Errata for “The homotopy groups π_{∗}(L_{2}S^{0}) at the prime 3”

Katsumi Shimomura and Xiangjun Wang December 15, 2002

p.1184 l. 14 B = {… n ≥ 0 … ↦ B = {… n > 0 …
l. 15 4 × 3^{n−2i−2} ≤ m} ↦ 4 × 3^{n−2i−2} ≤ m for n > 1}
l. 17 i ≠ k + 1 ↦ i < k + 1
l. -7 Add the condition “m ≠ 2 × 3^{n−1} if i = 1”
l. -5 2 × 3^{n−i−1} < 3^{i}m − 8 × 3^{n} ≤ 2 × 3^{n−i} ↦ 2 × 3^{n−i} < 3^{i}m − 8 × 3^{n} ≤ 2 × 3^{n−i+1}
l. -5 Add the condition “m ≠ 10 × 3^{n−1} if i = 1”
l. -3 Add condition “n > 0 if s = 3t + 1”
l. -3 s = 9t − 1 ↦ s = 3^{n+2}t − 1
l. -2 CI_{1} = {… 2 × 3^{n−i−1} < …} ↦ CI_{1} = {… n > 0, 4 × 3^{n−i−1} < …}
l. -1 8 × 3^{n−i} ↦ 8 × 3^{n−i} + 2 × 3^{n−2i},
l. -1 k + 1 > i if 3^{k}|t ↦ 8 × 3^{n−i} ≤ m if k + 1 ≤ i, and 4 × 3^{n−2i−1} ≤ m if k + 1 > i
for k such that 3^{k}|t and 3^{k+1} |̸t
p.1185 l. 1  4, 6, 7, ↦ 4, 6, 7, (9, 10 if n = 0),
l. 1  The condition “k + 1 > n if 3^{k}|t” should be replaced by “3^{k+1} |̸ (3^{n}m + 4 × 3^{n}) if 3^{k}|t”
l. -3 16a − 7 ↦ 16a − 3
p.1187 l. -10 v_{2}^{8×3^{n}}h_{10}/v_{1}^{8×3^{n}+1} ↦ v_{2}^{8×3^{n}}h_{10}/v_{1}^{10×3^{n}+1}
l. -9 v_{2}^{8}h_{10}/v_{1}^{9} ↦ v_{2}^{8}h_{10}/v_{1}^{11}
l. -7 F′′_{0} ↦ F′′_{0} ⊗ Λ(ζ_{2})
p.1189 l. 12 X_{1} = X_{1,1} ⊕ X_{1,2} ↦ X_{1} = X_{1,1} ⊕ X_{1,2} ⊕ X_{1,3}
l. 13 X_{1,1} = Z/3{… n ≥ 0 …, ↦ X_{1,1} = Z/3{… n > 0 …, and j ≤ 2 if n = 1}
l. 14 such that j > 4 × 3^{n−i−1} − 1 if 3^{i}|j} ↦
X_{1,2} = Z/3{v_{2}^{3^{n}(3t−1)}/v_{1}^{j}|n > 0, 3 |̸t ∈ Z, 0 < j < 4 × 3^{n−1},
and j ≤ 2 if n = 1, such that j > 4 × 3^{n−i−1} − 1 if 3^{i}|j}
l. 15 X_{1,2} = Z/3{v_{2}^{3^{n}(3^{k+2}u−1)}/v_{1}^{3^{k+1}m}| … 3 |̸u ∈ Z, …, ↦

X_{1,3} = Z/3{v_{2}^{3^{n}(3^{i+2}u−1)}/v_{1}^{3^{k}m}|n ≥ 0, 3 |̸mu ∈ Z, i ≥ k or 4 × 3^{n−k−1} ≤ 3^{k}m < 4 × 3^{n−1}}

l. 18 3 |̸m ↦ 3 |̸tm
l. 19 X_{2,2} = Z/3{v_{2}^{3^{n}(3^{i+2}u−1)}/v_{1}^{3^{k}m}|n ≥ 0, u ∈ Z, 0 ≤ i < k, 1 ≤ m ≤ 4 × 3^{n−2k−1}, 3 |̸mu}
l. -2 H_{2} is defined to be “H_{2} = Z/3{x = v_{2}^{3^{n}(9t−1)}h_{10}/v_{1}^{3^{i}k+1} | x ∉ H^{∗}, 3 |̸s,
0 < 3^{i}k ≤ 10 × 3^{n} + 1, 0 ≤ i ≤ n, and 3|k, i = n or 3^{i}k > 8 × 3^{n} + 2 × 3^{n−i}}”
p.1190 l. 3  HI_{2} is defined to be “HI_{2} = Z/3{x = v_{2}^{3^{n}(9t−1)}h_{10}/v_{1}^{j} | x ∉ H^{∗} ⊕ H_{2}, n ≥ 0, 0 ≤ j ≤ 10 × 3^{n}}
l. 6  H_{1}^{∗} = {… |n ≥ 0, …} ↦ H_{1}^{∗} = {… |n > 0, …}
l. 7  H_{2,1}^{∗} = {… j ≤ 4 × 3^{n−i−1} …} ↦ H_{2,1}^{∗} = {… j < 4 × 3^{n−i−1} …}
l. 8  H_{2,2}^{∗} = {… ≤ 4 × 3^{n+1} ↦ H_{2,2}^{∗} = {… < 4 × 3^{n+1}
l.-5  i ≠ k + 1 ↦ i > k + 1
l.-3  i = k + 1 ↦ i ≤ k + 1
p.1191 l. 3  s = 9t − 1 ↦ s = 3^{n+2}t − 1
l. 4  H̃I_{1} = {… 2 × 3^{n−i−1} < …} ↦ H̃I_{1} = {… 4 × 3^{n−i−1} < …}
l. 5  H̃I_{2} = {… | … < 8 × 3^{n−i} …} ↦
H̃I_{2} = {… |v_{2}^{3^{n}(9t−1)}h_{10}/v_{1}^{3^{i}m+1} ∈ HI_{2}, … < 8 × 3^{n−i} + 2 × 3^{n−2i} …}
l. 6  4, 6, 7, ↦ 4, 6, 7, (9, 10 if n = 0),
l. 7  The condition “k + 1 > n if 3^{k}|t” should be replaced by “3^{k+1} |̸ (3^{n}m + 4 × 3^{n}) if 3^{k}|t”
l. -14 X̃_{1}ζ_{2}X̃_{2}^{∗} ↦ X̃_{1}ζ_{2}, X̃_{2}^{∗}
l. -13 HIζ_{2}, X_{2}^{∗}ζ_{2} ↦ HIζ_{2}, H^{∗}ζ_{2}, X_{2}^{∗}ζ_{2}
p.1194 l. 7  Add the term 3t_{2} to the right hand side of the equation η_{R}(v_{2}) = . . ..
p.1196 l. 1  s ∈ I ↦ s with 3|s − 1 or 9|s + 1
l. 2  v_{2}^{3^{n}s}h_{10}ζ_{2}/v_{1} + v_{2}^{3^{n}s−2}b_{11}/v_{1} ↦ v_{2}^{3^{n}s}h_{10}ζ_{2}/v_{1}(+v_{2}^{s−2}b_{11}/v_{1} if n = 0)
l. 8  Add the condition i > 0 to 2.
l. 10 8 × 3^{n} ↦ 8 × 3^{n} + 2 × 3^{n−i}
l. 12 8 × 3^{n} ↦ 8 × 3^{n} + 2

The most impotant misprint is the statement of Prop. 3.1. in p.1191, l.-14 – l.-13. The statements of Prop.s 5.3 and 5.4 have some misprints (see above ‘p.1196 l.2– l.12’). The other misprints appear in the definitions of the modules, which can be recovered from the text.