We give the results of computations of root invariants in Ext over the Steenrod algebra through the 25-stem, with partial information through the 45-stem. This allows the computation of some new Steenrod operations as well

Bruner, Robert R

English

Digital Commons@Wayne State University

Wayne State University Mathematics Faculty Research Publications Mathematics 1-1-1998 Some root invariants and Steenrod operations in Ext_A(F2,F2) Robert R. Bruner Wayne State University, robert.bruner@wayne.edu Follow this and additional works at: https://digitalcommons.wayne.edu/mathfrp  Part of the Algebraic Geometry Commons, and the Geometry and Topology Commons Recommended Citation Bruner, R. R. Some root invariants and Steenrod operations in Ext_A(F2,F2). In M. Mahowald, S. Priddy, Eds. Homotopy Theory via Algebraic Geometry and Group Representations. Contemp. Math. 220 (1998), 27-33. Providence, RI: American Mathematical Society. This Article is brought to you for free and open access by the Mathematics at DigitalCommons@WayneState. It has been accepted for inclusion in Mathematics Faculty Research Publications by an authorized administrator of DigitalCommons@WayneState.    Some root invariants and Steenrod operations inExtA(F2, F2)ROBERT R. BRUNERAbstract. We give the results of computations of root invariants in Extover the Steenrod algebra through the 25-stem, with partial informationthrough the 45-stem. This allows the computation of some new Steenrodoperations as well.These calculations were started to provide evidence for or against the algebraicBredon-Löffler conjecture. This is the conjecture thatη∗k : Exts,tA (F2, F2) −→ Exts,tA (ΣL−k, F2)is a monomorphism for 0 < t − s < k/2. Here L is the Laurent series ringF2[x, x−1] with its usual Steenrod algebra action, L−k is the submodule of Lconsisting of those elements whose degree is greater than or equal to −k, andηk : ΣL−k −→ F2 is the nonzero homomorphism. The calculations of η∗k showed[5] that the conjecture holds for k ≤ 55. With those calculations in hand, weneeded only calculate the homomorphismsi∗k : Exts,tA (Σ1−kF2, F2) −→ Exts,tA (ΣL−k, F2)induced by the projections onto the bottom dimensions, ik : ΣL−k −→ Σ1−kF2,to calculate all the root invariants through the 25 stem, and about half of thosein the 26 through 45 stems. This extends the calculations of Mahowald and Shick[7], who found all the root invariants through the 16 stem. Our calculations weredone by first using the programs described in [1, 2] to compute resolutions of themodules L−k and the chain maps induced by ηk and ik. We then used MAGMAto compute the induced maps of Ext and the root invariants. A calculation ofExtA(F2, F2) for s < 40 and t ≤ 140 with complete information on the productstructure was calculated in the same manner: using MAGMA to process the1991Mathematics Subject Classification. Primary 55Q45, 55Q35, 55Q91; Secondary 55P42,55P91, 55T15 .   12 ROBERT R. BRUNERThese are the Sq0’s which are not ‘tautologous’. In addition, we have 12more which allow us to identify, in the resolution produced by the machine, theelements named. That is, we know by definition that e1 = Sq0(e0), for example.However, we have no way a priori to determine which of two elements in the 38stem is e1 since we have no general mechanical method of computing Steenrodoperations. Thus, the root invariant is quite useful, in that it unambiguouslyidentifies e1 for us. The elements identified in this manner are n1, p1, e1, e2, f1,and f2. We also determine m1 and t1 up to F2 indeterminacy.Finally, note that the root invariant R(h22g) = r1 shows that the Strong Alge-braic Bredon-Löffler Conjecture in [4] must have its (t − s)-intercept decreasedby at least 3. This is an inessential modification, as the key observation in thesecalculations supporting the conjecture is the slope -1/2.Conjecture 2. ( Strong Algebraic Bredon-Löffler Conjecture) The map η∗kis a monomorphism ifs < (k − n− 3)/2.KEY TO THE TABLES: Each row in the table lists the filtration s and stem n ofan element x ∈ ExtA(F2, F2), its root invariant R(x), the dimension k of the cellon which it occurs, whether Sq0(x) ∈ R(x), and a basis for the indeterminacyin R(x). The root invariant R(x) is in the n + k − 1 stem. By [6, Proposition2.5], we may test whether or not Sq0(x) ∈ R(x) by checking whether or not|Sq0(x)| = |R(x)|, that is, k = n+ s+ 1.(i) Sq0(Ph1) = h2g(ii) Sq0(Pc0) = h1y(iii) Sq0(Pd0) = gd1(iv) Sq0(i) = h2C(v) Sq0(k) = h2h5n+ ǫh3D2, where ǫ ∈ {0, 1}(vi) Sq0(q) = h2Q3(vii) Sq0(l) = h0x6,47(viii) Sq0(y) = h4Q3 + h2x1Theorem 1. In the cohomology of the mod 2 Steenrod algebraoutput of the minimal resolution programs [1, 2]. A preliminary version of thiscan be found in [3], where the notation for elements of ExtA(F2, F2) used here isdefined. Charts showing the results of the Ext calculations (but not the inducedhomomorphisms) can be found at http://www.math.wayne.edu/̃ rrb/cohom.By [6, Proposition 2.5], we may conclude that Sq0(x) ∈ R(x) when |Sq0(x)| =|R(x)|, that is, if k = n + s + 1. This allows us to use the calculations here todetermine a number of Steenrod operations.SOME ROOT INVARIANTS AND STEENROD OPERATIONS IN ExtA(F2, F2) 3k s n x R(x) Sq0 ∈ R(x)? Indeterminacy2 1 0 h0 h1 yes3 2 0 h20h21yes4 3 0 h30h20h2 yes8 4 0 h40h30h3 no10 5 0 h50Ph1 no11 6 0 h60h1Ph1 no12 7 0 h70h20Ph2 no16 8 0 h80h70h4 no18 9 0 h90P 2h1 no19 10 0 h100h1P2h1 no20 11 0 h110h20P 2h2 no24 12 0 h120h50i no26 13 0 h130P 3h1 no27 14 0 h140h1P3h1 no28 15 0 h150h20P 3h2 no32 16 0 h160h150h5 no34 17 0 h170P 4h1 no35 18 0 h180h1P4h1 no36 19 0 h190h20P 4h2 no3 1 1 h1 h2 yes5 2 2 h21h22yes5 1 3 h2 h3 yes6 2 3 h0h2 h1h3 yes7 3 3 h20h2 h21h3 yes9 2 6 h22h23yes9 1 7 h3 h4 yes10 2 7 h0h3 h1h4 yes11 3 7 h20h3 h21h4 yes12 4 7 h30h3 h20h2h4 yes11 2 8 h1h3 h2h4 yes12 3 8 c0 c1 yes13 3 9 h21h3 h22h4 yes14 4 9 h1c0 h2c1 yes15 5 9 Ph1 h2g yes21 6 10 h1Ph1 r no20 5 11 Ph2 h30h24no22 6 11 h0Ph2 q no23 7 11 h20Ph2 h1q no17 2 14 h23h24yes18 3 14 h0h23h1h24yes19 4 14 d0 d1 yes4 ROBERT R. BRUNERk s n x R(x) Sq0 ∈ R(x)? Indeterminacy20 5 14 h0d0 h0p yes25 6 14 h20d0 y no h1x17 1 15 h4 h5 yes18 2 15 h0h4 h1h5 yes19 3 15 h20h4 h21h5 yes20 4 15 h30h4 h20h2h5 yes21 5 15 h1d0 h2d1 yes24 5 15 h40h4 h30h3h5 no26 6 15 h50h4 h5Ph1 no h20f127 7 15 h60h4 h1h5Ph1 no28 8 15 h70h4 h20h5Ph2 no19 2 16 h1h4 h2h5 yes23 6 16 h21d0 h1x yes24 7 16 Pc0 h1y yes21 3 17 h21h4 h22h5 yes22 4 17 e0 e1 yes23 5 17 h0e0 h1e1 yes24 6 17 h20e0 h20f1 yes31 7 17 h30e0 h21h5d0 no30 8 17 h1Pc0 N no32 9 17 P 2h1 h1h5Pc0 no21 2 18 h2h4 h3h5 yes22 3 18 h0h2h4 h1h3h5 yes23 4 18 f0 f1 yes23 4 18 h20h2h4 h21h3h5 yes24 5 18 h0f0 h20c2 yes37 10 18 h1P2h1 R1 no h20h5i23 3 19 c1 c2 yes35 9 19 P 2h2 h5Pd0 no36 10 19 h0P2h2 h20h5i no39 11 19 h20P 2h2 h1Q1 no25 4 20 g g2 yes26 5 20 h0g h1g2 yes28 6 20 h20g h0h2g2 no25 3 21 h22h4 h23h5 yes27 5 21 h1g h2g2 yes27 4 22 h2c1 h3c2 yes31 8 22 Pd0 gd1 yes36 9 22 h0Pd0 h20Q2 no41 10 22 h20Pd0 x10,27 no x10,28, h1X128 4 23 h4c0 h5c1 yesSOME ROOT INVARIANTS AND STEENROD OPERATIONS IN ExtA(F2, F2) 5k s n x R(x) Sq0 ∈ R(x)? Indeterminacy29 5 23 h2g h3g2 yes30 6 23 h0h2g h1h3g2 yes31 7 23 i h2C yes36 8 23 h0i h20D2 no38 9 23 h20i h20B3 no40 9 23 h20i+ h1Pd0 h70h25no42 10 23 h30i x10,32 no h20h3Q243 11 23 h40i h1x10,32 no h0B2344 12 23 h50i h20PD2 no30 5 24 h1h4c0 h2h5c1 yes43 10 24 h21Pd0 B5 + PD2 no44 11 24 P 2c0 x11,35 + h20x9,40 no38 8 25 Pe0 x8,32 + x8,33 no39 9 25 h0Pe0 h1x8,32 no h20x7,33, h80h640 10 25 h20Pe0 h20h3Q2 no45 11 25 h30Pe0 h2B5 no h0PA46 12 25 h1P2c0 h2x11,35 no47 13 25 P 3h1 x13,34 no x13,3541 6 26 h22g r1 no36 7 26 j h0A′ no39 8 26 h0j h3Q2 no h0h2A, h20A′′40 9 26 h20j h20h3D2 no51 13 27 P 3h2 e0B4 no h60x7,5741 8 28 d20G21 no h0h3A′42 9 28 h0d20h1G21 no h0h2x7,4044 10 28 h20d20h0h2G21 no37 7 29 k h2h5n yes h3D240 8 29 h0k h0h3A′ no43 9 29 h20k h2G21 no33 2 30 h24h25yes34 3 30 h0h24h1h25yes35 4 30 h20h24h21h25yes36 5 30 h30h24h20h2h25yes37 6 30 r r1 yes39 7 30 h0r h22H1 + h3A′ no41 8 30 h20r h3x7,33 no44 9 30 h30r h20h4D2 no49 10 30 h40r P 2h25no e0A′50 11 30 h50r h1P2h25no51 12 30 P 2d0 x12,44 no33 1 31 h5 h6 yes6 ROBERT R. BRUNERk s n x R(x) Sq0 ∈ R(x)? Indeterminacy34 2 31 h0h5 h1h6 yes35 3 31 h20h5 h21h6 yes35 3 31 h1h24h2h25yes36 4 31 h30h5 h20h2h6 yes37 5 31 n n1 yes40 5 31 h40h5 h30h3h6 no42 6 31 h50h5 h6Ph1 no43 7 31 h60h5 h1h6Ph1 no h22Q340 8 31 d0e0 h3x7,33 yes44 8 31 h70h5 h20h6Ph2 no45 9 31 h0d0e0 h3G21 no h20x7,5348 9 31 h80h5 h70h4h6 no46 10 31 h20d0e0 h0x9,51 no50 10 31 h90h5 h6P2h1 no51 11 31 h100h5 h1h6P2h1 no h0gA′35 2 32 h1h5 h2h6 yes37 4 32 d1 d2 yes39 6 32 q h2Q3 yes40 7 32 l h0x6,47 yes43 8 32 h0l x8,51 no h20h6Ph244 9 32 h20l h20x7,53 no37 3 33 h21h5 h22h6 yes38 4 33 p p1 yes39 5 33 h0p h1p1 yes45 7 33 h1q x7,57 no h1x6,53, h20h6d037 2 34 h2h5 h3h6 yes38 3 34 h0h2h5 h1h3h6 yes39 4 34 h20h2h5 h21h3h6 yes41 6 34 h2n h3n1 yes h6Ph243 8 34 d0g h1x7,53 yes h20x6,5344 9 34 h0d0g h20m1 yes48 11 34 Pj h0gA′ no41 5 35 h2d1 h3d2 yes43 7 35 m m1 yes h20h6d044 8 35 h0m h20t1 yes47 9 35 h20m h3x8,51 no43 6 36 t t1 yes h40h4h641 3 37 h22h5 h23h6 yes43 5 37 x x1 yes44 6 37 h0x h20e2 yes45 7 37 h1t h2t1 yesSOME ROOT INVARIANTS AND STEENROD OPERATIONS IN ExtA(F2, F2) 7k s n x R(x) Sq0 ∈ R(x)? Indeterminacy46 7 37 h20x h0h2x1 no46 8 37 e0g e1g2 yes48 8 37 h30x h40f2 no41 2 38 h3h5 h4h6 yes42 3 38 h0h3h5 h1h4h6 yes43 4 38 h20h3h5 h21h4h6 yes43 4 38 e1 e2 yes44 5 38 h30h3h5 h20h2h4h6 yes45 6 38 h1x h2x1 yes45 6 38 y h4Q3 + h2x1 yes46 7 38 h0y h1h4Q3 yes h20h6g48 8 38 h20y h50c3 no43 3 39 h1h3h5 h2h4h6 yes44 4 39 h5c0 h6c1 yes45 5 39 h1e1 h2e2 yes49 7 39 h1y x7,74 no45 4 40 h21h3h5 h22h4h6 yes45 4 40 f1 f2 yes46 5 40 h0f1 h1f2 yes46 5 40 h1h5c0 h2h6c1 yes47 6 40 h20f1 h21f2 yes47 6 40 h5Ph1 h2h6g yes45 3 41 c2 c3 yes46 4 41 h0c2 h1c3 yes47 5 41 h20c2 h21c3 yes49 4 44 g2 g3 yes49 3 45 h23h5 h24h6 yesReferences1. Robert R. Bruner, “Calculation of large Ext modules”, Computers in Geometry andTopology (M. C. Tangora, ed.), Marcel Dekker, New York, 1989, 79-104.2. R.R.Bruner, “Ext in the nineties”, pp. 71-90 in Algebraic Topology, Oaxtepec 1991,Contemp. Math. 146, Amer. Math. Soc., Providence, 1993.3. R.R.Bruner, “The Cohomology of the Mod 2 Steenrod Algebra”, preprint, April 1997.4. Robert R. Bruner, “Some remarks on the root invariant”, Proceedings of the FieldsInstitute Workshops on Homotopy Theory, January 1996, Contemp. Math., to ap-pear.5. Robert Bruner and John Greenlees, “The Bredon-Löffler conjecture”, Exper. Math.4 (1995), 101-109.6. M.E.Mahowald and D.C.Ravenel, “The root invariant in homotopy theory”, Topology32 (1993), 865-898.7. M.E.Mahowald and Paul Shick, “Some root invariants in ExtA(Z/2, Z/2) ”, Proceed-ings of the Northwestern Homotopy Theory Conference, Contemp. Math. 19 (1983),227-231.8 ROBERT R. BRUNERMathematics Department, Wayne State University, Detroit, Michigan, 48202E-mail address: rrb@math.wayne.edu

Some root invariants and Steenrod operations in Ext_A(F2,F2)

https://digitalcommons.wayne.edu/cgi/viewcontent.cgi?article=1141&context=mathfrp

Some root invariants and Steenrod operations in Ext_A(F2,F2)

Abstract

Similar works

Full text

Available Versions

Digital Commons@Wayne State University