We prove two theorems concerning the action of the Steenrod algebra in cohomology and homology. (i) Let A denote a finitely generated graded F<sub>p</sub> polynomial algebra over the Steenrod algebra whose generators have dimensions not divisible by p. The possible sets of dimensions of the generators for such A are known. It was conjectured that if we replaced the polynomial algebra A by a polynomial algebra truncated at some height greater than p over the Steenrod algebras, the sets of all possible dimensions would coincide with the former list. We show that the conjecture is false. For example F<sub>11</sub>[x<sub>6</sub>,x<sub>10</sub>]<sup>12</sup> truncated at height 12 supports an action of the Steenrod algebra but F<sub>11</sub>[x<sub>6</sub>,x<sub>10</sub>] does not. (ii) Let V be an elementary abelian 2-group of rank 3. The problem of determining a minimal set of generators for H*(BV,F<sub>2</sub>) over the Steenrod algebra was an unresolved problem for many years. (A solution was announced by Kameko in June 1990, but is not yet published.) A dual problem is to determine the subring M of the Pontrjagin ring H*(BV,F<sub>2</sub>). We determine this ring completely and in particular give a verification that the minimum number of generators needed in each dimension in cohomology is as announced by Kameko, but by using completely different techniques. Let v ε V - (0) and denote by a_5(v) ε H*(BV,F<sub>2</sub>) the image of the non-zero class in H<sub>2s-1</sub>(RP<sup>∞</sup>,F<sub>2</sub>) imeq F<sub>2</sub> under the homomorphism induced by the inclusion of F<sub>2 → V onto (0,v). We show that M is isomorphic to the ring generated by (a</sub>_s(v),s ≥ 1, v ε V - (0)) except in dimensions of the form 2^r+3 + 2^r+1 + 2^r - 3, r ≥ 0, where we need to adjoin our additional generator.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:279407 |
| Date | January 1991 |
| Creators | Alghamdi, Mohamed A. M. A. |
| Publisher | University of Aberdeen |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://digitool.abdn.ac.uk:80/webclient/DeliveryManager?pid=166808 |
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