By Mosher R.E., Tangora M.C.

Cohomology operations are on the heart of an enormous quarter of task in algebraic topology. This remedy explores the only most crucial number of operations, the Steenrod squares. It constructs those operations, proves their significant houses, and offers a number of functions, together with numerous assorted ideas of homotopy conception necessary for computation.

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I. N. E. Steenrod [3,5,6]. 2. _ _ and D. B. A. Epstein [I]. CHAPTER 3 PROPERTIES OF THE SQUARES We assemble the fundamental properties of the squaring operations in an omnibus theorem. Theorem 1 The operations Sqi, defined (for i > 0) in the previous chapter, have the following properties: o. 1. 2. 3. 4. 5. 6. 7. Sqi is a natural homomorphism HP(K,L; 2 2 ) -+ HP+ i(K,L; 2 2 ) If i > p, Sqi(X) = 0 for all x E HP(K,L; 2 2 ) Sqi(X) = x 2 for all x E H\K,L; 2 2 ) SqO is the identity homomorphism Sql is the Bockstein homomorphism b*Sqi = Sqib* where b*: H*(L;2 2 ) -+H*(K,L; 2 2 ) Cartanformula: Sqi(xy) = IiSqjx)(Sqi- jy ) Adem relations: For a<2b, SqQSqb=Ic(~=~~l)SqQ+b-CSqcwhere the binomial coefficient is taken mod 2 We remark that the above properties completely characterize the squaring operations and may be taken as axioms, as is done in the book of Steenrod and Epstein.

COHOMOLOGY OPERATIONS 30 Lemma 5 Let R be an Adem relation. If R(y) = 0 for every class y of dimension p, then R(z) = 0 for every class z of dimension (p - 1). PROOF: Let u denote the generator of H l(S 1 ; Z 2)' Clearly Sqiu = 0 for all i > O. Therefore, by the Cartan formula, R(u x z) = u x R(z). But u x z has dimension p; hence R(u x z) = 0, and so R(z) = O. The Adem relations follow easily from Lemma 4 and Lemma 5 by induction on dimension. It remains to prove Lemma 2. We begin by recalling the formula = + (P;l), which holds for all p,q except for the case p = q = O.

We will prove the first form and deduce the second as a corollary. Consider the composition W®K®L~"", W® W®K®L~"", W®K® W®L ~K®~L"",K®K®L®L2"",K®L®K®L where r: W -----+ W ® W was defined in Chapter 2 and T permutes the second and third factors (we are not concerned with sign changes because we want conclusions in Z2 coefficients). This composition, which we denote by <{JK ® L, is easily seen to be suitable for computations of cup-i products and hence Sqi, in K ® L. V sing the same letters to denote (co-)homology classes or their representatives, and writing p,q,n for dim u, dim v, and p + q - i, respectively, we compute as follows: Sqi(U X v)(a ® b) = ((u ® v) un (u ® v»(a ® b) = (u ® v ® u® V)<{JK ® L(dn ® a ® b) = (u ® u ® v ® v)I <(JK(dj ® a) ® Tj<{JL(dn - j ® b) 25 PROPERTIES OF THE SQUARES the last step using the definition of r, and the summation being over j, o