The Homotopy Groups of K - Indiana Universitymmandell/talks/NotreDame-141030.pdfIntroduction...
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The Homotopy Groups of K (S)
Michael A. Mandell
Indiana University
Notre Dame Topology Seminar
October 30, 2014
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 1 / 15
![Page 2: The Homotopy Groups of K - Indiana Universitymmandell/talks/NotreDame-141030.pdfIntroduction Waldhausen’s Algebraic K-Theory of Spaces Algebraic K-theory of spaces ties algebraic](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20c1b1778ae456533b8ce9/html5/thumbnails/2.jpg)
Overview
Rognes calculated the homotopy groups of K (S) at regular primes.What happens at irregular primes?
Joint work with Andrew BlumbergPreprint arXiv:1408.0133
Outline
1 Introduction and main result2 Topological cyclic homology3 K -theory and étale cohomology4 Main theorem (reprise)
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 2 / 15
![Page 3: The Homotopy Groups of K - Indiana Universitymmandell/talks/NotreDame-141030.pdfIntroduction Waldhausen’s Algebraic K-Theory of Spaces Algebraic K-theory of spaces ties algebraic](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20c1b1778ae456533b8ce9/html5/thumbnails/3.jpg)
Overview
Rognes calculated the homotopy groups of K (S) at regular primes.What happens at irregular primes?
Joint work with Andrew BlumbergPreprint arXiv:1408.0133
Outline
1 Introduction and main result2 Topological cyclic homology3 K -theory and étale cohomology4 Main theorem (reprise)
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 2 / 15
![Page 4: The Homotopy Groups of K - Indiana Universitymmandell/talks/NotreDame-141030.pdfIntroduction Waldhausen’s Algebraic K-Theory of Spaces Algebraic K-theory of spaces ties algebraic](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20c1b1778ae456533b8ce9/html5/thumbnails/4.jpg)
Overview
Rognes calculated the homotopy groups of K (S) at regular primes.What happens at irregular primes?
Joint work with Andrew BlumbergPreprint arXiv:1408.0133
Outline1 Introduction and main result
2 Topological cyclic homology3 K -theory and étale cohomology4 Main theorem (reprise)
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 2 / 15
![Page 5: The Homotopy Groups of K - Indiana Universitymmandell/talks/NotreDame-141030.pdfIntroduction Waldhausen’s Algebraic K-Theory of Spaces Algebraic K-theory of spaces ties algebraic](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20c1b1778ae456533b8ce9/html5/thumbnails/5.jpg)
Overview
Rognes calculated the homotopy groups of K (S) at regular primes.What happens at irregular primes?
Joint work with Andrew BlumbergPreprint arXiv:1408.0133
Outline1 Introduction and main result2 Topological cyclic homology
3 K -theory and étale cohomology4 Main theorem (reprise)
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 2 / 15
![Page 6: The Homotopy Groups of K - Indiana Universitymmandell/talks/NotreDame-141030.pdfIntroduction Waldhausen’s Algebraic K-Theory of Spaces Algebraic K-theory of spaces ties algebraic](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20c1b1778ae456533b8ce9/html5/thumbnails/6.jpg)
Overview
Rognes calculated the homotopy groups of K (S) at regular primes.What happens at irregular primes?
Joint work with Andrew BlumbergPreprint arXiv:1408.0133
Outline1 Introduction and main result2 Topological cyclic homology3 K -theory and étale cohomology
4 Main theorem (reprise)
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 2 / 15
![Page 7: The Homotopy Groups of K - Indiana Universitymmandell/talks/NotreDame-141030.pdfIntroduction Waldhausen’s Algebraic K-Theory of Spaces Algebraic K-theory of spaces ties algebraic](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20c1b1778ae456533b8ce9/html5/thumbnails/7.jpg)
Overview
Rognes calculated the homotopy groups of K (S) at regular primes.What happens at irregular primes?
Joint work with Andrew BlumbergPreprint arXiv:1408.0133
Outline1 Introduction and main result2 Topological cyclic homology3 K -theory and étale cohomology4 Main theorem (reprise)
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 2 / 15
![Page 8: The Homotopy Groups of K - Indiana Universitymmandell/talks/NotreDame-141030.pdfIntroduction Waldhausen’s Algebraic K-Theory of Spaces Algebraic K-theory of spaces ties algebraic](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20c1b1778ae456533b8ce9/html5/thumbnails/8.jpg)
Introduction
Waldhausen’s Algebraic K -Theory of Spaces
Algebraic K -theory of spaces ties algebraic K -theory to differentialand PL topology:
A(X ) ' K (S[X ])
Smooth Whitehead space: Ω∞A(X ) ' Q+(X )×WhDiff(X )
Smooth stable concordance space: ΩWhDiff(X ) ' C Diff(X )
C Diff(X ) = colim(C(X )→ C(X × I)→ C(X × I2)→ · · · )
PL Whitehead space and PL stable concordance space
Ω∞(K (S) ∧ X+)→ Ω∞(A(X ))→WhPL(X )
Ω2Ω∞(K (S) ∧ X+)→ C Diff(X )→ C PL(X )
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 3 / 15
![Page 9: The Homotopy Groups of K - Indiana Universitymmandell/talks/NotreDame-141030.pdfIntroduction Waldhausen’s Algebraic K-Theory of Spaces Algebraic K-theory of spaces ties algebraic](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20c1b1778ae456533b8ce9/html5/thumbnails/9.jpg)
Introduction
Waldhausen’s Algebraic K -Theory of Spaces
Algebraic K -theory of spaces ties algebraic K -theory to differentialand PL topology:
A(X ) ' K (S[X ])
Smooth Whitehead space: Ω∞A(X ) ' Q+(X )×WhDiff(X )
Smooth stable concordance space: ΩWhDiff(X ) ' C Diff(X )
C Diff(X ) = colim(C(X )→ C(X × I)→ C(X × I2)→ · · · )
PL Whitehead space and PL stable concordance space
Ω∞(K (S) ∧ X+)→ Ω∞(A(X ))→WhPL(X )
Ω2Ω∞(K (S) ∧ X+)→ C Diff(X )→ C PL(X )
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 3 / 15
![Page 10: The Homotopy Groups of K - Indiana Universitymmandell/talks/NotreDame-141030.pdfIntroduction Waldhausen’s Algebraic K-Theory of Spaces Algebraic K-theory of spaces ties algebraic](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20c1b1778ae456533b8ce9/html5/thumbnails/10.jpg)
Introduction
Waldhausen’s Algebraic K -Theory of Spaces
Algebraic K -theory of spaces ties algebraic K -theory to differentialand PL topology:
A(X ) ' K (S[X ])
Smooth Whitehead space: Ω∞A(X ) ' Q+(X )×WhDiff(X )
Smooth stable concordance space: ΩWhDiff(X ) ' C Diff(X )
C Diff(X ) = colim(C(X )→ C(X × I)→ C(X × I2)→ · · · )
PL Whitehead space and PL stable concordance space
Ω∞(K (S) ∧ X+)→ Ω∞(A(X ))→WhPL(X )
Ω2Ω∞(K (S) ∧ X+)→ C Diff(X )→ C PL(X )
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 3 / 15
![Page 11: The Homotopy Groups of K - Indiana Universitymmandell/talks/NotreDame-141030.pdfIntroduction Waldhausen’s Algebraic K-Theory of Spaces Algebraic K-theory of spaces ties algebraic](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20c1b1778ae456533b8ce9/html5/thumbnails/11.jpg)
Introduction
Waldhausen’s Algebraic K -Theory of Spaces
Algebraic K -theory of spaces ties algebraic K -theory to differentialand PL topology:
A(X ) ' K (S[X ])
Smooth Whitehead space: Ω∞A(X ) ' Q+(X )×WhDiff(X )
Smooth stable concordance space: ΩWhDiff(X ) ' C Diff(X )
C Diff(X ) = colim(C(X )→ C(X × I)→ C(X × I2)→ · · · )
PL Whitehead space and PL stable concordance space
Ω∞(K (S) ∧ X+)→ Ω∞(A(X ))→WhPL(X )
Ω2Ω∞(K (S) ∧ X+)→ C Diff(X )→ C PL(X )
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 3 / 15
![Page 12: The Homotopy Groups of K - Indiana Universitymmandell/talks/NotreDame-141030.pdfIntroduction Waldhausen’s Algebraic K-Theory of Spaces Algebraic K-theory of spaces ties algebraic](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20c1b1778ae456533b8ce9/html5/thumbnails/12.jpg)
Introduction
Waldhausen’s Algebraic K -Theory of Spaces
Algebraic K -theory of spaces ties algebraic K -theory to differentialand PL topology:
A(X ) ' K (S[X ])
Smooth Whitehead space: Ω∞A(X ) ' Q+(X )×WhDiff(X )
Smooth stable concordance space: ΩWhDiff(X ) ' C Diff(X )
C Diff(X ) = colim(C(X )→ C(X × I)→ C(X × I2)→ · · · )
PL Whitehead space and PL stable concordance space
Ω∞(K (S) ∧ X+)→ Ω∞(A(X ))→WhPL(X )
Ω2Ω∞(K (S) ∧ X+)→ C Diff(X )→ C PL(X )
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 3 / 15
![Page 13: The Homotopy Groups of K - Indiana Universitymmandell/talks/NotreDame-141030.pdfIntroduction Waldhausen’s Algebraic K-Theory of Spaces Algebraic K-theory of spaces ties algebraic](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20c1b1778ae456533b8ce9/html5/thumbnails/13.jpg)
Introduction
Waldhausen’s Algebraic K -Theory of Spaces
Algebraic K -theory of spaces ties algebraic K -theory to differentialand PL topology:
A(X ) ' K (S[X ])
Smooth Whitehead space: Ω∞A(X ) ' Q+(X )×WhDiff(X )
Smooth stable concordance space: ΩWhDiff(X ) ' C Diff(X )
C Diff(X ) = colim(C(X )→ C(X × I)→ C(X × I2)→ · · · )
PL Whitehead space and PL stable concordance space
Ω∞(K (S) ∧ X+)→ Ω∞(A(X ))→WhPL(X )
Ω2Ω∞(K (S) ∧ X+)→ C Diff(X )→ C PL(X )
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 3 / 15
![Page 14: The Homotopy Groups of K - Indiana Universitymmandell/talks/NotreDame-141030.pdfIntroduction Waldhausen’s Algebraic K-Theory of Spaces Algebraic K-theory of spaces ties algebraic](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20c1b1778ae456533b8ce9/html5/thumbnails/14.jpg)
Introduction
Waldhausen’s Algebraic K -Theory of Spaces
Algebraic K -theory of spaces ties algebraic K -theory to differentialand PL topology:
A(X ) ' K (S[X ])
Smooth Whitehead space: Ω∞A(X ) ' Q+(X )×WhDiff(X )
Smooth stable concordance space: ΩWhDiff(X ) ' C Diff(X )
C Diff(X ) = colim(C(X )→ C(X × I)→ C(X × I2)→ · · · )
PL Whitehead space and PL stable concordance space
Ω∞(K (S) ∧ X+)→ Ω∞(A(X ))→WhPL(X )
Ω2Ω∞(K (S) ∧ X+)→ C Diff(X )→ C PL(X )
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 3 / 15
![Page 15: The Homotopy Groups of K - Indiana Universitymmandell/talks/NotreDame-141030.pdfIntroduction Waldhausen’s Algebraic K-Theory of Spaces Algebraic K-theory of spaces ties algebraic](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20c1b1778ae456533b8ce9/html5/thumbnails/15.jpg)
Introduction
Waldhausen’s Algebraic K -Theory of Spaces
Algebraic K -theory of spaces ties algebraic K -theory to differentialand PL topology:
A(X ) ' K (S[X ])
Smooth Whitehead space: Ω∞A(X ) ' Q+(X )×WhDiff(X )
Smooth stable concordance space: ΩWhDiff(X ) ' C Diff(X )
C Diff(X ) = colim(C(X )→ C(X × I)→ C(X × I2)→ · · · )
PL Whitehead space and PL stable concordance space
Ω∞(K (S) ∧ X+)→ Ω∞(A(X ))→WhPL(X )
Ω2Ω∞(K (S) ∧ X+)→ C Diff(X )→ C PL(X )
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 3 / 15
![Page 16: The Homotopy Groups of K - Indiana Universitymmandell/talks/NotreDame-141030.pdfIntroduction Waldhausen’s Algebraic K-Theory of Spaces Algebraic K-theory of spaces ties algebraic](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20c1b1778ae456533b8ce9/html5/thumbnails/16.jpg)
Introduction
Waldhausen’s Algebraic K -Theory of Spaces
Algebraic K -theory of spaces ties algebraic K -theory to differentialand PL topology:
A(X ) ' K (S[X ])
Smooth Whitehead space: Ω∞A(X ) ' Q+(X )×WhDiff(X )
Smooth stable concordance space: ΩWhDiff(X ) ' C Diff(X )
C Diff(X ) = colim(C(X )→ C(X × I)→ C(X × I2)→ · · · )
PL Whitehead space and PL stable concordance space
Ω∞(K (S) ∧ X+)→ Ω∞(A(X ))→WhPL(X )
Ω2Ω∞(K (S) ∧ X+)→ C Diff(X )→ C PL(X )
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 3 / 15
![Page 17: The Homotopy Groups of K - Indiana Universitymmandell/talks/NotreDame-141030.pdfIntroduction Waldhausen’s Algebraic K-Theory of Spaces Algebraic K-theory of spaces ties algebraic](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20c1b1778ae456533b8ce9/html5/thumbnails/17.jpg)
Introduction
Linearization Map
K (−) is functorial in
map
s
of ring spectra
Linearization map: S // Z
K K (S) K (Z)
TC TC(S) TC(Z)
Topological cyclic homology TC
cyclotomictrace
Theorem (Waldhausen)
The linearization map K (S)→ K (Z) is a rational equivalence.
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 4 / 15
![Page 18: The Homotopy Groups of K - Indiana Universitymmandell/talks/NotreDame-141030.pdfIntroduction Waldhausen’s Algebraic K-Theory of Spaces Algebraic K-theory of spaces ties algebraic](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20c1b1778ae456533b8ce9/html5/thumbnails/18.jpg)
Introduction
Linearization Map
K (−) is functorial in maps of ring spectra
Linearization map: S // Z
K
K (S) // K (Z)
TC TC(S) TC(Z)
Topological cyclic homology TC
cyclotomictrace
Theorem (Waldhausen)
The linearization map K (S)→ K (Z) is a rational equivalence.
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 4 / 15
![Page 19: The Homotopy Groups of K - Indiana Universitymmandell/talks/NotreDame-141030.pdfIntroduction Waldhausen’s Algebraic K-Theory of Spaces Algebraic K-theory of spaces ties algebraic](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20c1b1778ae456533b8ce9/html5/thumbnails/19.jpg)
Introduction
Linearization Map
K (−) is functorial in maps of ring spectra
Linearization map: S // Z
K
K (S) // K (Z)
TC TC(S) TC(Z)
Topological cyclic homology TC
cyclotomictrace
Theorem (Waldhausen)
The linearization map K (S)→ K (Z) is a rational equivalence.
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 4 / 15
![Page 20: The Homotopy Groups of K - Indiana Universitymmandell/talks/NotreDame-141030.pdfIntroduction Waldhausen’s Algebraic K-Theory of Spaces Algebraic K-theory of spaces ties algebraic](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20c1b1778ae456533b8ce9/html5/thumbnails/20.jpg)
Introduction
Linearization Map
K (−) is functorial in maps of ring spectra
Linearization map: S // Z
K
K (S) // K (Z)
TC TC(S) TC(Z)
Topological cyclic homology TC
cyclotomictrace
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 4 / 15
![Page 21: The Homotopy Groups of K - Indiana Universitymmandell/talks/NotreDame-141030.pdfIntroduction Waldhausen’s Algebraic K-Theory of Spaces Algebraic K-theory of spaces ties algebraic](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20c1b1778ae456533b8ce9/html5/thumbnails/21.jpg)
Introduction
Linearization / Cyclotomic Trace Square
K (−) is functorial in maps of ring spectra
Linearization map: S // Z
K
K (S) // K (Z)
TC
TC(S) // TC(Z)
Topological cyclic homology TC
cyclotomictrace
Theorem (Dundas)The linearization/cyclotomic trace square becomes homotopycartesian after p-completion.
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 4 / 15
![Page 22: The Homotopy Groups of K - Indiana Universitymmandell/talks/NotreDame-141030.pdfIntroduction Waldhausen’s Algebraic K-Theory of Spaces Algebraic K-theory of spaces ties algebraic](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20c1b1778ae456533b8ce9/html5/thumbnails/22.jpg)
Introduction
Linearization / Cyclotomic Trace Square
K (−) is functorial in maps of ring spectra
Linearization map: S // Z
K
K (S) //
K (Z)
TC TC(S) // TC(Z)
Topological cyclic homology TC
cyclotomictrace
Theorem (Dundas)The linearization/cyclotomic trace square becomes homotopycartesian after p-completion.
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 4 / 15
![Page 23: The Homotopy Groups of K - Indiana Universitymmandell/talks/NotreDame-141030.pdfIntroduction Waldhausen’s Algebraic K-Theory of Spaces Algebraic K-theory of spaces ties algebraic](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20c1b1778ae456533b8ce9/html5/thumbnails/23.jpg)
Introduction
Linearization / Cyclotomic Trace Square
K (−) is functorial in maps of ring spectra
Linearization map: S // Z
K
K (S) //
K (Z)
TC TC(S) // TC(Z)
Topological cyclic homology TC
cyclotomictrace
Theorem (Dundas)The linearization/cyclotomic trace square becomes homotopycartesian after p-completion.
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 4 / 15
![Page 24: The Homotopy Groups of K - Indiana Universitymmandell/talks/NotreDame-141030.pdfIntroduction Waldhausen’s Algebraic K-Theory of Spaces Algebraic K-theory of spaces ties algebraic](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20c1b1778ae456533b8ce9/html5/thumbnails/24.jpg)
Introduction
Main Theorem
Theorem (Dundas)The linearization/cyclotomic trace squarebecomes homotopy cartesian after p-completion.
K (S) //
K (Z)
TC(S) // TC(Z)
Consequence: Long exact sequence
· · · → πnK (S)∧p → πnK (Z)∧p⊕πn(TC(S)∧p )→ πn(TC(Z)∧p )→ πn−1K (S)∧p → · · ·
Theorem (Main Theorem)
The sequence πnK (S)∧p → πnK (Z)∧p ⊕ πn(TC(S)∧p )→ πn(TC(Z)∧p ) issplit short exact. (p > 2)
Corollary: p-torsion is split short exact.
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 5 / 15
![Page 25: The Homotopy Groups of K - Indiana Universitymmandell/talks/NotreDame-141030.pdfIntroduction Waldhausen’s Algebraic K-Theory of Spaces Algebraic K-theory of spaces ties algebraic](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20c1b1778ae456533b8ce9/html5/thumbnails/25.jpg)
Introduction
Main Theorem
Theorem (Dundas)The linearization/cyclotomic trace squarebecomes homotopy cartesian after p-completion.
K (S) //
K (Z)
TC(S) // TC(Z)
Consequence: Long exact sequence
· · · → πnK (S)∧p → πnK (Z)∧p⊕πn(TC(S)∧p )→ πn(TC(Z)∧p )→ πn−1K (S)∧p → · · ·
Theorem (Main Theorem)
The sequence πnK (S)∧p → πnK (Z)∧p ⊕ πn(TC(S)∧p )→ πn(TC(Z)∧p ) issplit short exact. (p > 2)
Corollary: p-torsion is split short exact.
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 5 / 15
![Page 26: The Homotopy Groups of K - Indiana Universitymmandell/talks/NotreDame-141030.pdfIntroduction Waldhausen’s Algebraic K-Theory of Spaces Algebraic K-theory of spaces ties algebraic](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20c1b1778ae456533b8ce9/html5/thumbnails/26.jpg)
Introduction
Main Theorem
Theorem (Dundas)The linearization/cyclotomic trace squarebecomes homotopy cartesian after p-completion.
K (S) //
K (Z)
TC(S) // TC(Z)
Consequence: Long exact sequence
· · · → πnK (S)∧p → πnK (Z)∧p⊕πn(TC(S)∧p )→ πn(TC(Z)∧p )→ πn−1K (S)∧p → · · ·
Theorem (Main Theorem)
The sequence πnK (S)∧p → πnK (Z)∧p ⊕ πn(TC(S)∧p )→ πn(TC(Z)∧p ) issplit short exact. (p > 2)
Corollary: p-torsion is split short exact.
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 5 / 15
![Page 27: The Homotopy Groups of K - Indiana Universitymmandell/talks/NotreDame-141030.pdfIntroduction Waldhausen’s Algebraic K-Theory of Spaces Algebraic K-theory of spaces ties algebraic](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20c1b1778ae456533b8ce9/html5/thumbnails/27.jpg)
Introduction
Main Theorem
Theorem (Dundas)The linearization/cyclotomic trace squarebecomes homotopy cartesian after p-completion.
K (S) //
K (Z)
TC(S) // TC(Z)
Consequence: Long exact sequence
· · · → πnK (S)∧p → πnK (Z)∧p⊕πn(TC(S)∧p )→ πn(TC(Z)∧p )→ πn−1K (S)∧p → · · ·
Theorem (Main Theorem)
The sequence πnK (S)∧p → πnK (Z)∧p ⊕ πn(TC(S)∧p )→ πn(TC(Z)∧p ) issplit short exact. (p > 2)
Corollary: p-torsion is split short exact.
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 5 / 15
![Page 28: The Homotopy Groups of K - Indiana Universitymmandell/talks/NotreDame-141030.pdfIntroduction Waldhausen’s Algebraic K-Theory of Spaces Algebraic K-theory of spaces ties algebraic](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20c1b1778ae456533b8ce9/html5/thumbnails/28.jpg)
Table: πnK (S) in low degrees
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 6 / 15
![Page 29: The Homotopy Groups of K - Indiana Universitymmandell/talks/NotreDame-141030.pdfIntroduction Waldhausen’s Algebraic K-Theory of Spaces Algebraic K-theory of spaces ties algebraic](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20c1b1778ae456533b8ce9/html5/thumbnails/29.jpg)
Topological Cyclic Homology
Topological Cyclic Homology
TC(R) is built from the fixed points of THH(R) andextra “cyclotomic” operators.
Theorem (Bökstedt-Hsiang-Madsen)
TC(S)∧p ' (S ∨ ΣCP∞−1)∧p
' S∧p ∨ ΣS∧p ∨ ΣCP∞−1
ΣCP∞−1 → ΣΣ∞+ CP∞ TrT−−→ S
Σ∞+ CP∞ ' S ∨ Σ∞CP =⇒ (CP∞−1)∧p ' S∧p ∨ CP∞−1
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 7 / 15
![Page 30: The Homotopy Groups of K - Indiana Universitymmandell/talks/NotreDame-141030.pdfIntroduction Waldhausen’s Algebraic K-Theory of Spaces Algebraic K-theory of spaces ties algebraic](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20c1b1778ae456533b8ce9/html5/thumbnails/30.jpg)
Topological Cyclic Homology
Topological Cyclic Homology
TC(R) is built from the fixed points of THH(R) andextra “cyclotomic” operators.
Theorem (Bökstedt-Hsiang-Madsen)
TC(S)∧p ' (S ∨ ΣCP∞−1)∧p
' S∧p ∨ ΣS∧p ∨ ΣCP∞−1
ΣCP∞−1 → ΣΣ∞+ CP∞ TrT−−→ S
Σ∞+ CP∞ ' S ∨ Σ∞CP =⇒ (CP∞−1)∧p ' S∧p ∨ CP∞−1
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 7 / 15
![Page 31: The Homotopy Groups of K - Indiana Universitymmandell/talks/NotreDame-141030.pdfIntroduction Waldhausen’s Algebraic K-Theory of Spaces Algebraic K-theory of spaces ties algebraic](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20c1b1778ae456533b8ce9/html5/thumbnails/31.jpg)
Topological Cyclic Homology
Topological Cyclic Homology
TC(R) is built from the fixed points of THH(R) andextra “cyclotomic” operators.
Theorem (Bökstedt-Hsiang-Madsen)
TC(S)∧p ' (S ∨ ΣCP∞−1)∧p
' S∧p ∨ ΣS∧p ∨ ΣCP∞−1
ΣCP∞−1 → ΣΣ∞+ CP∞ TrT−−→ S
Σ∞+ CP∞ ' S ∨ Σ∞CP =⇒ (CP∞−1)∧p ' S∧p ∨ CP∞−1
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 7 / 15
![Page 32: The Homotopy Groups of K - Indiana Universitymmandell/talks/NotreDame-141030.pdfIntroduction Waldhausen’s Algebraic K-Theory of Spaces Algebraic K-theory of spaces ties algebraic](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20c1b1778ae456533b8ce9/html5/thumbnails/32.jpg)
Topological Cyclic Homology
Topological Cyclic Homology
TC(R) is built from the fixed points of THH(R) andextra “cyclotomic” operators.
Theorem (Bökstedt-Hsiang-Madsen)
TC(S)∧p ' (S ∨ ΣCP∞−1)∧p
' S∧p ∨ ΣS∧p ∨ ΣCP∞−1
ΣCP∞−1 → ΣΣ∞+ CP∞ TrT−−→ S
Σ∞+ CP∞ ' S ∨ Σ∞CP =⇒ (CP∞−1)∧p ' S∧p ∨ CP∞−1
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 7 / 15
![Page 33: The Homotopy Groups of K - Indiana Universitymmandell/talks/NotreDame-141030.pdfIntroduction Waldhausen’s Algebraic K-Theory of Spaces Algebraic K-theory of spaces ties algebraic](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20c1b1778ae456533b8ce9/html5/thumbnails/33.jpg)
Topological Cyclic Homology
Topological Cyclic Homology
TC(R) is built from the fixed points of THH(R) andextra “cyclotomic” operators.
Theorem (Bökstedt-Hsiang-Madsen)
TC(S)∧p ' (S ∨ ΣCP∞−1)∧p
' S∧p ∨ ΣS∧p ∨ ΣCP∞−1
ΣCP∞−1 → ΣΣ∞+ CP∞ TrT−−→ S
Σ∞+ CP∞ ' S ∨ Σ∞CP =⇒ (CP∞−1)∧p ' S∧p ∨ CP∞−1
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 7 / 15
![Page 34: The Homotopy Groups of K - Indiana Universitymmandell/talks/NotreDame-141030.pdfIntroduction Waldhausen’s Algebraic K-Theory of Spaces Algebraic K-theory of spaces ties algebraic](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20c1b1778ae456533b8ce9/html5/thumbnails/34.jpg)
Topological Cyclic Homology
Topological Cyclic Homology
TC(R) is built from the fixed points of THH(R) andextra “cyclotomic” operators.
Theorem (Bökstedt-Hsiang-Madsen)
TC(S)∧p ' (S ∨ ΣCP∞−1)∧p ' S∧p ∨ ΣS∧p ∨ ΣCP∞−1
ΣCP∞−1 → ΣΣ∞+ CP∞ TrT−−→ S
Σ∞+ CP∞ ' S ∨ Σ∞CP =⇒ (CP∞−1)∧p ' S∧p ∨ CP∞−1
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 7 / 15
![Page 35: The Homotopy Groups of K - Indiana Universitymmandell/talks/NotreDame-141030.pdfIntroduction Waldhausen’s Algebraic K-Theory of Spaces Algebraic K-theory of spaces ties algebraic](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20c1b1778ae456533b8ce9/html5/thumbnails/35.jpg)
Topological Cyclic Homology
TC(Z)
Theorem (Bökstedt-Madsen) (p > 2)
TC(Z)∧p [0,∞) ' j ∨ Σj ∨ Σbu∧p
ku = KU[0,∞)bu = KU[2,∞), bu ' Σ2ku
ku∧p ' ` ∨ Σ2` ∨ · · · ∨ Σ2p−4`
Σbu∧p ' Σ3` ∨ · · · ∨ Σ2(p−2)−1` ∨ Σ2(p−1)−1` ∨ Σ2p−1`
j → ku∧pψk−1−−−→ bu∧p
j ' LK (1)S[0,∞)
TC(Z) ' j ∨ Σj ∨ (∨
i=0,2,...,p−2
Σ2i−1`) ∨ Σ2p−1`
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 8 / 15
![Page 36: The Homotopy Groups of K - Indiana Universitymmandell/talks/NotreDame-141030.pdfIntroduction Waldhausen’s Algebraic K-Theory of Spaces Algebraic K-theory of spaces ties algebraic](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20c1b1778ae456533b8ce9/html5/thumbnails/36.jpg)
Topological Cyclic Homology
TC(Z)
Theorem (Bökstedt-Madsen) (p > 2)
TC(Z)∧p [0,∞) ' j ∨ Σj ∨ Σbu∧p
ku = KU[0,∞)bu = KU[2,∞), bu ' Σ2ku
ku∧p ' ` ∨ Σ2` ∨ · · · ∨ Σ2p−4`
Σbu∧p ' Σ3` ∨ · · · ∨ Σ2(p−2)−1` ∨ Σ2(p−1)−1` ∨ Σ2p−1`
j → ku∧pψk−1−−−→ bu∧p
j ' LK (1)S[0,∞)
TC(Z) ' j ∨ Σj ∨ (∨
i=0,2,...,p−2
Σ2i−1`) ∨ Σ2p−1`
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 8 / 15
![Page 37: The Homotopy Groups of K - Indiana Universitymmandell/talks/NotreDame-141030.pdfIntroduction Waldhausen’s Algebraic K-Theory of Spaces Algebraic K-theory of spaces ties algebraic](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20c1b1778ae456533b8ce9/html5/thumbnails/37.jpg)
Topological Cyclic Homology
TC(Z)
Theorem (Bökstedt-Madsen) (p > 2)
TC(Z)∧p [0,∞) ' j ∨ Σj ∨ Σbu∧p
ku = KU[0,∞)bu = KU[2,∞), bu ' Σ2ku
ku∧p ' ` ∨ Σ2` ∨ · · · ∨ Σ2p−4`
Σbu∧p ' Σ3` ∨ · · · ∨ Σ2(p−2)−1` ∨ Σ2(p−1)−1` ∨ Σ2p−1`
j → ku∧pψk−1−−−→ bu∧p
j ' LK (1)S[0,∞)
TC(Z) ' j ∨ Σj ∨ (∨
i=0,2,...,p−2
Σ2i−1`) ∨ Σ2p−1`
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 8 / 15
![Page 38: The Homotopy Groups of K - Indiana Universitymmandell/talks/NotreDame-141030.pdfIntroduction Waldhausen’s Algebraic K-Theory of Spaces Algebraic K-theory of spaces ties algebraic](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20c1b1778ae456533b8ce9/html5/thumbnails/38.jpg)
Topological Cyclic Homology
TC(Z)
Theorem (Bökstedt-Madsen) (p > 2)
TC(Z)∧p [0,∞) ' j ∨ Σj ∨ Σbu∧p
ku = KU[0,∞)bu = KU[2,∞), bu ' Σ2ku
ku∧p ' ` ∨ Σ2` ∨ · · · ∨ Σ2p−4`
Σbu∧p ' Σ3` ∨ · · · ∨ Σ2(p−2)−1` ∨ Σ2(p−1)−1` ∨ Σ2p−1`
j → ku∧pψk−1−−−→ bu∧p
j ' LK (1)S[0,∞)
TC(Z) ' j ∨ Σj ∨ (∨
i=0,2,...,p−2
Σ2i−1`) ∨ Σ2p−1`
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 8 / 15
![Page 39: The Homotopy Groups of K - Indiana Universitymmandell/talks/NotreDame-141030.pdfIntroduction Waldhausen’s Algebraic K-Theory of Spaces Algebraic K-theory of spaces ties algebraic](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20c1b1778ae456533b8ce9/html5/thumbnails/39.jpg)
Topological Cyclic Homology
TC(Z)
Theorem (Bökstedt-Madsen) (p > 2)
TC(Z)∧p [0,∞) ' j ∨ Σj ∨ Σbu∧p
ku = KU[0,∞)bu = KU[2,∞), bu ' Σ2ku
ku∧p ' ` ∨ Σ2` ∨ · · · ∨ Σ2p−4`
Σbu∧p ' Σ3` ∨ · · · ∨ Σ2(p−2)−1` ∨ Σ2(p−1)−1` ∨ Σ2p−1`
j → ku∧pψk−1−−−→ bu∧p
j ' LK (1)S[0,∞)
TC(Z) ' j ∨ Σj ∨ (∨
i=0,2,...,p−2
Σ2i−1`) ∨ Σ2p−1`
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 8 / 15
![Page 40: The Homotopy Groups of K - Indiana Universitymmandell/talks/NotreDame-141030.pdfIntroduction Waldhausen’s Algebraic K-Theory of Spaces Algebraic K-theory of spaces ties algebraic](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20c1b1778ae456533b8ce9/html5/thumbnails/40.jpg)
Topological Cyclic Homology
TC(Z)
Theorem (Bökstedt-Madsen) (p > 2)
TC(Z)∧p [0,∞) ' j ∨ Σj ∨ Σbu∧p
ku = KU[0,∞)bu = KU[2,∞), bu ' Σ2ku
ku∧p ' ` ∨ Σ2` ∨ · · · ∨ Σ2p−4`
Σbu∧p ' Σ3` ∨ · · · ∨ Σ2(p−2)−1` ∨ Σ2(p−1)−1` ∨ Σ2p−1`
j → ku∧pψk−1−−−→ bu∧p
j ' LK (1)S[0,∞)
TC(Z) ' j ∨ Σj ∨ (∨
i=0,2,...,p−2
Σ2i−1`) ∨ Σ2p−1`
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 8 / 15
![Page 41: The Homotopy Groups of K - Indiana Universitymmandell/talks/NotreDame-141030.pdfIntroduction Waldhausen’s Algebraic K-Theory of Spaces Algebraic K-theory of spaces ties algebraic](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20c1b1778ae456533b8ce9/html5/thumbnails/41.jpg)
Topological Cyclic Homology
TC(Z)
Theorem (Bökstedt-Madsen) (p > 2)
TC(Z)∧p [0,∞) ' j ∨ Σj ∨ Σbu∧p
ku = KU[0,∞)bu = KU[2,∞), bu ' Σ2ku
ku∧p ' ` ∨ Σ2` ∨ · · · ∨ Σ2p−4`
Σbu∧p ' Σ3` ∨ · · · ∨ Σ2(p−2)−1` ∨ Σ2(p−1)−1` ∨ Σ2p−1`
j → ku∧pψk−1−−−→ bu∧p
j ' LK (1)S[0,∞)
TC(Z) ' j ∨ Σj ∨ (∨
i=0,2,...,p−2
Σ2i−1`) ∨ Σ2p−1`
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 8 / 15
![Page 42: The Homotopy Groups of K - Indiana Universitymmandell/talks/NotreDame-141030.pdfIntroduction Waldhausen’s Algebraic K-Theory of Spaces Algebraic K-theory of spaces ties algebraic](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20c1b1778ae456533b8ce9/html5/thumbnails/42.jpg)
Topological Cyclic Homology
TC of the Linearization Map
Originally studied by Klein and Rognes
What we need is easy using K (1)-localization:
TC(S)∧p //
'TC(Z)∧p'
S∧p ∨ ΣS∧p ∨ CP∞−1 j ∨ Σj ∨
∨(Σ2i−1`) ∨ Σ2p−1`
j ∨ Σj ∨∨
(Σ2i−1`) ∨ Σ2p−1`
J ∨ ΣJ ∨ Σ−1KU∧p J ∨ ΣJ ∨ Σ−1KU∧p
S(p) → Z(p) is (2p − 3)-connected=⇒ TC(S)∧p → TC(Z) is (2p − 3)-connected
v1 periodicity
=⇒ split surjection on π∗ except ∗ ≡ 1 (mod 2(p − 1))
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 9 / 15
![Page 43: The Homotopy Groups of K - Indiana Universitymmandell/talks/NotreDame-141030.pdfIntroduction Waldhausen’s Algebraic K-Theory of Spaces Algebraic K-theory of spaces ties algebraic](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20c1b1778ae456533b8ce9/html5/thumbnails/43.jpg)
Topological Cyclic Homology
TC of the Linearization Map
Originally studied by Klein and Rognes
What we need is easy using K (1)-localization:
TC(S)∧p //
'TC(Z)∧p'
S∧p ∨ ΣS∧p ∨ CP∞−1 j ∨ Σj ∨
∨(Σ2i−1`) ∨ Σ2p−1`
j ∨ Σj ∨∨
(Σ2i−1`) ∨ Σ2p−1`
J ∨ ΣJ ∨ Σ−1KU∧p J ∨ ΣJ ∨ Σ−1KU∧p
S(p) → Z(p) is (2p − 3)-connected=⇒ TC(S)∧p → TC(Z) is (2p − 3)-connected
v1 periodicity
=⇒ split surjection on π∗ except ∗ ≡ 1 (mod 2(p − 1))
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 9 / 15
![Page 44: The Homotopy Groups of K - Indiana Universitymmandell/talks/NotreDame-141030.pdfIntroduction Waldhausen’s Algebraic K-Theory of Spaces Algebraic K-theory of spaces ties algebraic](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20c1b1778ae456533b8ce9/html5/thumbnails/44.jpg)
Topological Cyclic Homology
TC of the Linearization Map
Originally studied by Klein and Rognes
What we need is easy using K (1)-localization:
TC(S)∧p //
'TC(Z)∧p'
S∧p ∨ ΣS∧p ∨ CP∞−1
**UUUUUUUUUUUUUUUUUUUUUj ∨ Σj ∨
∨(Σ2i−1`) ∨ Σ2p−1`
j ∨ Σj ∨∨
(Σ2i−1`) ∨ Σ2p−1`
J ∨ ΣJ ∨ Σ−1KU∧p
J ∨ ΣJ ∨ Σ−1KU∧p
S(p) → Z(p) is (2p − 3)-connected=⇒ TC(S)∧p → TC(Z) is (2p − 3)-connected
v1 periodicity
=⇒ split surjection on π∗ except ∗ ≡ 1 (mod 2(p − 1))
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 9 / 15
![Page 45: The Homotopy Groups of K - Indiana Universitymmandell/talks/NotreDame-141030.pdfIntroduction Waldhausen’s Algebraic K-Theory of Spaces Algebraic K-theory of spaces ties algebraic](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20c1b1778ae456533b8ce9/html5/thumbnails/45.jpg)
Topological Cyclic Homology
TC of the Linearization Map
Originally studied by Klein and Rognes
What we need is easy using K (1)-localization:
TC(S)∧p //
'TC(Z)∧p'
S∧p ∨ ΣS∧p ∨ CP∞−1
j ∨ Σj ∨∨
(Σ2i−1`) ∨ Σ2p−1`
j ∨ Σj ∨∨
(Σ2i−1`) ∨ Σ2p−1`
J ∨ ΣJ ∨ Σ−1KU∧p ???// J ∨ ΣJ ∨ Σ−1KU∧p
S(p) → Z(p) is (2p − 3)-connected=⇒ TC(S)∧p → TC(Z) is (2p − 3)-connected
v1 periodicity
=⇒ split surjection on π∗ except ∗ ≡ 1 (mod 2(p − 1))
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 9 / 15
![Page 46: The Homotopy Groups of K - Indiana Universitymmandell/talks/NotreDame-141030.pdfIntroduction Waldhausen’s Algebraic K-Theory of Spaces Algebraic K-theory of spaces ties algebraic](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20c1b1778ae456533b8ce9/html5/thumbnails/46.jpg)
Topological Cyclic Homology
TC of the Linearization Map
Originally studied by Klein and Rognes
What we need is easy using K (1)-localization:
TC(S)∧p //
'TC(Z)∧p'
S∧p ∨ ΣS∧p ∨ CP∞−1
j ∨ Σj ∨∨
(Σ2i−1`) ∨ Σ2p−1`
j ∨ Σj ∨∨
(Σ2i−1`) ∨ Σ2p−1`
J ∨ ΣJ ∨ Σ−1KU∧p ???// J ∨ ΣJ ∨ Σ−1KU∧p
S(p) → Z(p) is (2p − 3)-connected=⇒ TC(S)∧p → TC(Z) is (2p − 3)-connected
v1 periodicity
=⇒ split surjection on π∗ except ∗ ≡ 1 (mod 2(p − 1))
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 9 / 15
![Page 47: The Homotopy Groups of K - Indiana Universitymmandell/talks/NotreDame-141030.pdfIntroduction Waldhausen’s Algebraic K-Theory of Spaces Algebraic K-theory of spaces ties algebraic](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20c1b1778ae456533b8ce9/html5/thumbnails/47.jpg)
Topological Cyclic Homology
TC of the Linearization Map
Originally studied by Klein and Rognes
What we need is easy using K (1)-localization:
TC(S)∧p //
'TC(Z)∧p'
S∧p ∨ ΣS∧p ∨ CP∞−1
j ∨ Σj ∨∨
(Σ2i−1`) ∨ Σ2p−1`
j ∨ Σj ∨∨
(Σ2i−1`) ∨ Σ2p−1`
22dddddddddd
J ∨ ΣJ ∨ Σ−1KU∧p ???// J ∨ ΣJ ∨ Σ−1KU∧p
S(p) → Z(p) is (2p − 3)-connected=⇒ TC(S)∧p → TC(Z) is (2p − 3)-connected
v1 periodicity
=⇒ split surjection on π∗ except ∗ ≡ 1 (mod 2(p − 1))
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 9 / 15
![Page 48: The Homotopy Groups of K - Indiana Universitymmandell/talks/NotreDame-141030.pdfIntroduction Waldhausen’s Algebraic K-Theory of Spaces Algebraic K-theory of spaces ties algebraic](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20c1b1778ae456533b8ce9/html5/thumbnails/48.jpg)
Topological Cyclic Homology
TC of the Linearization Map
Originally studied by Klein and Rognes
What we need is easy using K (1)-localization:
TC(S)∧p //
'TC(Z)∧p'
S∧p ∨ ΣS∧p ∨ CP∞−1
j ∨ Σj ∨∨
(Σ2i−1`) ∨ Σ2p−1`
j ∨ Σj ∨∨
(Σ2i−1`) ∨ Σ2p−1`
22dddddddddd
J ∨ ΣJ ∨ Σ−1KU∧p ???// J ∨ ΣJ ∨ Σ−1KU∧p
S(p) → Z(p) is (2p − 3)-connected=⇒ TC(S)∧p → TC(Z) is (2p − 3)-connected
v1 periodicity
=⇒ split surjection on π∗ except ∗ ≡ 1 (mod 2(p − 1))
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 9 / 15
![Page 49: The Homotopy Groups of K - Indiana Universitymmandell/talks/NotreDame-141030.pdfIntroduction Waldhausen’s Algebraic K-Theory of Spaces Algebraic K-theory of spaces ties algebraic](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20c1b1778ae456533b8ce9/html5/thumbnails/49.jpg)
Topological Cyclic Homology
TC of the Linearization Map
Originally studied by Klein and Rognes
What we need is easy using K (1)-localization:
TC(S)∧p //
'TC(Z)∧p'
S∧p ∨ ΣS∧p ∨ CP∞−1
j ∨ Σj ∨∨
(Σ2i−1`) ∨ Σ2p−1`
j ∨ Σj ∨∨
(Σ2i−1`) ∨ Σ2p−1`
22dddddddddd
J ∨ ΣJ ∨ Σ−1KU∧p ???// J ∨ ΣJ ∨ Σ−1KU∧p
S(p) → Z(p) is (2p − 3)-connected=⇒ TC(S)∧p → TC(Z) is (2p − 3)-connected
v1 periodicity
=⇒ split surjection on π∗ except ∗ ≡ 1 (mod 2(p − 1))
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 9 / 15
![Page 50: The Homotopy Groups of K - Indiana Universitymmandell/talks/NotreDame-141030.pdfIntroduction Waldhausen’s Algebraic K-Theory of Spaces Algebraic K-theory of spaces ties algebraic](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20c1b1778ae456533b8ce9/html5/thumbnails/50.jpg)
Topological Cyclic Homology
TC of the Linearization Map
Originally studied by Klein and Rognes
What we need is easy using K (1)-localization:
TC(S)∧p //
'TC(Z)∧p'
S∧p ∨ ΣS∧p ∨ CP∞−1
j ∨ Σj ∨∨
(Σ2i−1`) ∨ Σ2p−1`
j ∨ Σj ∨∨
(Σ2i−1`) ∨ Σ2p−1`
22dddddddddd '∨'∨??∨ ???
22dddddddddd
J ∨ ΣJ ∨ Σ−1KU∧p ???// J ∨ ΣJ ∨ Σ−1KU∧p
S(p) → Z(p) is (2p − 3)-connected=⇒ TC(S)∧p → TC(Z) is (2p − 3)-connected
v1 periodicity
=⇒ split surjection on π∗ except ∗ ≡ 1 (mod 2(p − 1))
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 9 / 15
![Page 51: The Homotopy Groups of K - Indiana Universitymmandell/talks/NotreDame-141030.pdfIntroduction Waldhausen’s Algebraic K-Theory of Spaces Algebraic K-theory of spaces ties algebraic](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20c1b1778ae456533b8ce9/html5/thumbnails/51.jpg)
Topological Cyclic Homology
TC of the Linearization Map
Originally studied by Klein and Rognes
What we need is easy using K (1)-localization:
TC(S)∧p //
'TC(Z)∧p'
S∧p ∨ ΣS∧p ∨ CP∞−1
j ∨ Σj ∨∨
(Σ2i−1`) ∨ Σ2p−1`
j ∨ Σj ∨∨
(Σ2i−1`) ∨ Σ2p−1`
22dddddddddd '∨'∨??∨ ???
22dddddddddd
J ∨ ΣJ ∨ Σ−1KU∧p ???// J ∨ ΣJ ∨ Σ−1KU∧p
S(p) → Z(p) is (2p − 3)-connected=⇒ TC(S)∧p → TC(Z) is (2p − 3)-connected
v1 periodicity
=⇒ split surjection on π∗ except ∗ ≡ 1 (mod 2(p − 1))
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 9 / 15
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Topological Cyclic Homology
TC of the Linearization Map
Originally studied by Klein and Rognes
What we need is easy using K (1)-localization:
TC(S)∧p //
'TC(Z)∧p'
S∧p ∨ ΣS∧p ∨ CP∞−1
j ∨ Σj ∨∨
(Σ2i−1`) ∨ Σ2p−1`
j ∨ Σj ∨∨
(Σ2i−1`) ∨ Σ2p−1`
22dddddddddd '∨'∨'∨ ???
22dddddddddd
J ∨ ΣJ ∨ Σ−1KU∧p ???// J ∨ ΣJ ∨ Σ−1KU∧p
S(p) → Z(p) is (2p − 3)-connected=⇒ TC(S)∧p → TC(Z) is (2p − 3)-connected
v1 periodicity
=⇒ split surjection on π∗ except ∗ ≡ 1 (mod 2(p − 1))
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 9 / 15
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Topological Cyclic Homology
TC of the Linearization Map
Originally studied by Klein and Rognes
What we need is easy using K (1)-localization:
TC(S)∧p //
'TC(Z)∧p'
S∧p ∨ ΣS∧p ∨ CP∞−1
j ∨ Σj ∨∨
(Σ2i−1`) ∨ Σ2p−1`
j ∨ Σj ∨∨
(Σ2i−1`) ∨ Σ2p−1`
22dddddddddd '∨'∨'∨ ???
22dddddddddd
J ∨ ΣJ ∨ Σ−1KU∧p ???// J ∨ ΣJ ∨ Σ−1KU∧p
S(p) → Z(p) is (2p − 3)-connected=⇒ TC(S)∧p → TC(Z) is (2p − 3)-connected
v1 periodicity=⇒ split surjection on π∗ except ∗ ≡ 1 (mod 2(p − 1))
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 9 / 15
![Page 54: The Homotopy Groups of K - Indiana Universitymmandell/talks/NotreDame-141030.pdfIntroduction Waldhausen’s Algebraic K-Theory of Spaces Algebraic K-theory of spaces ties algebraic](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20c1b1778ae456533b8ce9/html5/thumbnails/54.jpg)
K -Theory and Étale Cohomology
Some Facts About K (Z), K (Z∧p)
Theorem (Hesselholt-Madsen)
K (Z)∧p //
K (Z∧p )∧p'
TC(Z)∧p [0,∞) '// TC(Z∧p )∧p [0,∞)
TC(Z)∧p [0,∞) ' j ∨ Σj ∨ Σbu∧p , LK (1)TC(Z) ' J ∨ ΣJ ∨ Σ−1KU
K (Z∧p )∧p → LK (1)K (Z∧p ) induces isomorphism on π∗ for ∗ > 1.
Theorem(?) (Quillen-Lichtenbaum Conjecture)
K (Z)∧p → LK (1)K (Z) induces an isomorphism of π∗ for ∗ > 1.
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 10 / 15
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K -Theory and Étale Cohomology
Some Facts About K (Z), K (Z∧p)
Theorem (Hesselholt-Madsen)
K (Z)∧p //
K (Z∧p )∧p'
TC(Z)∧p [0,∞) '// TC(Z∧p )∧p [0,∞)
TC(Z)∧p [0,∞) ' j ∨ Σj ∨ Σbu∧p , LK (1)TC(Z) ' J ∨ ΣJ ∨ Σ−1KU
K (Z∧p )∧p → LK (1)K (Z∧p ) induces isomorphism on π∗ for ∗ > 1.
Theorem(?) (Quillen-Lichtenbaum Conjecture)
K (Z)∧p → LK (1)K (Z) induces an isomorphism of π∗ for ∗ > 1.
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 10 / 15
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K -Theory and Étale Cohomology
Some Facts About K (Z), K (Z∧p)
Theorem (Hesselholt-Madsen)
K (Z)∧p //
K (Z∧p )∧p'
TC(Z)∧p [0,∞) '// TC(Z∧p )∧p [0,∞)
TC(Z)∧p [0,∞) ' j ∨ Σj ∨ Σbu∧p , LK (1)TC(Z) ' J ∨ ΣJ ∨ Σ−1KU
K (Z∧p )∧p → LK (1)K (Z∧p ) induces isomorphism on π∗ for ∗ > 1.
Theorem(?) (Quillen-Lichtenbaum Conjecture)
K (Z)∧p → LK (1)K (Z) induces an isomorphism of π∗ for ∗ > 1.
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 10 / 15
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K -Theory and Étale Cohomology
Some Facts About K (Z), K (Z∧p)
Theorem (Hesselholt-Madsen)
K (Z)∧p //
K (Z∧p )∧p'
TC(Z)∧p [0,∞) '// TC(Z∧p )∧p [0,∞)
TC(Z)∧p [0,∞) ' j ∨ Σj ∨ Σbu∧p , LK (1)TC(Z) ' J ∨ ΣJ ∨ Σ−1KU
K (Z∧p )∧p → LK (1)K (Z∧p ) induces isomorphism on π∗ for ∗ > 1.
Theorem(?) (Quillen-Lichtenbaum Conjecture)
K (Z)∧p → LK (1)K (Z) induces an isomorphism of π∗ for ∗ > 1.
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 10 / 15
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K -Theory and Étale Cohomology
Some Facts About LK (1)K (Z), LK (1)K (Z∧p)
Theorem (Thomason)
Let R = Z or Z∧p . Let M be a p-torsion group or a pro-p-group.
π2q(LK (K (R)); M) ∼= H0et (R[1/p]; M(q))⊕ H2
et (R[1/p]; M(q + 1))
π2q−1(LK (K (R)); M) ∼= H1et (R[1/p]; M(q))
Theorem (Poitou-Tate Duality)Exact sequenceH1
et (Z[1/p];Z∧p (q))→ H1et (Q
∧p ;Z∧p (q))→ (H1
et (Z[1/p],Z/p∞(1− q)))∗.
Look at q = m(p − 1) + 1
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 11 / 15
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K -Theory and Étale Cohomology
Some Facts About LK (1)K (Z), LK (1)K (Z∧p)
Theorem (Thomason)
Let R = Z or Z∧p . Let M be a p-torsion group or a pro-p-group.
π2q(LK (K (R)); M) ∼= H0et (R[1/p]; M(q))⊕ H2
et (R[1/p]; M(q + 1))
π2q−1(LK (K (R)); M) ∼= H1et (R[1/p]; M(q))
Theorem (Poitou-Tate Duality)Exact sequenceH1
et (Z[1/p];Z∧p (q))→ H1et (Q
∧p ;Z∧p (q))→ (H1
et (Z[1/p],Z/p∞(1− q)))∗.
Look at q = m(p − 1) + 1
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 11 / 15
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K -Theory and Étale Cohomology
A Fact About H1et
Theorem
H1et (Z[1/p];Z/p∞(−m(p − 1))) = 0
Ultimately boils down to (Cl(OQ(ζpn ))∧p )[1] = 0
Corollary
K (Z)∧p → TC(Z)∧p is surjective on πn for n ≡ 1 (mod 2(p − 1)).
Actually, K (Z)∧p splits K (Z)∧p ' j ∨ Σ2p−1` ∨ rest
K (Z)∧p
j ∨ rest ∨ Σ2p−1`
TC(Z)∧p j ∨ Σj ∨ (∨
Σ2i−1`) ∨ Σ2p−1`
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 12 / 15
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K -Theory and Étale Cohomology
A Fact About H1et
Theorem
H1et (Z[1/p];Z/p∞(−m(p − 1))) = 0
Ultimately boils down to (Cl(OQ(ζpn ))∧p )[1] = 0
Corollary
K (Z)∧p → TC(Z)∧p is surjective on πn for n ≡ 1 (mod 2(p − 1)).
Actually, K (Z)∧p splits K (Z)∧p ' j ∨ Σ2p−1` ∨ rest
K (Z)∧p
j ∨ rest ∨ Σ2p−1`
TC(Z)∧p j ∨ Σj ∨ (∨
Σ2i−1`) ∨ Σ2p−1`
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 12 / 15
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K -Theory and Étale Cohomology
A Fact About H1et
Theorem
H1et (Z[1/p];Z/p∞(−m(p − 1))) = 0
Ultimately boils down to (Cl(OQ(ζpn ))∧p )[1] = 0
Corollary
K (Z)∧p → TC(Z)∧p is surjective on πn for n ≡ 1 (mod 2(p − 1)).
Actually, K (Z)∧p splits K (Z)∧p ' j ∨ Σ2p−1` ∨ rest
K (Z)∧p
j ∨ rest ∨ Σ2p−1`
TC(Z)∧p j ∨ Σj ∨ (∨
Σ2i−1`) ∨ Σ2p−1`
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 12 / 15
![Page 63: The Homotopy Groups of K - Indiana Universitymmandell/talks/NotreDame-141030.pdfIntroduction Waldhausen’s Algebraic K-Theory of Spaces Algebraic K-theory of spaces ties algebraic](https://reader033.fdocuments.us/reader033/viewer/2022042406/5f20c1b1778ae456533b8ce9/html5/thumbnails/63.jpg)
K -Theory and Étale Cohomology
A Fact About H1et
Theorem
H1et (Z[1/p];Z/p∞(−m(p − 1))) = 0
Ultimately boils down to (Cl(OQ(ζpn ))∧p )[1] = 0
Corollary
K (Z)∧p → TC(Z)∧p is surjective on πn for n ≡ 1 (mod 2(p − 1)).
Actually, K (Z)∧p splits K (Z)∧p ' j ∨ Σ2p−1` ∨ rest
K (Z)∧p
j ∨ rest ∨ Σ2p−1`
TC(Z)∧p j ∨ Σj ∨ (∨
Σ2i−1`) ∨ Σ2p−1`
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 12 / 15
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The Main Theorem
The Linearization/Cyclotomic Trace Square
K (S)∧p //
K (Z)
S ∨ ΣS ∨ CP∞−1
// j ∨ Σj ∨ (∨
Σ2i−1`) ∨ Σ2p−1`
Let c = hofib(S∧p → j) “coker j”
Theorem (Main Theorem)
p-tors(K (S)) ∼= p-tors(S)⊕p-tors(Σc)⊕p-tors(CP∞−1)⊕p-tors(K red(Z))
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 13 / 15
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The Main Theorem
The Linearization/Cyclotomic Trace Square
K (S)∧p //
K (Z)
S ∨ ΣS ∨ CP∞−1
// j ∨ Σj ∨ (∨
Σ2i−1`) ∨ Σ2p−1`
Let c = hofib(S∧p → j) “coker j”
Theorem (Main Theorem)
p-tors(K (S)) ∼= p-tors(S)⊕p-tors(Σc)⊕p-tors(CP∞−1)⊕p-tors(K red(Z))
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 13 / 15
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The Main Theorem
The Linearization/Cyclotomic Trace Square
K (S)∧p //
K (Z)
S ∨ ΣS ∨ CP∞−1
// j ∨ Σj ∨ (∨
Σ2i−1`) ∨ Σ2p−1`
Let c = hofib(S∧p → j) “coker j”
Theorem (Main Theorem)
p-tors(K (S)) ∼= p-tors(S)⊕p-tors(Σc)⊕p-tors(CP∞−1)⊕p-tors(K red(Z))
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 13 / 15
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The Main Theorem
The Linearization/Cyclotomic Trace Square
K (S)∧p //
K (Z)
S ∨ ΣS ∨ CP∞−1
// j ∨ Σj ∨ (∨
Σ2i−1`) ∨ Σ2p−1`
Let c = hofib(S∧p → j) “coker j”
Theorem (Main Theorem)
p-tors(K (S)) ∼= p-tors(S)⊕p-tors(Σc)⊕p-tors(CP∞−1)⊕p-tors(K red(Z))
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 13 / 15
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The Main Theorem
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 14 / 15
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Nilpotence
Multiplication on π∗K (S) is Nilpotent
Suffices to consider rational part, p-torsion separately
Rational part is in odd degrees
Saw K (Z)∧p → TC(Z)∧p is surjective on πn for n ≡ 1 (mod 2p − 2).
Implies πnK (Z) = 0 for n ≡ 0 (mod 2p − 2) (Poitou-Tate).
Implies πn hofib(trc) = 0 for n ≡ 0 (mod 2p − 2).Also for p = 2, πn hofib(trc) = 0 for n ≡ 0 (mod 8).
So K (S)∧p → TC(S)∧p injective on πn for n ≡ 0 (mod 8p − 8).
Suffices to see π∗TC(S)∧p is nilpotent.
TC(S)∧p ' S∧p ∨ ΣCP∞−1
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 15 / 15
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Nilpotence
Multiplication on π∗K (S) is Nilpotent
Suffices to consider rational part, p-torsion separately
Rational part is in odd degrees
Saw K (Z)∧p → TC(Z)∧p is surjective on πn for n ≡ 1 (mod 2p − 2).
Implies πnK (Z) = 0 for n ≡ 0 (mod 2p − 2) (Poitou-Tate).
Implies πn hofib(trc) = 0 for n ≡ 0 (mod 2p − 2).Also for p = 2, πn hofib(trc) = 0 for n ≡ 0 (mod 8).
So K (S)∧p → TC(S)∧p injective on πn for n ≡ 0 (mod 8p − 8).
Suffices to see π∗TC(S)∧p is nilpotent.
TC(S)∧p ' S∧p ∨ ΣCP∞−1
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 15 / 15
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Nilpotence
Multiplication on π∗K (S) is Nilpotent
Suffices to consider rational part, p-torsion separately
Rational part is in odd degrees
Saw K (Z)∧p → TC(Z)∧p is surjective on πn for n ≡ 1 (mod 2p − 2).
Implies πnK (Z) = 0 for n ≡ 0 (mod 2p − 2) (Poitou-Tate).
Implies πn hofib(trc) = 0 for n ≡ 0 (mod 2p − 2).Also for p = 2, πn hofib(trc) = 0 for n ≡ 0 (mod 8).
So K (S)∧p → TC(S)∧p injective on πn for n ≡ 0 (mod 8p − 8).
Suffices to see π∗TC(S)∧p is nilpotent.
TC(S)∧p ' S∧p ∨ ΣCP∞−1
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 15 / 15
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Nilpotence
Multiplication on π∗K (S) is Nilpotent
Suffices to consider rational part, p-torsion separately
Rational part is in odd degrees
Saw K (Z)∧p → TC(Z)∧p is surjective on πn for n ≡ 1 (mod 2p − 2).
Implies πnK (Z) = 0 for n ≡ 0 (mod 2p − 2) (Poitou-Tate).
Implies πn hofib(trc) = 0 for n ≡ 0 (mod 2p − 2).Also for p = 2, πn hofib(trc) = 0 for n ≡ 0 (mod 8).
So K (S)∧p → TC(S)∧p injective on πn for n ≡ 0 (mod 8p − 8).
Suffices to see π∗TC(S)∧p is nilpotent.
TC(S)∧p ' S∧p ∨ ΣCP∞−1
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 15 / 15
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Nilpotence
Multiplication on π∗K (S) is Nilpotent
Suffices to consider rational part, p-torsion separately
Rational part is in odd degrees
Saw K (Z)∧p → TC(Z)∧p is surjective on πn for n ≡ 1 (mod 2p − 2).
Implies πnK (Z) = 0 for n ≡ 0 (mod 2p − 2) (Poitou-Tate).
Implies πn hofib(trc) = 0 for n ≡ 0 (mod 2p − 2).Also for p = 2, πn hofib(trc) = 0 for n ≡ 0 (mod 8).
So K (S)∧p → TC(S)∧p injective on πn for n ≡ 0 (mod 8p − 8).
Suffices to see π∗TC(S)∧p is nilpotent.
TC(S)∧p ' S∧p ∨ ΣCP∞−1
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 15 / 15
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Nilpotence
Multiplication on π∗K (S) is Nilpotent
Suffices to consider rational part, p-torsion separately
Rational part is in odd degrees
Saw K (Z)∧p → TC(Z)∧p is surjective on πn for n ≡ 1 (mod 2p − 2).
Implies πnK (Z) = 0 for n ≡ 0 (mod 2p − 2) (Poitou-Tate).
Implies πn hofib(trc) = 0 for n ≡ 0 (mod 2p − 2).Also for p = 2, πn hofib(trc) = 0 for n ≡ 0 (mod 8).
So K (S)∧p → TC(S)∧p injective on πn for n ≡ 0 (mod 8p − 8).
Suffices to see π∗TC(S)∧p is nilpotent.
TC(S)∧p ' S∧p ∨ ΣCP∞−1
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 15 / 15
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Nilpotence
Multiplication on π∗K (S) is Nilpotent
Suffices to consider rational part, p-torsion separately
Rational part is in odd degrees
Saw K (Z)∧p → TC(Z)∧p is surjective on πn for n ≡ 1 (mod 2p − 2).
Implies πnK (Z) = 0 for n ≡ 0 (mod 2p − 2) (Poitou-Tate).
Implies πn hofib(trc) = 0 for n ≡ 0 (mod 2p − 2).Also for p = 2, πn hofib(trc) = 0 for n ≡ 0 (mod 8).
So K (S)∧p → TC(S)∧p injective on πn for n ≡ 0 (mod 8p − 8).
Suffices to see π∗TC(S)∧p is nilpotent.
TC(S)∧p ' S∧p ∨ ΣCP∞−1
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 15 / 15
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Nilpotence
Multiplication on π∗K (S) is Nilpotent
Suffices to consider rational part, p-torsion separately
Rational part is in odd degrees
Saw K (Z)∧p → TC(Z)∧p is surjective on πn for n ≡ 1 (mod 2p − 2).
Implies πnK (Z) = 0 for n ≡ 0 (mod 2p − 2) (Poitou-Tate).
Implies πn hofib(trc) = 0 for n ≡ 0 (mod 2p − 2).Also for p = 2, πn hofib(trc) = 0 for n ≡ 0 (mod 8).
So K (S)∧p → TC(S)∧p injective on πn for n ≡ 0 (mod 8p − 8).
Suffices to see π∗TC(S)∧p is nilpotent.
TC(S)∧p ' S∧p ∨ ΣCP∞−1
M.A.Mandell (IU) The Homotopy Groups of K (S) Oct 2014 15 / 15