By Barnsley M.F.
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If G is a group then a unit in 7l.. (G) is an element with a two-sided multi plicative inverse. The elements of the group ± G = {gig E G } v { - gig E G} are called the trivial units, and all others are called the non-trivial units of 7l.. ( G) . 6) A . Suppose that G is an abelian group such that 7l.. (G) has non-trivial units . (G) matrix A which cannot be transformed to an identity matrix by any finite sequence of the operations (I)-(V). B. The group G = 7l.. 5 is an abelian group such that 7l..
9), determines the deformation (M, u K) A (M, u K) up L 2 L, L, = K Ll L2 , reI L 2 . J It follows directly from the second definition that f* is a group homo morphism. 6) it follows directly that g* f* = (gf) * . 2) There is a covariant functor from the category offinite CW complexes and cellular maps to the category of abelian groups and group homomorphisms given by L ...... Wh(L) and (f: L I -+ L2) ...... U* : Wh(L 1) -+ Wh(L 2 ))' Moreover iff ':::!. g then f* = g * . PROOF: The reader having done his duty, we need only verify that if f ':::!.
Matrices and formal deformations Given a homotopically trivial CW pair, we have shown that it can be transformed into a pair in simplified form. So consider a simplified pair (K, L) ; K = L u Uej u Uer + 1 where the ej are trivially attached at eO . W + 1 : 81' + 1 --+ L u U ej, where 'Pi is a characteristic map for er+ 1 . (K" L ; eO) in the homotopy r + exact sequence of the triple (K, K , L). Since, however, freely homotopic r attaching maps give (7. 1 ) the same result up to simple-homotopy type, we do not wish to be bound to homotopies keeping the base point fixed.