Steenrod Connections and Connectivity in H-Spaces by James P. Lin

By James P. Lin

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4. 1 is used to derive information about generators of filtration degree 2 +2 Sq k r. Assumption g(r) together with many factorizations of imply that there are many secondary operations defined on elements of filtration degree r . We merely need a systematic method to sort out the indeterminacy. This is provided by the arguments in [12, Chap. 6]. This chapter is essentially motivated by increasing all degrees by a factor of 2 r + 1 from the degrees in [12]. 1. Let v(s) = 25+1-i a(s,k) = 2s+2k b{s,k) = 2s+2k + 2v(s-l) +2v(s) a(s,k,r) = 2s+r+lfc + 2T v (s -1) = b(s,k,r) = 2s+r+lk +2rv{s) Then a (s ,k ,r ) =

Because x <2> ? must be cancelled by £>/ ( (l). In fact those variables that belong to Rodd have reduced coproduct in £ff* ® R odd , and the other variables that belong to D (l) have reducced coproduct in D{1)® D{1).

The basic techniques are the same as those used in [18,25]. There is a commutative diagram nicc X XX Since Df -2- of a fundamental class in jff* (K) corresponds to the reduced coproduct of the fundamental class, we have that im Df C £H* ® H* + D (l) ® £> (l). In fact those variables that belong to Rodd have reduced coproduct in £ff* ® R odd , and the other variables that belong to D (l) have reducced coproduct in D{1)® D{1). Therefore there exist generalized Eilenberg MacLane spaces A A2 and maps f {: X -> A{ such that Df factors as Df X X X /i X / Ax X A, K u STEENROD CONNECTIONS 35 where im f * C £H* + D {l) C D (1) and im / o C i ^ r f and w (m (fx X />)) = wi^ +/>(1) is null homotopic.

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