Classical and motivic Adams charts

0.66 MB·

by  Daniel C. Isaksen

CLASSICAL AND MOTIVIC ADAMS CHARTS
DANIELC.ISAKSEN
Abstract. Thisdocument contains large-formatAdamschartsthatcompute2-completestablehomotopy groups,
bothintheclassicalcontext andinthemotiviccontext overC. Thechartsareessentiallycompletethrough 6. The E -page of the motivic Adams spectral sequence
the59-stemandcontainpartialresultstothe70-stem. Intheclassicalcontext, webelievethatthesearethemost 4
accuratechartsoftheirkind. This chart indicates the Adams d and d differentials on the E -page of the motivic Adams spectral
4 5 4
WealsoincludeAdamschartsforthemotivichomotopy groupsofthecofiber ofτ.
sequence. The chartiscomplete throughthe 65-stem,with afew indicatedexceptions. Beyondthe 65-stem,
there are several unknown differentials.
Because of unknown d differentials, the actual E -page beyond the 65-stem is a subquotient of what is
3 4
This document contains large-format Adams charts that compute 2-complete stable homotopy groups,
shown.
both inthe classicalcontextandin the motivic context overC. The charts areessentially complete through
See Section 3 for instructions on interpreting the chart. In addition:
the 59-stemandcontainpartialresultstothe 70-stem. Inthe classicalcontext,webelievethatthese arethe
(1) Purple dots indicate copies of M /τ4.
most accurate charts of their kind. 2
(2) Blue lines of slope −4 and −5 indicate Adams d and d differentials.
We also include Adams charts for the motivic homotopy groups of the cofiber of τ. 4 5
(3) Magenta lines of slope −5 indicate that an Adams d differential hits τ times a generator. For
The charts are intended to be viewed electronically. The author can supply versions that are suitable for 5
example, d (τPh e )=τd z in the 55-stem.
printing. 5 5 0 0
(4) Orangelinesofslope−4and−5indicatethatanAdamsd ord differentialhitsτk timesagenerator
Justifications for these calculations appear in [1]. 4 5
for k ≥2.
1. The classical Adams spectral sequence (5) Dashed lines indicate plausible Adams d3, d4, and d5 differentials, but we have not independently
verified their existence.
This chartshows the classicalAdams spectral sequence. The E -page is complete through the 70-stem,as
2 (6) For clarity, the unknown Adams d differentials are also shown on this chart.
3
are the Adams d differentials. The Adams d , d , and d differentials are complete through the 65-stem,
2 3 4 5
with a few indicated exceptions. 7. The E∞-page of the motivic Adams spectral sequence
(1) Black dots indicate copies of F .
2 This chartindicates the E∞-page of the motivic Adams spectralsequence. The chartis complete through
(2) Vertical lines indicate h multiplications.
0 the 59-stem. Because of unknown differentials, the actual E∞-page beyond the 59-stem is a subquotient of
(3) Lines of slope 1 indicate h multiplications.
1 what is shown.
(4) Lines of slope 1/3 indicate h multiplications.
2 See Section 3 for instructions on interpreting the chart. In addition:
(5) Light blue lines of slope −2 indicate Adams d differentials.
2 (1) Purple dots indicate copies of M /τ4.
(6) Red lines of slope −3 indicate Adams d differentials. 2
3 (2) For clarity, some of the unknown Adams d , d , and d differentials are also shown on this chart.
(7) Green lines of slope −4 indicate Adams d differentials. 3 4 5
4
(8) Blue lines of slope −5 indicate Adams d5 differentials. 8. The E2-page of the motivic Adams spectral sequence for the cofiber of τ
(9) Dashed lines indicate plausible Adams d , d , and d differentials, but we have not independently
3 4 5
This chartindicates the E -page of the motivic Adams spectralsequence for the cofiber of τ. The chartis
verified their existence. 2
complete through the 70-stem, with indicated exceptions.
2. The E∞-page of the classical Adams spectral sequence (1) Black dots indicate copies of M /τ that are in the image of the inclusion E (S0,0)→E (Cτ) of the
2 2 2
bottom cell.
This chart indicates the E∞-page of the classical Adams spectral sequence. The chart is essentially
(2) Reddots indicate copies ofM /τ that aredetected by the projectionE (Cτ)→E (S0,0) to the top
complete throughthe 59-stem. Because ofunknowndifferentials, the actual E∞-pagebeyondthe 59-stemis 2 2 2
cell.
a subquotient of what is shown.
(3) Blacklinesindicateextensionsbyh , h ,andh thatareintheimageofthe inclusionofthebottom
See Section 1 for instructions on interpreting the chart. In addition: 0 1 2
cell.
(1) Olive lines indicate hidden 2 extensions.
(4) Red lines indicate extensions by h , h , and h that are detected by projection to the top cell.
(2) Purple lines indicate hidden η extensions. 0 1 2
(5) Blue lines indicate extensions by h , h , and h that are hidden in the sense that they are not
(3) Brown lines indicate hidden ν extensions. 0 1 2
detected by the top cell or the bottom cell.
(4) Dashed olive, purple, and brown lines indicate possible hidden extensions.
(6) Arrows of slope 1 indicate infinite towers of h extensions.
(5) For clarity, some of the unknown Adams d , d , and d differentials are also shown on this chart. 1
3 4 5 (7) Dashed blue lines indicate unknown hidden extensions.
3. The cohomology of the motivic Steenrod algebra (8) Light blue lines indicate Adams differentials.
(9) Dashed light blue lines indicate unknown Adams differentials. These unknown differentials are also
This chart shows the cohomology of the motivic Steenrod algebra over C. The chart is complete through
indicated on later charts.
the 70-stem.
(1) Black dots indicate copies of M2. 9. The E3-page of the motivic Adams spectral sequence for the cofiber of τ
(2) Red dots indicate copies of M /τ.
2 Thischartindicatesthe E -pageofthe motivic Adamsspectralsequence forthe cofiberofτ. The E -page
(3) Blue dots indicate copies of M /τ2. 3 3
2 is complete through the 70-stem, but the Adams d differentials are complete only through the 64-stem.
(4) Green dots indicate copies of M /τ3. 3
2 Beyond the 64-stem, there are a number of unknown differentials.
(5) Vertical lines indicate h multiplications. These lines might be black, red, blue, or green,depending
0 See Section 8 for instructions on interpreting the chart.
on the τ torsion of the target.
(6) Linesofslope1indicateh multiplications. Theselinesmightbeblack,red,blue,orgreen,depending 10. The E -page of the motivic Adams spectral sequence for the cofiber of τ
1 4
on the τ torsion of the target.
Thischartindicatesthe E -pageofthe motivic Adamsspectralsequence forthe cofiberofτ. The E -page
(7) Lines of slope 1/3 indicate h multiplications. These lines might be black, red, blue, or green, 4 4
2 iscompletethroughthe64-stem. Beyondthe64-stem,theactualE -pageisasubquotientofwhatisshown.
depending on the τ torsion of the target. 4
The chart shows both Adams d and d differentials.
(8) Red arrows indicate infinite towers of h multiplications, all of which are annihilated by τ. 4 5
1 See Section 8 for instructions on interpreting the chart.
(9) Magenta lines indicate that an extension hits τ times a generator. For example, h ·h h = τh3 in
0 0 2 1
the 3-stem. 11. The E∞-page of the motivic Adams spectral sequence for the cofiber of τ
(10) Orange lines indicate that an extension hits τ2 times a generator. For example, h ·h3x=τ2h e g
0 0 0 0
in the 37-stem. This chart indicates the E∞-page of the motivic Adams spectral sequence for the cofiber of τ. The chart
is complete through the 63-stem, but hidden extensions are only shown through the 59-stem. Beyond the
(11) Dotted lines indicate that the extension is hidden in the May spectral sequence.
(12) Squaresindicatethatthereisaτ extensionthatishiddenintheMayspectralsequence. Forexample, 63-stem, the actual E∞-page is a subquotient of what is shown.
τ ·Pc0d0 =h50r in the 30-stem. (1) Black dots indicate copies of M2/τ that are in the image of the inclusion π∗,∗ → π∗,∗(Cτ) of the
bottom cell.
4. The E2-page of the motivic Adams spectral sequence (2) Red dots indicate copies of M2/τ that are detected by the projection π∗,∗(Cτ) → π∗−1,∗+1 to the
top cell.
This chart indicates the Adams d differentials on the E -page of the motivic Adams spectral sequence.
2 2
(3) Blacklinesindicateextensionsbyh , h ,andh thatareintheimageofthe inclusionofthebottom
The chart is complete through the 70-stem. See Section 3 for instructions on interpreting the chart. In 0 1 2
cell.
addition:
(4) Red lines indicate extensions by h , h , and h that are detected by projection to the top cell.
(1) Blue lines of slope −2 indicate Adams d differentials. 0 1 2
(2) Magenta lines of slope −2 indicate tha2t an Adams d differential hits τ times a generator. For (5) Bluelinesindicateextensionsby2,η,andν thatarenothiddenintheE∞-pagebutarenotdetected
2 by the top cell or the bottom cell.
example, d (h c )=τh2e in the 40-stem.
2 0 2 1 1 (6) Arrows of slope 1 indicate infinite towers of h extensions.
(3) Orange lines of slope −2 indicate that an Adams d differential hits τ2 times a generator. For 1
example, d (h y)=τ2h e g in the 37-stem. 2 (7) Olive lines indicate 2 extensions that are hidden in the E∞-page.
2 0 0 0 (8) Purple lines indicate η extensions that are hidden in the E∞-page.
5. The E -page of the motivic Adams spectral sequence (9) Brown lines indicate ν extensions that are hidden in the E∞-page.
3
(10) Dashed lines indicate possible extensions.
This chart indicates the Adams d differentials on the E -page of the motivic Adams spectral sequence.
3 3 (11) Dashed light blue lines indicate unknown Adams differentials. These unknown differentials are
Thechartiscompletethroughthe65-stem,withindicatedexceptions. Beyondthe65-stem,thereareseveral
included for clarity.
unknown differentials.
See Section 3 for instructions on interpreting the chart. In addition: References
(1) Blue lines of slope −3 indicate Adams d3 differentials. [1] DanielC.Isaksen, Stable stems (2014), preprint,availableatarXiv:1407.8418.
(2) Magenta lines of slope −3 indicate that an Adams d differential hits τ times a generator. For
3
example, d (r)=τh d2 in the 29-stem.
3 1 0
(3) Orange lines of slope −3 indicate that an Adams d differential hits τk times a generator for k≥2.
3
For example, d (Q )=τ2gt in the 56-stem, and d (τW )=τ4e4 in the 68-stem.
3 2 3 1 0
(4) Dashed lines indicate plausible Adams d differentials, but we have not independently verified their
3
existence.
1
4102 ceD 61 ]TA.htam[ 3v3894.1041:viXra
2
18
36
The classical Adams spectral sequence
17
34
P8h1 P8h2
16
32 P7d0
P7c0
15
30
P7h1 P7h2
14
28 P6d0 P6e0 P5d20
P6c0
P5j P4d0i
13
26
P6h1 P6h2
12
24 P5d0 P5e0 P4d20 P4d0e0 P3d30 P3d20e0
P5c0
P4i P4j P3d0i P3d0j P2d20i P2d20j+h60R′1
11
22
P2i2 P2ij Pd0i2
P5h1 P5h2 P3u P3v P2d0u
10
20 P4d0 P4e0 P3d20 P3d0e0 P2d30 P2d20e0 Pd40 Pd30e0 d50
P4c0
P3j P2d0i P2d0j Pd20i Pd20j+h90R P2d0m d30j
9
18 Pij d0i2 d0ij Pd20r P2x′ d20z
P4h1 P4h2 P2u P2v Pd0u Pd0v d20u d20v R′1
8
16 P3d0 P3e0 P2d20 P2d0e0 Pd30 Pd20e0 d40 d30e0 d20e20 d0e30 e40
P3c0 P2i P2j Pd0i Pd0j d20i d20j+h50R1 Pd0m d20l Ph1x′ d20m d0e0m d0gm
7
14 i2 ij Pd0r il d20r jm d0gr U lm d0x′ e0x′m2
gv
P3h1 P3h2 Q′ Pu Pv d0u d0v e0v gw R2 G11 W1
6
12 P2d0 P2e0 Pd20 Pd0e0 d30 d20e0 d0e20 d0e0g e20g e0g2 g3 Ph5j nm
P2c0 Pj d0i d0j d0k d0l d0m e0m gm nr C11
10 5 z d0r e0r gr x′ R1 gt Q1 B21 B22R q1 B23 B5D2′ P(A+A′)
P2h1 P2h2 u v w gn Ph5d0 Ph5e0 B4 X1 X3 C′′
8 4 Pd0 Pe0 d20 d0e0+h70h5 e20 e0g g2 N Ph5c0 d1g h5i h5j E1E1+C0 h3Q2 G21 D3′ d1e1
Pc0
i j k l m B1 B2 Q2 B3 C′ X2 h3D2 G0 h3(A+A′)
3
6 r q t y Ph1h5 Ph2h5 C G D2 A′A+A′ h5n A′′ r1 h3H1
Ph1 Ph2 n x h5d0 h5e0 h5f0 h3g2 D1 H1 n1 Q3
2
4 d0 e0 f0 g h4c0 d1 p e1 h5c0 f1 g2 h3c2 h5c1 D3 d2 p′ p1
c0 c1 c2 h23h5
1
2
h23 h24 h3h5 h25 h3h6
h0 h1 h2 h3 h4 h5 h6
0
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70
3
18
36
The E -page of the classical Adams spectral sequence
∞
17
34
P8h1 P8h2
16
32
P7c0
15
30
P7h1 P7h2
14
28
P6c0
13
26
h205h6
P6h1 P6h2 P4h20i
12
24
P5c0
11
22
P5h1 P5h2
10
20
h70Q′
P4c0
9
18
P4h1 P4h2 P2h20i d20u
8
16
d20e20
P3c0 d0e0m
d20l
7 h1W1
14 d0e0r d0x′e0gr
d0w
P3h1 P3h2 Pu d0u gw
6
12 d30 d0e20 e20g g3 Ph5j h0h4x′
e0m C11 h30G21
P2c0 h100h5 d0l
5
10 z e0r x′ B21 h1X1 R q1 B23 B5+D2′
P2h1 P2h2 h20i u w gn h0h5i X3 h3(E1+C0)
8 4 Pd0 d20 e20 g2 N Ph5c0 d1g h1Q2 E1+C0 h3Q2 D3′ d1e1
Pc0 B1 B2 B3 X2 C′ k1 h22H1h3A′ h1h3H1
3
6 q t Ph1h5 Ph2h5 C A′ h5n h1H1 r1 h2Q3
Ph1 Ph2 n x h3d1 h5d0 h3g2 Q3
2
4 d0 h30h4 g h4c0 d1 p h20h3h5 h5c0 f1 g2 h5c1 D3 d2 p′ p1
c0 c1 h0h2h5 h22h5 h1h3h5 h23h5 h22h6
1
2 h23 h1h4 h2h4 h24 h1h5 h25 h1h6 h3h6
h0 h1 h2 h3
0
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70
4
18
36
The cohomology of the motivic Steenrod algebra
17
34
P8h1 P8h2
16
32 P7d0
P7c0
P6c0d0
15
30
P7h1 P7h2
14
28 P6d0 P6e0 P5d20
P6c0 P5c0d0 P5c0e0 P5j P4c0d20 P4d0i
13
26
P6h1 P6h2
24 12 P5d0 P5e0 P4d20 P4d0e0 P3d30 P3d20e0
P5c0 P4c0d0 P4i P4c0e0 P4j P3c0d20 P3d0i P3c0d0e0P3d0j P2c0d30 P2d20i P2c0d20Pe02d20j P3u′
11
22
P2i2 P2ij P2ik
P5h1 P5h2 P3u P3v P2d0u
10
20 P4d0 P4e0 P3d20 P3d0e0 P2d30 P2d20e0 Pd40 Pd30e0 d50
P4c0 P3c0d0 P3c0e0 P3j P2c0d20 P2d0i P2c0d0e0 P2d0j Pc0d30 Pd20i Pc0d20e0 Pd20j P2u′ c0d40 d30iP2v′ c0d30e0 dP30jd0u′ c0d20e20
9 d20z
18
Pij d0i2 d0ij d0ik P2x′
P4h1 P4h2 P2u P2v Pd0u Pd0v d20u Pc0xR′′1 d20v
8
16 P3d0 P3e0 P2d20 P2d0e0 Pd30 Pd20e0 d40 d30e0 d20e20 d0e30 e40
P3c0 P2c0d0 P2i P2c0e0 P2j Pc0d20 Pd0i Pc0d0e0 Pd0j c0d30 d20i c0d20e0 d20j Pu′ c0d0e20 d20kPv′ c0e30 d20l d0u′ Phc10xe20′g d20me0u′ d0e0me0v′ e20m
7
14 i2 ij ik il im jm U km d0x′ lm e0x′m2
P3h1 P3h2 Q′ Pu Pv d0u d0v e0v h0g3 c0x′ τe0hw2g3 R2 τgw B8d0 h3gG311 τW1 B8e0
6
12 P2d0 P2e0 Pd20 Pd0e0 d30 d20e0 d0e20 e30 e20g τe0g2 τg3 Ph5j nm
P2c0 Pc0d0 Pc0e0 Pj c0d20 d0i c0d0e0 d0j c0e20 d0k c0e0g d0lu′ e0lv′ e0m gm c1g2 nr C11h2B23 h2B5
10 5 z d0r e0r gr x′ R1 gt Q1 B21 B22 R q1 τB23c0Q2 τB5D2′ P(A+A′)
P2h1 P2h2 u v h2g2 τw h3g2 G3 gn B8 Ph5e0 D11 B4 X1 X3 C′′
8 4 Pd0 Pe0 d20 d0e0 e20 e0g τg2 N Ph5c0 d1g h5c0d0 h5i h5j e1g E1C0 d21 h3Q2 G21 h3B7 h2G0D3′ d1e1
Pc0
c0d0 i c0e0 j k l m c1g B1 B2 i1 B6 Q2 j1 B3 B7 C′ X2 h3D2 k1 τG0 h3(A+A′)
3
6 r q t y Ph1h5 Ph2h5 C τG D4 h23g2 D2 A′A+A′ h5n A′′ r1 h0Q3 h3H1 h2Q3
Ph1 Ph2 h2g h3g n x h5d0 h5e0 h5f0 h3g2 D1 H1 n1 τQ3
2
4 d0 e0 f0 τg h4c0 d1 p e1 h5c0 f1 g2 h3c2 h5c1 D3 d2 p′ p1
c0 c1 c2 h23h5
1
2
h23 h24 h3h5 h25 h3h6
h0 h1 h2 h3 h4 h5 h6
0
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70
5
18
36
The E -page of the motivic Adams spectral sequence
2
17
34
P8h1 P8h2
16
32 P7d0
P7c0
P6c0d0
15
30
P7h1 P7h2
14
28 P6d0 P6e0 P5d20
P6c0 P5c0d0 P5c0e0 P5j P4c0d20 P4d0i
13
26
P6h1 P6h2
24 12 P5d0 P5e0 P4d20 P4d0e0 P3d30 P3d20e0
P5c0 P4c0d0 P4i P4c0e0 P4j P3c0d20 P3d0i P3c0d0e0P3d0j P2c0d30 P2d20i P2c0d20Pe02d20j P3u′
11
22
P2i2 P2ij P2ik
P5h1 P5h2 P3u P3v P2d0u
10
20 P4d0 P4e0 P3d20 P3d0e0 P2d30 P2d20e0 Pd40 Pd30e0 d50
P4c0 P3c0d0 P3c0e0 P3j P2c0d20 P2d0i P2c0d0e0 P2d0j Pc0d30 Pd20i Pc0d20e0 Pd20j P2u′ c0d40 d30iP2v′ c0d30e0 dP30jd0u′ c0d20e20
9 d20z
18
Pij d0i2 d0ij d0ik P2x′
P4h1 P4h2 P2u P2v Pd0u Pd0v d20u Pc0xR′′1 d20v
8
16 P3d0 P3e0 P2d20 P2d0e0 Pd30 Pd20e0 d40 d30e0 d20e20 d0e30 e40
P3c0 P2c0d0 P2i P2c0e0 P2j Pc0d20 Pd0i Pc0d0e0 Pd0j c0d30 d20i c0d20e0 d20j Pu′ c0d0e20 d20kPv′ c0e30 d20l d0u′ Phc10xe20′g d20me0u′ d0e0me0v′ e20m
7
14 i2 ij ik il im jm U km d0x′ lm e0x′m2
P3h1 P3h2 Q′ Pu Pv d0u d0v e0v h0g3 c0x′ τe0hw2g3 R2 τgw B8d0 h3gG311 τW1 B8e0
6
12 P2d0 P2e0 Pd20 Pd0e0 d30 d20e0 d0e20 e30 e20g τe0g2 τg3 Ph5j nm
P2c0 Pc0d0 Pc0e0 Pj c0d20 d0i c0d0e0 d0j c0e20 d0k c0e0g d0lu′ e0lv′ e0m gm c1g2 nr C11h2B23 h2B5
10 5 z d0r e0r gr x′ R1 gt Q1 B21 B22 R q1 τB23c0Q2 τB5D2′ P(A+A′)
P2h1 P2h2 u v h2g2 τw h3g2 G3 gn B8 Ph5e0 D11 B4 X1 X3 C′′
8 4 Pd0 Pe0 d20 d0e0 e20 e0g τg2 N Ph5c0 d1g h5c0d0 h5i h5j e1g E1C0 d21 h3Q2 G21 h3B7 h2G0D3′ d1e1
Pc0
c0d0 i c0e0 j k l m c1g B1 B2 i1 B6 Q2 j1 B3 B7 C′ X2 h3D2 k1 τG0 h3(A+A′)
3
6 r q t y Ph1h5 Ph2h5 C τG D4 h23g2 D2 A′A+A′ h5n A′′ r1 h0Q3 h3H1 h2Q3
Ph1 Ph2 h2g h3g n x h5d0 h5e0 h5f0 h3g2 D1 H1 n1 τQ3
2
4 d0 e0 f0 τg h4c0 d1 p e1 h5c0 f1 g2 h3c2 h5c1 D3 d2 p′ p1
c0 c1 c2 h23h5
1
2
h23 h24 h3h5 h25 h3h6
h0 h1 h2 h3 h4 h5 h6
0
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70
6
18
36
The E -page of the motivic Adams spectral sequence
3
17
34
P8h1 P8h2
16
32 P7d0
P7c0
P6c0d0
15
30
P7h1 P7h2
14
28 P6d0 P5d20
P6c0 P5c0d0 P4c0d20
13
26
P6h1 P6h2 P4h20i
12
24 P5d0 P4d20 τP4d0e0 P3d30 τP3d20e0
P5c0 P4c0d0 P3c0d20 P2c0d30 P3u′+τP2d20j
11
22
P2i2 P2ij Pd0i2
P5h1 P5h2 P3u P2d0u
10
20 P4d0 P3d20 τP3d0e0 P2d30 τP2d20e0 Pd40 τPd30e0 d50
P4c0 P3c0d0 P2c0d20 Pc0d30 P2u′+τPd20j c0d40 τ2P2d0m Pd0u′+τd30j c0d20e20
9 d20z
18
h50Q′ Pij d0i2 d0ij Pd20r P2x′
P4h1 P4h2 P2h20i P2u Pd0u τPd0v d20u Pc0x′ τd20v
8
16 P3d0 P2d20 τP2d0e0 Pd30 τPd20e0 d40 τd30e0 d20e20 τd0e30 e40
P3c0 P2c0d0 Pc0d20 c0d30 Pu′+τd20j c0d0e20 τ2Pd0m d0u′+τd20l Ph1x′ τ2d20m τ2d0e0m τ2d0gm
7
14 i2 ij Pd0r d0z d20r d0e0r e20r d0x′e0gr h1h3g3 m2 τe0x′
P3h1 P3h2 Pu d0u τd0vh2e20g τd0w h0g3 c0x′ τgv h2g3 τgw B8d0 τW1
6
12 P2d0 Pd20 τPd0e0 d30 τd20e0 d0e20 τd0e0g τe20g τe0g2 τg3 Ph5j nm h0h4x′
P2c0 Pc0d0 c0d20 c0e20 τd0l+u′ τ2d0m τ2e0m τ2gm c1g2 nr h2B22 C11h2B23 τh30G2h12B5
10 5 z d0r e0r h1h3g2 gr h1G3 x′ gt B21 h1X1+ττBh212X1R q1 τB23c0Q2 τ2B5+D2′
P2h1 P2h2 h20i u h0g2 h2g2 τw gn B8 Ph5e0 D11 h31D4 h21X2 X3 C′′ h3E1
8 4 Pd0 d20 τd0e0 e20 τe0g τg2 N Ph5c0 d1g h5i h5c0e0 h5j e1g EC10 h70h6 d21 h3Q2 τD3′ d1e1
Pc0 c0d0 c1g B1 B2 i1 B6 Q2 j1 B3 τX2 C′ k1 h22H1h3A′ τh1h3H1
3
6 h1h3g r q t Ph1h5 Ph2h5 h1h5e0 C τ2G h23g2 A′ h5n τh1H1 r1 h0Q3 h2Q3
Ph1 Ph2 h0g h2g n x h5d0 h3g2 τQ3
2
4 d0 τg h4c0 h30h5 d1 p e1 h5c0 f1 g2 h5c1 D3 d2 p′ p1
c0 c1 h23h5 h22h6
1
2 h23 h0h4 h1h4 h2h4 h24 h1h5 h2h5 h3h5 h25 h1h6 h3h6
h0 h1 h2 h3
0
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70
7
18
36
The E -page of the motivic Adams spectral sequence
4
17
34
P8h1 P8h2
16
32 P7d0
P7c0
15
30
P7h1 P7h2
14
28 P6d0 P5d20
P6c0
13
26
P6h1 P6h2 P4h20i τ2P4h1d0e0
12
24 P5d0 P4d20 P3d30 τ2P3d20e0
P5c0
P2h0i2 P3u′+τP2d20j
11
22
P2ij
P5h1 P5h2 P3u P2d0u
10
20 P4d0 h70Q′ P3d20 τ2P3d0e0 P2d30 τ2P2d20e0 Pd40 τ2Pd30e0 d50
P4c0
P2u′+τPd20j h108h6 Pd0u′+τd30j
9 d20z
18
Pij d0ij τ2Pd20r P2x′
P4h1 P4h2 P2h20i P2u Pd0u τ2Pd0v d20u τ2d20v
8
16 P3d0 P2d20 τ2P2d0e0 Pd30 τ2Pd20e0 d40 τ2d30e0 d20e20 τ2d0e30 e40
P3c0 Pu′+τd20j d0u′+τd20l Ph1x′ τ3d20m τ2d0e0m τ3d0gm
14 7 ij d0z τ2d20r d0e0r τ2d0gr d0x′e0gr h1h3g3 τh1Wτ21e0xτ′2m2
P3h1 P3h2 Pu d0u τ2d0vh2e20g τd0w h0g3 τ2gv h2g3 τgw+h41X1 B8d0
12 6 P2d0 Pd20 τ2Pd0e0 d30 τ2d20e0 d0e20 τ2d0e0g τe20g τ2e0g2 Ph31h5τeg03+h41h5c0e0 Ph5j h0h4x′
P2c0 c0e20 τd0l+u′ τ2e0m h61h5e0 τ3gm c1g2 τ2h1B22 h2B22 C11h2B23 τh30Gτ2h12B5
5 h2C′′
10 z e0r h1h3g2 τ2gr h1G3 x′ gt B21 τh1X1 R q1 τB23c0Q2 τ2B5+D2′
P2h1 P2h2 h20i u h0g2 h2g2 τw gn B8 h0h5i τPh5e0 D11 h3d1g h31D4 h21X2 X3 h3(E1+C0)
8 4 Pd0 d20 τ2d0e0+h70h5 e20 τ2e0g τg2 N Ph5c0 d1g h1Q2 E1+C0 h3Q2d21 τD3′ d1e1
Pc0 c0d0 c1g B1 B2 i1 j1 B3 τX2 C′ k1 h22H1h3A′ τh1h3H1
3
6 h1h3g q t Ph1h5 Ph2h5 C h23g2 A′ h5n τh1H1 r1 h0Q3 h2Q3
Ph1 Ph2 h0g h2g n x h3d1 h5d0 h3g2 τQ3
2
4 d0 h30h4 τg h4c0 d1 p h5c0 f1 g2 h5c1 D3 d2 p′ p1
c0 c1 h0h2h5 h22h5 h23h5 h22h6
1
2 h23 h1h4 h2h4 h24 h1h5 h3h5 h25 h1h6 h3h6
h0 h1 h2 h3
0
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70
8
18
36
The E -page of the motivic Adams spectral sequence
∞
17
34
P8h1 P8h2
16
32
P7c0
15
30
P7h1 P7h2
14
28
P6c0
13
26
h205h6
P6h1 P6h2 P4h20i
12
24
P5c0
11
22
P5h1 P5h2
10
20
h70Q′
P4c0
9
18
P4h1 P4h2 P2h20i d20u
8
16 d20e20 e40
P3c0 τ2d0e0m
d0u′+τd20l
7 τh1W1
14 d0z d0e0r d0x′e0gr h1h3g3
τd0w
P3h1 P3h2 Pu d0u h2e20g h0g3 h2g3 τgw+h41X1 B8d0
12 6 d30 d0e20 τe20g Ph31h5τeg03+h41h5c0e0 Ph5j h0h4x′
P2c0 h100h5 c0e20 τd0l+u′ τ2e0m h61h5e0 c1g2 τh21X1 h2B22 C11h2B23 τh30G21
5 h2C′′
10 z e0r h1h3g2 h1G3 x′ gt B21 τh1X1 R q1 τB23c0Q2 τ2B5+D2′
P2h1 P2h2 h20i u h0g2 h2g2 τw gn B8 h0h5i D11 h3d1g h31D4 h21X2 X3 hh223C(E′ 1+C0)
8 4 Pd0 d20 e20 τg2 N Ph5c0 d1g h1Q2 E1+C0 h3Q2d21 τD3′ d1e1
Pc0 c0d0 c1g B1 B2 i1 j1 B3 τX2 C′ k1 h22H1h3A′ τh1h3H1
3
6 h1h3g q t Ph1h5 Ph2h5 C h23g2 A′ h5n τh1H1 r1 h0Q3 h2Q3
Ph1 Ph2 h0g h2g n x h3d1 h5d0 h3g2 τQ3
2
4 d0 h30h4 τg h4c0 d1 p h20h3h5 h5c0 f1 g2 h5c1 D3 d2 p′ p1
c0 c1 h0h2h5 h22h5 h1h3h5 h23h5 h22h6
1
2 h23 h1h4 h2h4 h24 h1h5 h25 h1h6 h3h6
h0 h1 h2 h3
0
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70
9
18
36
The E -page of the motivic Adams spectral sequence for the cofiber of τ
2
P8h41
17
34
P8h1 P8h2
16
32 P7d0
P7h21c0
P7c0
P7h41 P6d0h41 P6c0d0
15
30
P7h1 P7h2 P6h21e0
14
28 P6h21c0 P6d0 P6e0 P5d20
P6h41 P6c0 P5d0h41 P5c0d0 P5h1c0d0 P5c0e0 P5j P4c0d20 P4d0i
13
26
P5c0e0 P4d0c0d0
P6h1 P6h2 P5h21e0 P4c0c0d0 P4c0c0e0 P3c0d0c0d0
24 12 P5h21c0 P5d0 P5e0 P4d20 P4d0e0 P3d30 P3d20e0
P5h41 P5c0 P4d0h41 P4c0d0 P4i P4c0e0 P4j P3c0d20 P3d0i P3c0d0e0P3d0j P2c0dP302d20i P3h31u P2c0d20Pe02d20j P3u′
11
22
P4c0d0 P4c0e0 P3d0c0d0 P2i2 P3e0c0d0 P2ij P2d20c0d0 P2ik P2d0e0c0d0
P3h1v P2d0u
P5h1 P5h2 P4h21e0 P3c0c0d0 P3c0c0e0 P2c0d0c0d0 P3u P2c0e0c0d0 P3v Pc0d20c0d0 Pc0d0e0c0d0
10
20 P4h21c0 P4d0 P4e0 P3d20 P3d0e0 P2d30 P2d20e0 τP2h0d20e0 Pd40 τPh0d40 Pd30e0Pd0τh0d20e0 d50
P4h41 P4c0 P3d0h41 P3c0d0 P3h1c0d0P3c0e0 P3j P2c0d20 P2d0i P2c0d0e0 P2d0j Pc0d30 PdP20i2h31u Pc0d20e0 Pd20j P2u′ c0d40 P2h1ud′30i P2v′ c0d30e0 dP30jd0u′ c0d20e20
18 9 P3c0e0 P2d0c0d0 P2e0c0d0 Pij Pd20c0d0 d0i2 Pd0e0c0d0 d0ij d30c0d0 d0ik P2v′ d20e0c0d0 P2x′d20zPd0u′
P2h1vPd0u c0e20c0d0 Pe0v Pc0x′
P4h1 P4h2 P3h21e0 P2c0c0d0 P2c0c0e0 Pc0d0c0d0 P2u Pc0e0c0d0 P2v c0d20c0d0 c0d0e0c0d0 Ph1d0u Pd0v x′Ph41 Pd0h1v c0e20c0e0 d0h1d0uR′1 d20v
8
16 P3h21c0 P3d0 P3e0 P2d20 P2d0e0 Pd30 Pd20e0 d40 τh0d40 d30e0 d0τh0d20e0 d20e20 e0τh0d20e0 d0e30 e0τh0d0e20 e40 e0τh0e30 Pc0x′
P3h41 P3c0 P2d0h41 P2c0d0 P2i P2c0e0 P2j Pc0d20 Pd0i Pc0d0e0 Pd0j c0d30 d20iPh31u c0d20e0 d20j Pu′ c0Pd0he120u′ d20kPv′ c0e30 d20l d0u′ Phc10xe20′g d20me0u′ d0e0me0v′ h21U e20m h21d0x′
14 7 P2c0d0 P2c0e0 Pd0c0d0 i2 Pe0c0d0 ij d20c0d0 ik d0e0c0d0 il e20c0d0 im Pv′ e20c0e0 jmd0u′ e0gc0e0U km e0u′ d0x′ lm e0v′ e0x′m2
P3h1 P3h2 P2h21e0 Pc0c0d0 Pc0c0e0 c0d0c0d0 Q′ Pu c0e0c0d0 Pv c0e0c0e0 Ph1v d0u e20h21e0 h1d0u d0ve0gh21e0 dx0′hh141v e0v h0g3 c0x′ e0h1vτhe20gw3h1c0x′ h2gBτ2h10hg412R2 τgw B8dh03g3τh22Gg131 τWB11c0d0 B8e0
6
12 P2h21c0 P2d0 P2e0 Pd20 Pd0e0 d30 d20e0 τh0d20e0 d0e20 τh0d0e20 e30 τh0e30 e20g d0τh0g2 τe0g2 e0τh0g2 τg3 Ph5j nm τB8d0 h3g3
P2h41 P2c0 Pd0h41 Pc0d0 Ph1c0d0 Pc0e0 Pj c0d20 d0i c0d0e0 d0j c0e20 d0k h31u c0e0g d0l u′ e0l v′ e0m gm τh0gm c1g2 τh2gm nr Ph5c0d0 τh2c1g2 h5Pc0e0 h21B22 C1h12B23 h2B5
10 5 Pc0e0 d0c0d0 e0c0d0 z e0c0e0 d0r c0e0g e0r u′ gr v′ x′ R1 gt Q1 h21B8 Bh215Ph21e0 B22 R q1 τB23c0Q2 τB5D2′ P(A+A′)
P2h1 P2h2 Ph21e0 c0c0d0 d0h21e0 e0h21e0 u h21e0g v h2g2 h1v τw h3g2τh22g2 h31Bh15Ph21c0G3 gn B8 τh2gn Ph5e0 D11 h3G3 B4 X1 c0Q2 XC3′′ h1X3
8 4 Ph21c0 Pd0 Pe0 d20 d0e0 e20 e0g τg2 τh0g2 τ2h1g2 τ2h2g2 N Ph5c0 h3g2 d1Gg3 h5c0d0 h1d1g h5i τh2d1g h5j e1g d1h3gh21Q2 E1C0 h5d0ed021 h3Q2 c0D4 e1h3g G21 h3B7 h2G0D3′ d1eh123B6
Ph41 Pc0 d0h41 c0d0 i c0e0 j k l m c1g τh2c1g B1 B2 i1 h5c0d0 Bh61i1 h5c0e0 Q2 j1 B3 B7 CX′ 2 h3B6 h3D2 k1 τG0 h3j1 h3(A+A′)
3
6 c0d0 c0e0 r q t y Ph1h5 Ph2h5 h23h3g C h5h21e0 τG B6 D4 h23g2 D2 j1 A′A+A′ h5n A′′ h3D4 r1 h0Q3 h1r1 h3H1 h2Q3
Ph1 Ph2 h21e0 h2g h4h21c0 h3g n h3h3g x h5h21c0 h5d0 h21g2 h5e0 h5f0 h3g2 D1 D4 h23g2 H1 n1 τQ3
2
4 h21c0 d0 e0 f0 τg h4c0 h3g d1 p h5h41 e1 h5c0 f1 g2 h3c2 h5c1 D3 d2 h6h41 p′ p1
h41 c0 c1 c2 h23h5
1
2
h23 h24 h3h5 h25 h3h6
h0 h1 h2 h3 h4 h5 h6
0
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70
10
18
36
The E -page of the motivic Adams spectral sequence for the cofiber of τ
3
P8h41
17
34
P8h1 P8h2
16
32
P7h21c0
P7c0
P7h41
15
30
P7h1 P7h2
14
28 P6d0
P6h21c0
P6c0
P6h41
13
26
P6h1 P6h2 P4h20i
12
24
P5h21c0
P5c0
P5h41
11
22 P2i2
P5h1 P5h2
10
20
P4h21c0 P4d0 τP2h0d20e0
P4c0
P4h41
9
18
d20u
P4h1 P4h2 P2h20i h40Q′
8
16 P3h21c0 d20e20 e40
P3c0 c0d0e20
P3h41 d0u′
14 7 kmh21c0x′+U d0x′ lm h1h3g3 m2
P3h1 P3h2 d0u h2e20g x′h41 h0g3 c0xτ′h20g3 τgvh2g3 τh0h2g3 τgw B8d0 τh22g3 τWh11h3g3
6
12 P2h21c0 P2d0 d0e20 τh0e30 τe0g2 τg3 Ph5j nm τB8d0
P2h41 P2c0 c0e20 u′ c1g2 nrPh5c0d0 τh2c1g2 h2B22 Ph5c0e0 C11h2B23h30G21 h2B5
10 5 d0r e0r h1h3g2 gr h1G3 x′ gt Q1 h21B8 BP21h21h5e0 R h1X1 q1 c0Q2 τB5D2′ h21D4c0 P(A+A′)
P2h1 P2h2 h20i u h0g2 τh20g2 h2g2 τh0h2g2 τw τh22g2 h1h3gh231BP1h21h5c0 gn B8 h1G3 τh2gn Ph5e0 D11 h3G3 h21X2 c0Q2 XC3′′ h3E1 h1X3
8 4 Ph21c0 d20 e20 τg2 τ2h2g2 N Ph5c0 d1g h1d1g h5i h5c0e0τh2d1g h5j e1g h21D4 h3d1hg21Q2 h31D4 E1C0 hh705hd60e0 d21 h3Q2h0h3D2 h3e1g d1eh123B6
Ph41 Pc0 c0d0 h0y c1g τh2c1g B1 B2 i1 h5c0d0 B6? h5c0e0 Q2 h0D2 j1 B3 C′ h3B6 k1 h3j1 h3(A+A′)
3
6 h1h3g r q t Ph1h5 Ph2h5 h3h23g h1h5e0 C h21h5e0 B6 h23g2 j1 A′ h5n h2H1 r1 h0Q3 h1r1 h2Q3
h23g x
Ph1 Ph2 h0g h2g h21h4c0 h1h3g n h21h5c0 h5d0 h21g2 h3g2 h23g2 τQ3
2
4 h21c0 d0 τg h4c0 h30h5 d1 p h41h5 e1 h5c0 f1 h0c2 g2 h5c1 D3 d2 h41h6 p′ p1
h41 c0 c1 h23h5 h0h2h6 h22h6
1
2
h23 h0h4 h1h4 h2h4 h24 h1h5 h2h5 h3h5 h25 h1h6 h3h6
h0 h1 h2 h3
0
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70