Cohomology Operations and Applications to Homotopy Theory by Mosher R.E., Tangora M.C. PDF

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

Cohomology operations are on the heart of an immense sector of task in algebraic topology. This therapy explores the only most crucial number of operations, the Steenrod squares. It constructs those operations, proves their significant houses, and offers quite a few purposes, together with a number of various options of homotopy conception worthwhile for computation.

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Suppose the cocycle u E CZP(X;Z) satisfies bu = 2a for some a. i. Show that u V o u + U VI U is a cocycle mod 4. 11. Define a natural operation, the Pontrjagin square, P z : HZp( ;Zz) ~ H 4p( ;Z4) CONSTRUCTION OF THE STEENROD SQUARES 2t iii. Show that pP2(u) = u U u, where p: H*( ;Z4) -+ H*( ;Z2) denotes reduction mod 2. iv. Show that P 2 (u + v) = Piu) + P 2 (v) + u u v, where u u v is computed with the non-trivial pairing Z2 <8> Z2 -+ Z4. REFERENCES General Other definitions of the squares I.

X"' where Xi is the non-trivial one-dimensional class of the ith copy of K(Z2,l). In this polynomial ring Zz[xj, . ,0",,] where O"j is the elementary symmetric function of degree j (for example, 0"1 =X 1 + ... + X,,). Proposition 3 In H*(K,,; Z 2)' Sqi(O",,) = 0",,0" i (l < i :s;; n). PROOF: Sq(O",,) = Sq([1 xJ = [1 Sq(x;) = [1 0"" :D=o O"i' The result follows. (Xi + X;) = 0",,([1 (I + Xi)) = Corollary 1 In H*(Zz,n; Zz), Sqi ,,, =I- 0 for 0 :::; i < n. PROOF: By Theorem I of Chapter I, we can find a map f of K n into K(Zz,n) such that f*(I,,) = 0"".

Now Pn,k is also a union of cells, of the same dimensions as in (Vn,k)"' Moreover, we have a map ({J: Pn,k~ Vn,k' just as in the case k = 1 discussed in Proposition 1; this map is a homeomorphism into, and its image is evidently (Vn,k)"' This proves the corollary. THE COHOMOLOGY OF Pn,k We turn now to consideration of the cohomology of these spaces. Recall that the ring H*(P n;Z2) is the truncated polynomial ring Z2[a]/a n. We have the sequencePn_k ~Pn -4Pn,k = Pn/Pn- k, where i is the injection andjthe identification map.

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Cohomology Operations and Applications to Homotopy Theory by Mosher R.E., Tangora M.C.

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