0. BASIC HOMOTOPY NOTIONS

7

one on each M(a) in the following way: Let [M] G Hm(\M\,d\M\) be the

fundamental class. Let A|M| = U {M(a) :\a\^n- 2}. Then the image of [M]

under

Hm(\M\,d\M\) h Hm^{d\M\) h Hm-^dlMl A\M\)

= 0 Hrn^1(\diMld\diM\)

(where j * is the inclusion map) shall be denoted by E i K n ( ~ ^ ) 1 ^ ^ ] ' Now by

induction we obtain fundamental classes for each M(a)\ the usual combinatorial

argument which shows that d2 = 0 in a simplicial complex demonstrates that

the class so obtained depends only on a (and not on any choice of construction).