — the elements ±g for g in G, viewed as 1x1 matrices. Quotienting them out removes the part of K_1 forced by the obvious invertible elements, leaving the geometrically meaningful obstructions. For G trivial, Wh(G) = K_1(Z) / {±1} = 0.
Computing Wh(G) is hard and deeply arithmetic. Whitehead's conjecture, proved in many cases, asserts Wh(G) = 0 for G a torsion-free group satisfying suitable conditions (it is a special case of the Farrell-Jones conjecture). For finite groups Wh(G) is a finitely generated abelian group whose rank is r - q, where r is the number of irreducible real representations and q the number of irreducible rational representations; its torsion subgroup relates to SK_1(Z[G]) and class groups of the relevant orders.
For the cyclic group G = Z/5Z, Wh(G) is infinite cyclic, Wh(Z/5Z) = Z, reflecting the unit (1 + t + t^2)(...) phenomena in Z[Z/5Z]. For infinite cyclic G = Z, Wh(Z) = 0, so homotopy equivalences over the circle carry no Whitehead torsion.
Wh(Z/5Z) = Z, but Wh(Z) = 0.
Whitehead torsion underlies the s-cobordism theorem, which says an h-cobordism on a manifold of dimension at least 5 is a product exactly when its torsion in Wh(pi_1) vanishes. This is the precise reason the Whitehead group, an algebraic K-theory object, controls a central question in geometric topology.