By Alex Heller, Myles Tierney
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Extra resources for Algebra, Topology, and Category Theory. A Collection of Papers in Honor of Samuel Eilenberg
B) If [G: E] = s < oo, then the h o m o m o r p h i s m Hn(E; can be indentified with Ζ Proof. Z) Hn(G; Z ) I. By a Shapiro lemma and Poincaré duality we have Hn(E; Z ) ^ Hn(G; Z[G]®m] I) * H°(G; Z [ G ] ® 2 [ ]£ Z)^ (Z[G]®2 G [ ]£ Z) . SOME PROPERTIES OF T W O - D I M E N S I O N A L P O I N C A R É D U A L I T Y G R O U P S 47 The module Z [ G ] ® z [ f :] Ζ can be indentified with the free abelian group on the cosets G/E with G-module structure arising from the usual action of G on G/E. T h u s the module can be identified with integer valued functions on G/E with finite support.
2. If M is a C-set, and Μα is an injection for all α e C, OP have then ZM is projective if and only if the components of (Y, M) right zeros. Proof. If M is the disjoint union of indécomposables Mx, then ZM is the coproduct of the ZMF and the (Y, M,) are the components of (Y, M). OP Hence it suffices to assume that M is indecomposable. Then if (Y, M) has a right zero, M is a retract of C(C, ) for some C, and so ZM is a retract of ZC(C, ). Hence ZM is projective. Conversely, assume that ZM is projective where M is indecomposable.
Then e is idempotent, and it is easy to see that C has a right zero if and only if its idempotent completion has an initial object. If M is a op C-set, then (Y, M ) has a right zero if and only if M is a retract of a representable. A C-set M is indecomposable if it is nonempty (that is, if at least one of its values is nonempty), and if it cannot be written as the disjoint union (coproduct) of two nonempty sub C-sets. Any C-set M is the coproduct of its indecomposable sub C-sets, and if M, are these indécomposables, then ( y , Mi) are the components of (Y, M).
Algebra, Topology, and Category Theory. A Collection of Papers in Honor of Samuel Eilenberg by Alex Heller, Myles Tierney