I know a topological example. Let X be a connected Hausdorf compact space and let a_1, , a_2 be two points from X. Let C,C' be complexes of cochains of Alexander-Spenier of X and a point respectively. Let f:C --> C' be a difference f_1-f_2 , where f_1,f_2 are induced by the inclusions i_1,i_2 of the points a_1,a_2 to X. Then a result is: 0 --> H(C') --> H(C' + C) --> H(C) --> 0 splits if and only if i_1,i_2 induce equal homomorphisms in Steenrod homology of X. This is in my paper Batanin M.A., Mappings of spectral sequences and the generalized homotopy axiom, Siberian Math. Journal, 5, (1987), 22-31. An example when i_1,i_2 induce different homomorphisms was constructed by U.Karimov in "On generalised homotopy axiom" , Reports of Tadjikistan Academie of Science, v.XXII,9, 521-524.,1979. In this example X is a 2-adic solenoid i.e. inverse limit of S^1 <--- S^1 <--- .... where all the maps are two-fold coverings and a_1, a_2 are in different linear connected components. Hope this will help. Michael Batanin. on 26/4/02 11:51 PM, Michael Barr at barr@barrs.org wrote:
Suppose you have an exact sequence of chain complexes 0 --> C' --> C' + C --> C -->0 in which the middle term has boundary operator given by the matrix [ d -f ] [ 0 d ] for some f: C --> C'. This is more or less the mapping cone of f. Suppose in addition that f(Z(C)) is included in B(C'). It then follows immediately that 0 --> H(C') --> H(C' + C) --> H(C) --> 0 is exact. Is it split?
The answer is yes if this is a sequence of abelian groups (or modules over a hereditary ring) and C is projective. Anyone know of a counter-example?
Michael
29-Apr-2002 16:12:19 -0300,1369;000000000000-00000000