By Barnsley M.F.

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**Example text**

Then = idA®e; therefore, cp ® 1 is a monomorphism, and the map iP ® 1 splits the exact sequence of tensor products. 0 Exercise. Prove that the sequence 0 -+ Z ® Z2 exact sequence 0 -+ Z -+ Q is not exact. 3. The Groups Tor and Ext. The homology and cohomology groups with arbitrary coefficients can be expressed in terms of integral homology groups. The expressions include operations Tor and Ext, which assign Abelian groups Tor(A, B) and Ext(A, B) to pairs of Abelian groups A and B. In cmputations, it is sufficient to know Tor(A, B) and Ext(A, B) for A, B = Z, Zm, Q, and JR.

Suppose that A = F / R, where F is a free group. The group F is a direct sum of a number of copies of Z; hence it is a subgroup of the group G defined as the direct sum of the same number of copies of Q. The group G has a subgroup corresponding to R. The quotient group G / R contains A as a subgroup. The group Ext(A, B) can be defined not only by using a projective resolution of A but also by using an injective resolution of B. Namely, let o - B - G - H - 0 be an injective resolution of the group B.

BlIBis cycles on a handle Figure 5. BlIBis cycles on the polygon must take the quotient group of zm modulo the subgroup generated by the element (2, ... ,2). Consider el - (1,0, ... ,0), ... , em-l = (0, ... ,0,1,0), and em = (1, ... ,1). We have The condition (al, ... , am) E Imih is equivalent to al - am = 0, ... , am-lam = 0 and am == 0 (mod 2). Therefore, the quotient group zm / 1m ih is isomorphic to zm-l EB Z2; the group zm-l is generated by el. , em-I. and the group Z2 is generated by em.