# Symmetric Powers of Spheres

Ηλεκτρονική - Συσκευές

13 Οκτ 2013 (πριν από 4 χρόνια και 9 μήνες)

110 εμφανίσεις

Symmetric Powers of Spheres
Neil Strickland
(with Johann Sigurdsson)
August 9,2007
Overview of homotopy theory

(S

)
H

(S
0
)

S

(S
0
)
MU

(S
0
)
Formal groups

k+1
S
1
E

k+2
S
2
E

k+3
S
3
E

k+4
S
4

k
(QS
0
)=
S
k
(S
0
)

k+2
S
3
H

k+3
S
5
H

k+4
S
7
H
EHPSS
S
0
=X(1)
X(2)
X(3)
X(4)
X(1)=MU
X(n)=Thom(
SU(n)!
SU=BU)
X(n)=X(n;0)
X(n;1)
X(n;2)
X(n;2)
X(n;1)=X(n+1)
X(n;k) from the James ltration on
(SU(n+1)=SU(n))=
S
2n+1
=JS
2n
Nilpotence ltration
Koszul ltration
S
0
=SP
1
(S
0
)
SP
2
(S
0
)
SP
3
(S
0
)
SP
4
(S
0
)
SP
1
(S
0
)=H
SP
n
(S
0
)= prespectrum with k'th space (S
k
)
n
=
n
S
0
=SP
1
(S
0
)
SP
p
(S
0
)
SP
p
2
(S
0
)
SP
p
3
(S
0
)
SP
1
(S
0
)=H
SP
n
(S
0
)= prespectrum with k'th space (S
k
)
n
=
n
S
0
=SP
1
(S
0
)
L(0)
SP
p
(S
0
)
L(1)
SP
p
2
(S
0
)

2
L(2)
SP
p
3
(S
0
)

3
L(3)
SP
1
(S
0
)=H

1
L() is a DGA up to homotopy,chain equivalent to Z (Whitehead,Kuhn,Priddy)
Symmetric power ltration
S
MU
MU
MU^
MU
MU
(2)
MU^
MU
(2)
MU
(3)
MU^
MU
(3)
MU
(4)
Algebraic NSS
Unstable Adams SS,Lambda algebra,central series for simplicial groups
X
1
=
S
1
=Z
QS
0
X
2
X
3

1
Q(2)
h
2
X
4

1
Q(3)
h
3
X
5

1
Q(4)
h
4
Q(n) is a certain nite 
n
-spectrum,with H

Q(n) = H
0
Q(n) = Lie(n).
It is dened using combinatorics of partitions of n points,and is related to SP
n
(S
0
)=SP
n1
(S
0
)
There is a similar tower for
S
k+1
,with bres

1
(S
nk
^ Q(n))
h
n
(Goodwillie,Johnson,Arone,Mahowald)
EHPSS
Goodwillie tower
Algebraic NSS
Unstable Adams SS,Lambda algebra,central series for simplicial groups
KU

(S
0
)
Ell

(S
0
)
K(n)

(S
0
)

(L
K(n)
S
0
)
v
1
n

(S

)
Bouseld-Kuhn

n
:Spaces!Spectra 
n
(

1
X) = L
K(n)
(X)
Overview of homotopy theory

(S

)
H

(S
0
)

S

(S
0
)
MU

(S
0
)
Formal groups

k+1
S
1
E

k+2
S
2
E

k+3
S
3
E

k+4
S
4

k
(QS
0
)=
S
k
(S
0
)

k+2
S
3
H

k+3
S
5
H

k+4
S
7
H
EHPSS
S
0
=X(1)
X(2)
X(3)
X(4)
X(1)=MU
X(n)=Thom(
SU(n)!
SU=BU)
X(n)=X(n;0)
X(n;1)
X(n;2)
X(n;2)
X(n;1)=X(n+1)
X(n;k) from the James ltration on
(SU(n+1)=SU(n))=
S
2n+1
=JS
2n
Nilpotence ltration
Koszul ltration
S
0
=SP
1
(S
0
)
SP
2
(S
0
)
SP
3
(S
0
)
SP
4
(S
0
)
SP
1
(S
0
)=H
SP
n
(S
0
)= prespectrum with k'th space (S
k
)
n
=
n
S
0
=SP
1
(S
0
)
SP
p
(S
0
)
SP
p
2
(S
0
)
SP
p
3
(S
0
)
SP
1
(S
0
)=H
SP
n
(S
0
)= prespectrum with k'th space (S
k
)
n
=
n
S
0
=SP
1
(S
0
)
L(0)
SP
p
(S
0
)
L(1)
SP
p
2
(S
0
)

2
L(2)
SP
p
3
(S
0
)

3
L(3)
SP
1
(S
0
)=H

1
L() is a DGA up to homotopy,chain equivalent to Z (Whitehead,Kuhn,Priddy)
Symmetric power ltration
S
MU
MU
MU^
MU
MU
(2)
MU^
MU
(2)
MU
(3)
MU^
MU
(3)
MU
(4)
Algebraic NSS
Unstable Adams SS,Lambda algebra,central series for simplicial groups
X
1
=
S
1
=Z
QS
0
X
2
X
3

1
Q(2)
h
2
X
4

1
Q(3)
h
3
X
5

1
Q(4)
h
4
Q(n) is a certain nite 
n
-spectrum,with H

Q(n) = H
0
Q(n) = Lie(n).
It is dened using combinatorics of partitions of n points,and is related to SP
n
(S
0
)=SP
n1
(S
0
)
There is a similar tower for
S
k+1
,with bres

1
(S
nk
^ Q(n))
h
n
(Goodwillie,Johnson,Arone,Mahowald)
EHPSS
Goodwillie tower
Algebraic NSS
Unstable Adams SS,Lambda algebra,central series for simplicial groups
KU

(S
0
)
Ell

(S
0
)
K(n)

(S
0
)

(L
K(n)
S
0
)
v
1
n

(S

)
Bouseld-Kuhn

n
:Spaces!Spectra 
n
(

1
X) = L
K(n)
(X)
Overview of homotopy theory

(S

)
H

(S
0
)

S

(S
0
)
MU

(S
0
)
Formal groups

k+1
S
1
E

k+2
S
2
E

k+3
S
3
E

k+4
S
4

k
(QS
0
)=
S
k
(S
0
)

k+2
S
3
H

k+3
S
5
H

k+4
S
7
H
EHPSS
S
0
=X(1)
X(2)
X(3)
X(4)
X(1)=MU
X(n)=Thom(
SU(n)!
SU=BU)
X(n)=X(n;0)
X(n;1)
X(n;2)
X(n;2)
X(n;1)=X(n+1)
X(n;k) from the James ltration on
(SU(n+1)=SU(n))=
S
2n+1
=JS
2n
Nilpotence ltration
Koszul ltration
S
0
=SP
1
(S
0
)
SP
2
(S
0
)
SP
3
(S
0
)
SP
4
(S
0
)
SP
1
(S
0
)=H
SP
n
(S
0
)= prespectrum with k'th space (S
k
)
n
=
n
S
0
=SP
1
(S
0
)
SP
p
(S
0
)
SP
p
2
(S
0
)
SP
p
3
(S
0
)
SP
1
(S
0
)=H
SP
n
(S
0
)= prespectrum with k'th space (S
k
)
n
=
n
S
0
=SP
1
(S
0
)
L(0)
SP
p
(S
0
)
L(1)
SP
p
2
(S
0
)

2
L(2)
SP
p
3
(S
0
)

3
L(3)
SP
1
(S
0
)=H

1
L() is a DGA up to homotopy,chain equivalent to Z (Whitehead,Kuhn,Priddy)
Symmetric power ltration
S
MU
MU
MU^
MU
MU
(2)
MU^
MU
(2)
MU
(3)
MU^
MU
(3)
MU
(4)
Algebraic NSS
Unstable Adams SS,Lambda algebra,central series for simplicial groups
X
1
=
S
1
=Z
QS
0
X
2
X
3

1
Q(2)
h
2
X
4

1
Q(3)
h
3
X
5

1
Q(4)
h
4
Q(n) is a certain nite 
n
-spectrum,with H

Q(n) = H
0
Q(n) = Lie(n).
It is dened using combinatorics of partitions of n points,and is related to SP
n
(S
0
)=SP
n1
(S
0
)
There is a similar tower for
S
k+1
,with bres

1
(S
nk
^ Q(n))
h
n
(Goodwillie,Johnson,Arone,Mahowald)
EHPSS
Goodwillie tower
Algebraic NSS
Unstable Adams SS,Lambda algebra,central series for simplicial groups
KU

(S
0
)
Ell

(S
0
)
K(n)

(S
0
)

(L
K(n)
S
0
)
v
1
n

(S

)
Bouseld-Kuhn

n
:Spaces!Spectra 
n
(

1
X) = L
K(n)
(X)
Overview of homotopy theory

(S

)
H

(S
0
)

S

(S
0
)
MU

(S
0
)
Formal groups

k+1
S
1
E

k+2
S
2
E

k+3
S
3
E

k+4
S
4

k
(QS
0
)=
S
k
(S
0
)

k+2
S
3
H

k+3
S
5
H

k+4
S
7
H
EHPSS
S
0
=X(1)
X(2)
X(3)
X(4)
X(1)=MU
X(n)=Thom(
SU(n)!
SU=BU)
X(n)=X(n;0)
X(n;1)
X(n;2)
X(n;2)
X(n;1)=X(n+1)
X(n;k) from the James ltration on
(SU(n+1)=SU(n))=
S
2n+1
=JS
2n
Nilpotence ltration
Koszul ltration
S
0
=SP
1
(S
0
)
SP
2
(S
0
)
SP
3
(S
0
)
SP
4
(S
0
)
SP
1
(S
0
)=H
SP
n
(S
0
)= prespectrum with k'th space (S
k
)
n
=
n
S
0
=SP
1
(S
0
)
SP
p
(S
0
)
SP
p
2
(S
0
)
SP
p
3
(S
0
)
SP
1
(S
0
)=H
SP
n
(S
0
)= prespectrum with k'th space (S
k
)
n
=
n
S
0
=SP
1
(S
0
)
L(0)
SP
p
(S
0
)
L(1)
SP
p
2
(S
0
)

2
L(2)
SP
p
3
(S
0
)

3
L(3)
SP
1
(S
0
)=H

1
L() is a DGA up to homotopy,chain equivalent to Z (Whitehead,Kuhn,Priddy)
Symmetric power ltration
S
MU
MU
MU^
MU
MU
(2)
MU^
MU
(2)
MU
(3)
MU^
MU
(3)
MU
(4)
Algebraic NSS
Unstable Adams SS,Lambda algebra,central series for simplicial groups
X
1
=
S
1
=Z
QS
0
X
2
X
3

1
Q(2)
h
2
X
4

1
Q(3)
h
3
X
5

1
Q(4)
h
4
Q(n) is a certain nite 
n
-spectrum,with H

Q(n) = H
0
Q(n) = Lie(n).
It is dened using combinatorics of partitions of n points,and is related to SP
n
(S
0
)=SP
n1
(S
0
)
There is a similar tower for
S
k+1
,with bres

1
(S
nk
^ Q(n))
h
n
(Goodwillie,Johnson,Arone,Mahowald)
EHPSS
Goodwillie tower
Algebraic NSS
Unstable Adams SS,Lambda algebra,central series for simplicial groups
KU

(S
0
)
Ell

(S
0
)
K(n)

(S
0
)

(L
K(n)
S
0
)
v
1
n

(S

)
Bouseld-Kuhn

n
:Spaces!Spectra 
n
(

1
X) = L
K(n)
(X)
Overview of homotopy theory

(S

)
H

(S
0
)

S

(S
0
)
MU

(S
0
)
Formal groups

k+1
S
1
E

k+2
S
2
E

k+3
S
3
E

k+4
S
4

k
(QS
0
)=
S
k
(S
0
)

k+2
S
3
H

k+3
S
5
H

k+4
S
7
H
EHPSS
S
0
=X(1)
X(2)
X(3)
X(4)
X(1)=MU
X(n)=Thom(
SU(n)!
SU=BU)
X(n)=X(n;0)
X(n;1)
X(n;2)
X(n;2)
X(n;1)=X(n+1)
X(n;k) from the James ltration on
(SU(n+1)=SU(n))=
S
2n+1
=JS
2n
Nilpotence ltration
Koszul ltration
S
0
=SP
1
(S
0
)
SP
2
(S
0
)
SP
3
(S
0
)
SP
4
(S
0
)
SP
1
(S
0
)=H
SP
n
(S
0
)= prespectrum with k'th space (S
k
)
n
=
n
S
0
=SP
1
(S
0
)
SP
p
(S
0
)
SP
p
2
(S
0
)
SP
p
3
(S
0
)
SP
1
(S
0
)=H
SP
n
(S
0
)= prespectrum with k'th space (S
k
)
n
=
n
S
0
=SP
1
(S
0
)
L(0)
SP
p
(S
0
)
L(1)
SP
p
2
(S
0
)

2
L(2)
SP
p
3
(S
0
)

3
L(3)
SP
1
(S
0
)=H

1
L() is a DGA up to homotopy,chain equivalent to Z (Whitehead,Kuhn,Priddy)
Symmetric power ltration
S
MU
MU
MU^
MU
MU
(2)
MU^
MU
(2)
MU
(3)
MU^
MU
(3)
MU
(4)
Algebraic NSS
Unstable Adams SS,Lambda algebra,central series for simplicial groups
X
1
=
S
1
=Z
QS
0
X
2
X
3

1
Q(2)
h
2
X
4

1
Q(3)
h
3
X
5

1
Q(4)
h
4
Q(n) is a certain nite 
n
-spectrum,with H

Q(n) = H
0
Q(n) = Lie(n).
It is dened using combinatorics of partitions of n points,and is related to SP
n
(S
0
)=SP
n1
(S
0
)
There is a similar tower for
S
k+1
,with bres

1
(S
nk
^ Q(n))
h
n
(Goodwillie,Johnson,Arone,Mahowald)
EHPSS
Goodwillie tower
Algebraic NSS
Unstable Adams SS,Lambda algebra,central series for simplicial groups
KU

(S
0
)
Ell

(S
0
)
K(n)

(S
0
)

(L
K(n)
S
0
)
v
1
n

(S

)
Bouseld-Kuhn

n
:Spaces!Spectra 
n
(

1
X) = L
K(n)
(X)
Overview of homotopy theory

(S

)
H

(S
0
)

S

(S
0
)
MU

(S
0
)
Formal groups

k+1
S
1
E

k+2
S
2
E

k+3
S
3
E

k+4
S
4

k
(QS
0
)=
S
k
(S
0
)

k+2
S
3
H

k+3
S
5
H

k+4
S
7
H
EHPSS
S
0
=X(1)
X(2)
X(3)
X(4)
X(1)=MU
X(n)=Thom(
SU(n)!
SU=BU)
X(n)=X(n;0)
X(n;1)
X(n;2)
X(n;2)
X(n;1)=X(n+1)
X(n;k) from the James ltration on
(SU(n+1)=SU(n))=
S
2n+1
=JS
2n
Nilpotence ltration
Koszul ltration
S
0
=SP
1
(S
0
)
SP
2
(S
0
)
SP
3
(S
0
)
SP
4
(S
0
)
SP
1
(S
0
)=H
SP
n
(S
0
)= prespectrum with k'th space (S
k
)
n
=
n
S
0
=SP
1
(S
0
)
SP
p
(S
0
)
SP
p
2
(S
0
)
SP
p
3
(S
0
)
SP
1
(S
0
)=H
SP
n
(S
0
)= prespectrum with k'th space (S
k
)
n
=
n
S
0
=SP
1
(S
0
)
L(0)
SP
p
(S
0
)
L(1)
SP
p
2
(S
0
)

2
L(2)
SP
p
3
(S
0
)

3
L(3)
SP
1
(S
0
)=H

1
L() is a DGA up to homotopy,chain equivalent to Z (Whitehead,Kuhn,Priddy)
Symmetric power ltration
S
MU
MU
MU^
MU
MU
(2)
MU^
MU
(2)
MU
(3)
MU^
MU
(3)
MU
(4)
Algebraic NSS
Unstable Adams SS,Lambda algebra,central series for simplicial groups
X
1
=
S
1
=Z
QS
0
X
2
X
3

1
Q(2)
h
2
X
4

1
Q(3)
h
3
X
5

1
Q(4)
h
4
Q(n) is a certain nite 
n
-spectrum,with H

Q(n) = H
0
Q(n) = Lie(n).
It is dened using combinatorics of partitions of n points,and is related to SP
n
(S
0
)=SP
n1
(S
0
)
There is a similar tower for
S
k+1
,with bres

1
(S
nk
^ Q(n))
h
n
(Goodwillie,Johnson,Arone,Mahowald)
EHPSS
Goodwillie tower
Algebraic NSS
Unstable Adams SS,Lambda algebra,central series for simplicial groups
KU

(S
0
)
Ell

(S
0
)
K(n)

(S
0
)

(L
K(n)
S
0
)
v
1
n

(S

)
Bouseld-Kuhn

n
:Spaces!Spectra 
n
(

1
X) = L
K(n)
(X)
Overview of homotopy theory

(S

)
H

(S
0
)

S

(S
0
)
MU

(S
0
)
Formal groups

k+1
S
1
E

k+2
S
2
E

k+3
S
3
E

k+4
S
4

k
(QS
0
)=
S
k
(S
0
)

k+2
S
3
H

k+3
S
5
H

k+4
S
7
H
EHPSS
S
0
=X(1)
X(2)
X(3)
X(4)
X(1)=MU
X(n)=Thom(
SU(n)!
SU=BU)
X(n)=X(n;0)
X(n;1)
X(n;2)
X(n;2)
X(n;1)=X(n+1)
X(n;k) from the James ltration on
(SU(n+1)=SU(n))=
S
2n+1
=JS
2n
Nilpotence ltration
Koszul ltration
S
0
=SP
1
(S
0
)
SP
2
(S
0
)
SP
3
(S
0
)
SP
4
(S
0
)
SP
1
(S
0
)=H
SP
n
(S
0
)= prespectrum with k'th space (S
k
)
n
=
n
S
0
=SP
1
(S
0
)
SP
p
(S
0
)
SP
p
2
(S
0
)
SP
p
3
(S
0
)
SP
1
(S
0
)=H
SP
n
(S
0
)= prespectrum with k'th space (S
k
)
n
=
n
S
0
=SP
1
(S
0
)
L(0)
SP
p
(S
0
)
L(1)
SP
p
2
(S
0
)

2
L(2)
SP
p
3
(S
0
)

3
L(3)
SP
1
(S
0
)=H

1
L() is a DGA up to homotopy,chain equivalent to Z (Whitehead,Kuhn,Priddy)
Symmetric power ltration
S
MU
MU
MU^
MU
MU
(2)
MU^
MU
(2)
MU
(3)
MU^
MU
(3)
MU
(4)
Algebraic NSS
Unstable Adams SS,Lambda algebra,central series for simplicial groups
X
1
=
S
1
=Z
QS
0
X
2
X
3

1
Q(2)
h
2
X
4

1
Q(3)
h
3
X
5

1
Q(4)
h
4
Q(n) is a certain nite 
n
-spectrum,with H

Q(n) = H
0
Q(n) = Lie(n).
It is dened using combinatorics of partitions of n points,and is related to SP
n
(S
0
)=SP
n1
(S
0
)
There is a similar tower for
S
k+1
,with bres

1
(S
nk
^ Q(n))
h
n
(Goodwillie,Johnson,Arone,Mahowald)
EHPSS
Goodwillie tower
Algebraic NSS
Unstable Adams SS,Lambda algebra,central series for simplicial groups
KU

(S
0
)
Ell

(S
0
)
K(n)

(S
0
)

(L
K(n)
S
0
)
v
1
n

(S

)
Bouseld-Kuhn

n
:Spaces!Spectra 
n
(

1
X) = L
K(n)
(X)
Overview of homotopy theory

(S

)
H

(S
0
)

S

(S
0
)
MU

(S
0
)
Formal groups

k+1
S
1
E

k+2
S
2
E

k+3
S
3
E

k+4
S
4

k
(QS
0
)=
S
k
(S
0
)

k+2
S
3
H

k+3
S
5
H

k+4
S
7
H
EHPSS
S
0
=X(1)
X(2)
X(3)
X(4)
X(1)=MU
X(n)=Thom(
SU(n)!
SU=BU)
X(n)=X(n;0)
X(n;1)
X(n;2)
X(n;2)
X(n;1)=X(n+1)
X(n;k) from the James ltration on
(SU(n+1)=SU(n))=
S
2n+1
=JS
2n
Nilpotence ltration
Koszul ltration
S
0
=SP
1
(S
0
)
SP
2
(S
0
)
SP
3
(S
0
)
SP
4
(S
0
)
SP
1
(S
0
)=H
SP
n
(S
0
)= prespectrum with k'th space (S
k
)
n
=
n
S
0
=SP
1
(S
0
)
SP
p
(S
0
)
SP
p
2
(S
0
)
SP
p
3
(S
0
)
SP
1
(S
0
)=H
SP
n
(S
0
)= prespectrum with k'th space (S
k
)
n
=
n
S
0
=SP
1
(S
0
)
L(0)
SP
p
(S
0
)
L(1)
SP
p
2
(S
0
)

2
L(2)
SP
p
3
(S
0
)

3
L(3)
SP
1
(S
0
)=H

1
L() is a DGA up to homotopy,chain equivalent to Z (Whitehead,Kuhn,Priddy)
Symmetric power ltration
S
MU
MU
MU^
MU
MU
(2)
MU^
MU
(2)
MU
(3)
MU^
MU
(3)
MU
(4)
Algebraic NSS
Unstable Adams SS,Lambda algebra,central series for simplicial groups
X
1
=
S
1
=Z
QS
0
X
2
X
3

1
Q(2)
h
2
X
4

1
Q(3)
h
3
X
5

1
Q(4)
h
4
Q(n) is a certain nite 
n
-spectrum,with H

Q(n) = H
0
Q(n) = Lie(n).
It is dened using combinatorics of partitions of n points,and is related to SP
n
(S
0
)=SP
n1
(S
0
)
There is a similar tower for
S
k+1
,with bres

1
(S
nk
^ Q(n))
h
n
(Goodwillie,Johnson,Arone,Mahowald)
EHPSS
Goodwillie tower
Algebraic NSS
Unstable Adams SS,Lambda algebra,central series for simplicial groups
KU

(S
0
)
Ell

(S
0
)
K(n)

(S
0
)

(L
K(n)
S
0
)
v
1
n

(S

)
Bouseld-Kuhn

n
:Spaces!Spectra 
n
(

1
X) = L
K(n)
(X)
Overview of homotopy theory

(S

)
H

(S
0
)

S

(S
0
)
MU

(S
0
)
Formal groups

k+1
S
1
E

k+2
S
2
E

k+3
S
3
E

k+4
S
4

k
(QS
0
)=
S
k
(S
0
)

k+2
S
3
H

k+3
S
5
H

k+4
S
7
H
EHPSS
S
0
=X(1)
X(2)
X(3)
X(4)
X(1)=MU
X(n)=Thom(
SU(n)!
SU=BU)
X(n)=X(n;0)
X(n;1)
X(n;2)
X(n;2)
X(n;1)=X(n+1)
X(n;k) from the James ltration on
(SU(n+1)=SU(n))=
S
2n+1
=JS
2n
Nilpotence ltration
Koszul ltration
S
0
=SP
1
(S
0
)
SP
2
(S
0
)
SP
3
(S
0
)
SP
4
(S
0
)
SP
1
(S
0
)=H
SP
n
(S
0
)= prespectrum with k'th space (S
k
)
n
=
n
S
0
=SP
1
(S
0
)
SP
p
(S
0
)
SP
p
2
(S
0
)
SP
p
3
(S
0
)
SP
1
(S
0
)=H
SP
n
(S
0
)= prespectrum with k'th space (S
k
)
n
=
n
S
0
=SP
1
(S
0
)
L(0)
SP
p
(S
0
)
L(1)
SP
p
2
(S
0
)

2
L(2)
SP
p
3
(S
0
)

3
L(3)
SP
1
(S
0
)=H

1
L() is a DGA up to homotopy,chain equivalent to Z (Whitehead,Kuhn,Priddy)
Symmetric power ltration
S
MU
MU
MU^
MU
MU
(2)
MU^
MU
(2)
MU
(3)
MU^
MU
(3)
MU
(4)
Algebraic NSS
Unstable Adams SS,Lambda algebra,central series for simplicial groups
X
1
=
S
1
=Z
QS
0
X
2
X
3

1
Q(2)
h
2
X
4

1
Q(3)
h
3
X
5

1
Q(4)
h
4
Q(n) is a certain nite 
n
-spectrum,with H

Q(n) = H
0
Q(n) = Lie(n).
It is dened using combinatorics of partitions of n points,and is related to SP
n
(S
0
)=SP
n1
(S
0
)
There is a similar tower for
S
k+1
,with bres

1
(S
nk
^ Q(n))
h
n
(Goodwillie,Johnson,Arone,Mahowald)
EHPSS
Goodwillie tower
Algebraic NSS
Unstable Adams SS,Lambda algebra,central series for simplicial groups
KU

(S
0
)
Ell

(S
0
)
K(n)

(S
0
)

(L
K(n)
S
0
)
v
1
n

(S

)
Bouseld-Kuhn

n
:Spaces!Spectra 
n
(

1
X) = L
K(n)
(X)
Overview of homotopy theory

(S

)
H

(S
0
)

S

(S
0
)
MU

(S
0
)
Formal groups

k+1
S
1
E

k+2
S
2
E

k+3
S
3
E

k+4
S
4

k
(QS
0
)=
S
k
(S
0
)

k+2
S
3
H

k+3
S
5
H

k+4
S
7
H
EHPSS
S
0
=X(1)
X(2)
X(3)
X(4)
X(1)=MU
X(n)=Thom(
SU(n)!
SU=BU)
X(n)=X(n;0)
X(n;1)
X(n;2)
X(n;2)
X(n;1)=X(n+1)
X(n;k) from the James ltration on
(SU(n+1)=SU(n))=
S
2n+1
=JS
2n
Nilpotence ltration
Koszul ltration
S
0
=SP
1
(S
0
)
SP
2
(S
0
)
SP
3
(S
0
)
SP
4
(S
0
)
SP
1
(S
0
)=H
SP
n
(S
0
)= prespectrum with k'th space (S
k
)
n
=
n
S
0
=SP
1
(S
0
)
SP
p
(S
0
)
SP
p
2
(S
0
)
SP
p
3
(S
0
)
SP
1
(S
0
)=H
SP
n
(S
0
)= prespectrum with k'th space (S
k
)
n
=
n
S
0
=SP
1
(S
0
)
L(0)
SP
p
(S
0
)
L(1)
SP
p
2
(S
0
)

2
L(2)
SP
p
3
(S
0
)

3
L(3)
SP
1
(S
0
)=H

1
L() is a DGA up to homotopy,chain equivalent to Z (Whitehead,Kuhn,Priddy)
Symmetric power ltration
S
MU
MU
MU^
MU
MU
(2)
MU^
MU
(2)
MU
(3)
MU^
MU
(3)
MU
(4)
Algebraic NSS
Unstable Adams SS,Lambda algebra,central series for simplicial groups
X
1
=
S
1
=Z
QS
0
X
2
X
3

1
Q(2)
h
2
X
4

1
Q(3)
h
3
X
5

1
Q(4)
h
4
Q(n) is a certain nite 
n
-spectrum,with H

Q(n) = H
0
Q(n) = Lie(n).
It is dened using combinatorics of partitions of n points,and is related to SP
n
(S
0
)=SP
n1
(S
0
)
There is a similar tower for
S
k+1
,with bres

1
(S
nk
^ Q(n))
h
n
(Goodwillie,Johnson,Arone,Mahowald)
EHPSS
Goodwillie tower
Algebraic NSS
Unstable Adams SS,Lambda algebra,central series for simplicial groups
KU

(S
0
)
Ell

(S
0
)
K(n)

(S
0
)

(L
K(n)
S
0
)
v
1
n

(S

)
Bouseld-Kuhn

n
:Spaces!Spectra 
n
(

1
X) = L
K(n)
(X)
Overview of homotopy theory

(S

)
H

(S
0
)

S

(S
0
)
MU

(S
0
)
Formal groups

k+1
S
1
E

k+2
S
2
E

k+3
S
3
E

k+4
S
4

k
(QS
0
)=
S
k
(S
0
)

k+2
S
3
H

k+3
S
5
H

k+4
S
7
H
EHPSS
S
0
=X(1)
X(2)
X(3)
X(4)
X(1)=MU
X(n)=Thom(
SU(n)!
SU=BU)
X(n)=X(n;0)
X(n;1)
X(n;2)
X(n;2)
X(n;1)=X(n+1)
X(n;k) from the James ltration on
(SU(n+1)=SU(n))=
S
2n+1
=JS
2n
Nilpotence ltration
Koszul ltration
S
0
=SP
1
(S
0
)
SP
2
(S
0
)
SP
3
(S
0
)
SP
4
(S
0
)
SP
1
(S
0
)=H
SP
n
(S
0
)= prespectrum with k'th space (S
k
)
n
=
n
S
0
=SP
1
(S
0
)
SP
p
(S
0
)
SP
p
2
(S
0
)
SP
p
3
(S
0
)
SP
1
(S
0
)=H
SP
n
(S
0
)= prespectrum with k'th space (S
k
)
n
=
n
S
0
=SP
1
(S
0
)
L(0)
SP
p
(S
0
)
L(1)
SP
p
2
(S
0
)

2
L(2)
SP
p
3
(S
0
)

3
L(3)
SP
1
(S
0
)=H

1
L() is a DGA up to homotopy,chain equivalent to Z (Whitehead,Kuhn,Priddy)
Symmetric power ltration
S
MU
MU
MU^
MU
MU
(2)
MU^
MU
(2)
MU
(3)
MU^
MU
(3)
MU
(4)
Algebraic NSS
Unstable Adams SS,Lambda algebra,central series for simplicial groups
X
1
=
S
1
=Z
QS
0
X
2
X
3

1
Q(2)
h
2
X
4

1
Q(3)
h
3
X
5

1
Q(4)
h
4
Q(n) is a certain nite 
n
-spectrum,with H

Q(n) = H
0
Q(n) = Lie(n).
It is dened using combinatorics of partitions of n points,and is related to SP
n
(S
0
)=SP
n1
(S
0
)
There is a similar tower for
S
k+1
,with bres

1
(S
nk
^ Q(n))
h
n
(Goodwillie,Johnson,Arone,Mahowald)
EHPSS
Goodwillie tower
Algebraic NSS
Unstable Adams SS,Lambda algebra,central series for simplicial groups
KU

(S
0
)
Ell

(S
0
)
K(n)

(S
0
)

(L
K(n)
S
0
)
v
1
n

(S

)
Bouseld-Kuhn

n
:Spaces!Spectra 
n
(

1
X) = L
K(n)
(X)
Overview of homotopy theory

(S

)
H

(S
0
)

S

(S
0
)
MU

(S
0
)
Formal groups

k+1
S
1
E

k+2
S
2
E

k+3
S
3
E

k+4
S
4

k
(QS
0
)=
S
k
(S
0
)

k+2
S
3
H

k+3
S
5
H

k+4
S
7
H
EHPSS
S
0
=X(1)
X(2)
X(3)
X(4)
X(1)=MU
X(n)=Thom(
SU(n)!
SU=BU)
X(n)=X(n;0)
X(n;1)
X(n;2)
X(n;2)
X(n;1)=X(n+1)
X(n;k) from the James ltration on
(SU(n+1)=SU(n))=
S
2n+1
=JS
2n
Nilpotence ltration
Koszul ltration
S
0
=SP
1
(S
0
)
SP
2
(S
0
)
SP
3
(S
0
)
SP
4
(S
0
)
SP
1
(S
0
)=H
SP
n
(S
0
)= prespectrum with k'th space (S
k
)
n
=
n
S
0
=SP
1
(S
0
)
SP
p
(S
0
)
SP
p
2
(S
0
)
SP
p
3
(S
0
)
SP
1
(S
0
)=H
SP
n
(S
0
)= prespectrum with k'th space (S
k
)
n
=
n
S
0
=SP
1
(S
0
)
L(0)
SP
p
(S
0
)
L(1)
SP
p
2
(S
0
)

2
L(2)
SP
p
3
(S
0
)

3
L(3)
SP
1
(S
0
)=H

1
L() is a DGA up to homotopy,chain equivalent to Z (Whitehead,Kuhn,Priddy)
Symmetric power ltration
S
MU
MU
MU^
MU
MU
(2)
MU^
MU
(2)
MU
(3)
MU^
MU
(3)
MU
(4)
Algebraic NSS
Unstable Adams SS,Lambda algebra,central series for simplicial groups
X
1
=
S
1
=Z
QS
0
X
2
X
3

1
Q(2)
h
2
X
4

1
Q(3)
h
3
X
5

1
Q(4)
h
4
Q(n) is a certain nite 
n
-spectrum,with H

Q(n) = H
0
Q(n) = Lie(n).
It is dened using combinatorics of partitions of n points,and is related to SP
n
(S
0
)=SP
n1
(S
0
)
There is a similar tower for
S
k+1
,with bres

1
(S
nk
^ Q(n))
h
n
(Goodwillie,Johnson,Arone,Mahowald)
EHPSS
Goodwillie tower
Algebraic NSS
Unstable Adams SS,Lambda algebra,central series for simplicial groups
KU

(S
0
)
Ell

(S
0
)
K(n)

(S
0
)

(L
K(n)
S
0
)
v
1
n

(S

)
Bouseld-Kuhn

n
:Spaces!Spectra 
n
(

1
X) = L
K(n)
(X)
Overview of homotopy theory

(S

)
H

(S
0
)

S

(S
0
)
MU

(S
0
)
Formal groups

k+1
S
1
E

k+2
S
2
E

k+3
S
3
E

k+4
S
4

k
(QS
0
)=
S
k
(S
0
)

k+2
S
3
H

k+3
S
5
H

k+4
S
7
H
EHPSS
S
0
=X(1)
X(2)
X(3)
X(4)
X(1)=MU
X(n)=Thom(
SU(n)!
SU=BU)
X(n)=X(n;0)
X(n;1)
X(n;2)
X(n;2)
X(n;1)=X(n+1)
X(n;k) from the James ltration on
(SU(n+1)=SU(n))=
S
2n+1
=JS
2n
Nilpotence ltration
Koszul ltration
S
0
=SP
1
(S
0
)
SP
2
(S
0
)
SP
3
(S
0
)
SP
4
(S
0
)
SP
1
(S
0
)=H
SP
n
(S
0
)= prespectrum with k'th space (S
k
)
n
=
n
S
0
=SP
1
(S
0
)
SP
p
(S
0
)
SP
p
2
(S
0
)
SP
p
3
(S
0
)
SP
1
(S
0
)=H
SP
n
(S
0
)= prespectrum with k'th space (S
k
)
n
=
n
S
0
=SP
1
(S
0
)
L(0)
SP
p
(S
0
)
L(1)
SP
p
2
(S
0
)

2
L(2)
SP
p
3
(S
0
)

3
L(3)
SP
1
(S
0
)=H

1
L() is a DGA up to homotopy,chain equivalent to Z (Whitehead,Kuhn,Priddy)
Symmetric power ltration
S
MU
MU
MU^
MU
MU
(2)
MU^
MU
(2)
MU
(3)
MU^
MU
(3)
MU
(4)
Algebraic NSS
Unstable Adams SS,Lambda algebra,central series for simplicial groups
X
1
=
S
1
=Z
QS
0
X
2
X
3

1
Q(2)
h
2
X
4

1
Q(3)
h
3
X
5

1
Q(4)
h
4
Q(n) is a certain nite 
n
-spectrum,with H

Q(n) = H
0
Q(n) = Lie(n).
It is dened using combinatorics of partitions of n points,and is related to SP
n
(S
0
)=SP
n1
(S
0
)
There is a similar tower for
S
k+1
,with bres

1
(S
nk
^ Q(n))
h
n
(Goodwillie,Johnson,Arone,Mahowald)
EHPSS
Goodwillie tower
Algebraic NSS
Unstable Adams SS,Lambda algebra,central series for simplicial groups
KU

(S
0
)
Ell

(S
0
)
K(n)

(S
0
)

(L
K(n)
S
0
)
v
1
n

(S

)
Bouseld-Kuhn

n
:Spaces!Spectra 
n
(

1
X) = L
K(n)
(X)
Overview of homotopy theory

(S

)
H

(S
0
)

S

(S
0
)
MU

(S
0
)
Formal groups

k+1
S
1
E

k+2
S
2
E

k+3
S
3
E

k+4
S
4

k
(QS
0
)=
S
k
(S
0
)

k+2
S
3
H

k+3
S
5
H

k+4
S
7
H
EHPSS
S
0
=X(1)
X(2)
X(3)
X(4)
X(1)=MU
X(n)=Thom(
SU(n)!
SU=BU)
X(n)=X(n;0)
X(n;1)
X(n;2)
X(n;2)
X(n;1)=X(n+1)
X(n;k) from the James ltration on
(SU(n+1)=SU(n))=
S
2n+1
=JS
2n
Nilpotence ltration
Koszul ltration
S
0
=SP
1
(S
0
)
SP
2
(S
0
)
SP
3
(S
0
)
SP
4
(S
0
)
SP
1
(S
0
)=H
SP
n
(S
0
)= prespectrum with k'th space (S
k
)
n
=
n
S
0
=SP
1
(S
0
)
SP
p
(S
0
)
SP
p
2
(S
0
)
SP
p
3
(S
0
)
SP
1
(S
0
)=H
SP
n
(S
0
)= prespectrum with k'th space (S
k
)
n
=
n
S
0
=SP
1
(S
0
)
L(0)
SP
p
(S
0
)
L(1)
SP
p
2
(S
0
)

2
L(2)
SP
p
3
(S
0
)

3
L(3)
SP
1
(S
0
)=H

1
L() is a DGA up to homotopy,chain equivalent to Z (Whitehead,Kuhn,Priddy)
Symmetric power ltration
S
MU
MU
MU^
MU
MU
(2)
MU^
MU
(2)
MU
(3)
MU^
MU
(3)
MU
(4)
Algebraic NSS
Unstable Adams SS,Lambda algebra,central series for simplicial groups
X
1
=
S
1
=Z
QS
0
X
2
X
3

1
Q(2)
h
2
X
4

1
Q(3)
h
3
X
5

1
Q(4)
h
4
Q(n) is a certain nite 
n
-spectrum,with H

Q(n) = H
0
Q(n) = Lie(n).
It is dened using combinatorics of partitions of n points,and is related to SP
n
(S
0
)=SP
n1
(S
0
)
There is a similar tower for
S
k+1
,with bres

1
(S
nk
^ Q(n))
h
n
(Goodwillie,Johnson,Arone,Mahowald)
EHPSS
Goodwillie tower
Algebraic NSS
Unstable Adams SS,Lambda algebra,central series for simplicial groups
KU

(S
0
)
Ell

(S
0
)
K(n)

(S
0
)

(L
K(n)
S
0
)
v
1
n

(S

)
Bouseld-Kuhn

n
:Spaces!Spectra 
n
(

1
X) = L
K(n)
(X)
Overview of homotopy theory

(S

)
H

(S
0
)

S

(S
0
)
MU

(S
0
)
Formal groups

k+1
S
1
E

k+2
S
2
E

k+3
S
3
E

k+4
S
4

k
(QS
0
)=
S
k
(S
0
)

k+2
S
3
H

k+3
S
5
H

k+4
S
7
H
EHPSS
S
0
=X(1)
X(2)
X(3)
X(4)
X(1)=MU
X(n)=Thom(
SU(n)!
SU=BU)
X(n)=X(n;0)
X(n;1)
X(n;2)
X(n;2)
X(n;1)=X(n+1)
X(n;k) from the James ltration on
(SU(n+1)=SU(n))=
S
2n+1
=JS
2n
Nilpotence ltration
Koszul ltration
S
0
=SP
1
(S
0
)
SP
2
(S
0
)
SP
3
(S
0
)
SP
4
(S
0
)
SP
1
(S
0
)=H
SP
n
(S
0
)= prespectrum with k'th space (S
k
)
n
=
n
S
0
=SP
1
(S
0
)
SP
p
(S
0
)
SP
p
2
(S
0
)
SP
p
3
(S
0
)
SP
1
(S
0
)=H
SP
n
(S
0
)= prespectrum with k'th space (S
k
)
n
=
n
S
0
=SP
1
(S
0
)
L(0)
SP
p
(S
0
)
L(1)
SP
p
2
(S
0
)

2
L(2)
SP
p
3
(S
0
)

3
L(3)
SP
1
(S
0
)=H

1
L() is a DGA up to homotopy,chain equivalent to Z (Whitehead,Kuhn,Priddy)
Symmetric power ltration
S
MU
MU
MU^
MU
MU
(2)
MU^
MU
(2)
MU
(3)
MU^
MU
(3)
MU
(4)
Algebraic NSS
Unstable Adams SS,Lambda algebra,central series for simplicial groups
X
1
=
S
1
=Z
QS
0
X
2
X
3

1
Q(2)
h
2
X
4

1
Q(3)
h
3
X
5

1
Q(4)
h
4
Q(n) is a certain nite 
n
-spectrum,with H

Q(n) = H
0
Q(n) = Lie(n).
It is dened using combinatorics of partitions of n points,and is related to SP
n
(S
0
)=SP
n1
(S
0
)
There is a similar tower for
S
k+1
,with bres

1
(S
nk
^ Q(n))
h
n
(Goodwillie,Johnson,Arone,Mahowald)
EHPSS
Goodwillie tower
Algebraic NSS
Unstable Adams SS,Lambda algebra,central series for simplicial groups
KU

(S
0
)
Ell

(S
0
)
K(n)

(S
0
)

(L
K(n)
S
0
)
v
1
n

(S

)
Bouseld-Kuhn

n
:Spaces!Spectra 
n
(

1
X) = L
K(n)
(X)
Overview of homotopy theory

(S

)
H

(S
0
)

S

(S
0
)
MU

(S
0
)
Formal groups

k+1
S
1
E

k+2
S
2
E

k+3
S
3
E

k+4
S
4

k
(QS
0
)=
S
k
(S
0
)

k+2
S
3
H

k+3
S
5
H

k+4
S
7
H
EHPSS
S
0
=X(1)
X(2)
X(3)
X(4)
X(1)=MU
X(n)=Thom(
SU(n)!
SU=BU)
X(n)=X(n;0)
X(n;1)
X(n;2)
X(n;2)
X(n;1)=X(n+1)
X(n;k) from the James ltration on
(SU(n+1)=SU(n))=
S
2n+1
=JS
2n
Nilpotence ltration
Koszul ltration
S
0
=SP
1
(S
0
)
SP
2
(S
0
)
SP
3
(S
0
)
SP
4
(S
0
)
SP
1
(S
0
)=H
SP
n
(S
0
)= prespectrum with k'th space (S
k
)
n
=
n
S
0
=SP
1
(S
0
)
SP
p
(S
0
)
SP
p
2
(S
0
)
SP
p
3
(S
0
)
SP
1
(S
0
)=H
SP
n
(S
0
)= prespectrum with k'th space (S
k
)
n
=
n
S
0
=SP
1
(S
0
)
L(0)
SP
p
(S
0
)
L(1)
SP
p
2
(S
0
)

2
L(2)
SP
p
3
(S
0
)

3
L(3)
SP
1
(S
0
)=H

1
L() is a DGA up to homotopy,chain equivalent to Z (Whitehead,Kuhn,Priddy)
Symmetric power ltration
S
MU
MU
MU^
MU
MU
(2)
MU^
MU
(2)
MU
(3)
MU^
MU
(3)
MU
(4)
Algebraic NSS
Unstable Adams SS,Lambda algebra,central series for simplicial groups
X
1
=
S
1
=Z
QS
0
X
2
X
3

1
Q(2)
h
2
X
4

1
Q(3)
h
3
X
5

1
Q(4)
h
4
Q(n) is a certain nite 
n
-spectrum,with H

Q(n) = H
0
Q(n) = Lie(n).
It is dened using combinatorics of partitions of n points,and is related to SP
n
(S
0
)=SP
n1
(S
0
)
There is a similar tower for
S
k+1
,with bres

1
(S
nk
^ Q(n))
h
n
(Goodwillie,Johnson,Arone,Mahowald)
EHPSS
Goodwillie tower
Algebraic NSS
Unstable Adams SS,Lambda algebra,central series for simplicial groups
KU

(S
0
)
Ell

(S
0
)
K(n)

(S
0
)

(L
K(n)
S
0
)
v
1
n

(S

)
Bouseld-Kuhn

n
:Spaces!Spectra 
n
(

1
X) = L
K(n)
(X)
Overview of homotopy theory

(S

)
H

(S
0
)

S

(S
0
)
MU

(S
0
)
Formal groups

k+1
S
1
E

k+2
S
2
E

k+3
S
3
E

k+4
S
4

k
(QS
0
)=
S
k
(S
0
)

k+2
S
3
H

k+3
S
5
H

k+4
S
7
H
EHPSS
S
0
=X(1)
X(2)
X(3)
X(4)
X(1)=MU
X(n)=Thom(
SU(n)!
SU=BU)
X(n)=X(n;0)
X(n;1)
X(n;2)
X(n;2)
X(n;1)=X(n+1)
X(n;k) from the James ltration on
(SU(n+1)=SU(n))=
S
2n+1
=JS
2n
Nilpotence ltration
Koszul ltration
S
0
=SP
1
(S
0
)
SP
2
(S
0
)
SP
3
(S
0
)
SP
4
(S
0
)
SP
1
(S
0
)=H
SP
n
(S
0
)= prespectrum with k'th space (S
k
)
n
=
n
S
0
=SP
1
(S
0
)
SP
p
(S
0
)
SP
p
2
(S
0
)
SP
p
3
(S
0
)
SP
1
(S
0
)=H
SP
n
(S
0
)= prespectrum with k'th space (S
k
)
n
=
n
S
0
=SP
1
(S
0
)
L(0)
SP
p
(S
0
)
L(1)
SP
p
2
(S
0
)

2
L(2)
SP
p
3
(S
0
)

3
L(3)
SP
1
(S
0
)=H

1
L() is a DGA up to homotopy,chain equivalent to Z (Whitehead,Kuhn,Priddy)
Symmetric power ltration
S
MU
MU
MU^
MU
MU
(2)
MU^
MU
(2)
MU
(3)
MU^
MU
(3)
MU
(4)
Algebraic NSS
Unstable Adams SS,Lambda algebra,central series for simplicial groups
X
1
=
S
1
=Z
QS
0
X
2
X
3

1
Q(2)
h
2
X
4

1
Q(3)
h
3
X
5

1
Q(4)
h
4
Q(n) is a certain nite 
n
-spectrum,with H

Q(n) = H
0
Q(n) = Lie(n).
It is dened using combinatorics of partitions of n points,and is related to SP
n
(S
0
)=SP
n1
(S
0
)
There is a similar tower for
S
k+1
,with bres

1
(S
nk
^ Q(n))
h
n
(Goodwillie,Johnson,Arone,Mahowald)
EHPSS
Goodwillie tower
Algebraic NSS
Unstable Adams SS,Lambda algebra,central series for simplicial groups
KU

(S
0
)
Ell

(S
0
)
K(n)

(S
0
)

(L
K(n)
S
0
)
v
1
n

(S

)
Bouseld-Kuhn

n
:Spaces!Spectra 
n
(

1
X) = L
K(n)
(X)
Overview of homotopy theory

(S

)
H

(S
0
)

S

(S
0
)
MU

(S
0
)
Formal groups

k+1
S
1
E

k+2
S
2
E

k+3
S
3
E

k+4
S
4

k
(QS
0
)=
S
k
(S
0
)

k+2
S
3
H

k+3
S
5
H

k+4
S
7
H
EHPSS
S
0
=X(1)
X(2)
X(3)
X(4)
X(1)=MU
X(n)=Thom(
SU(n)!
SU=BU)
X(n)=X(n;0)
X(n;1)
X(n;2)
X(n;2)
X(n;1)=X(n+1)
X(n;k) from the James ltration on
(SU(n+1)=SU(n))=
S
2n+1
=JS
2n
Nilpotence ltration
Koszul ltration
S
0
=SP
1
(S
0
)
SP
2
(S
0
)
SP
3
(S
0
)
SP
4
(S
0
)
SP
1
(S
0
)=H
SP
n
(S
0
)= prespectrum with k'th space (S
k
)
n
=
n
S
0
=SP
1
(S
0
)
SP
p
(S
0
)
SP
p
2
(S
0
)
SP
p
3
(S
0
)
SP
1
(S
0
)=H
SP
n
(S
0
)= prespectrum with k'th space (S
k
)
n
=
n
S
0
=SP
1
(S
0
)
L(0)
SP
p
(S
0
)
L(1)
SP
p
2
(S
0
)

2
L(2)
SP
p
3
(S
0
)

3
L(3)
SP
1
(S
0
)=H

1
L() is a DGA up to homotopy,chain equivalent to Z (Whitehead,Kuhn,Priddy)
Symmetric power ltration
S
MU
MU
MU^
MU
MU
(2)
MU^
MU
(2)
MU
(3)
MU^
MU
(3)
MU
(4)
Algebraic NSS
Unstable Adams SS,Lambda algebra,central series for simplicial groups
X
1
=
S
1
=Z
QS
0
X
2
X
3

1
Q(2)
h
2
X
4

1
Q(3)
h
3
X
5

1
Q(4)
h
4
Q(n) is a certain nite 
n
-spectrum,with H

Q(n) = H
0
Q(n) = Lie(n).
It is dened using combinatorics of partitions of n points,and is related to SP
n
(S
0
)=SP
n1
(S
0
)
There is a similar tower for
S
k+1
,with bres

1
(S
nk
^ Q(n))
h
n
(Goodwillie,Johnson,Arone,Mahowald)
EHPSS
Goodwillie tower
Algebraic NSS
Unstable Adams SS,Lambda algebra,central series for simplicial groups
KU

(S
0
)
Ell

(S
0
)
K(n)

(S
0
)

(L
K(n)
S
0
)
v
1
n

(S

)
Bouseld-Kuhn

n
:Spaces!Spectra 
n
(

1
X) = L
K(n)
(X)
Overview of homotopy theory

(S

)
H

(S
0
)

S

(S
0
)
MU

(S
0
)
Formal groups

k+1
S
1
E

k+2
S
2
E

k+3
S
3
E

k+4
S
4

k
(QS
0
)=
S
k
(S
0
)

k+2
S
3
H

k+3
S
5
H

k+4
S
7
H
EHPSS
S
0
=X(1)
X(2)
X(3)
X(4)
X(1)=MU
X(n)=Thom(
SU(n)!
SU=BU)
X(n)=X(n;0)
X(n;1)
X(n;2)
X(n;2)
X(n;1)=X(n+1)
X(n;k) from the James ltration on
(SU(n+1)=SU(n))=
S
2n+1
=JS
2n
Nilpotence ltration
Koszul ltration
S
0
=SP
1
(S
0
)
SP
2
(S
0
)
SP
3
(S
0
)
SP
4
(S
0
)
SP
1
(S
0
)=H
SP
n
(S
0
)= prespectrum with k'th space (S
k
)
n
=
n
S
0
=SP
1
(S
0
)
SP
p
(S
0
)
SP
p
2
(S
0
)
SP
p
3
(S
0
)
SP
1
(S
0
)=H
SP
n
(S
0
)= prespectrum with k'th space (S
k
)
n
=
n
S
0
=SP
1
(S
0
)
L(0)
SP
p
(S
0
)
L(1)
SP
p
2
(S
0
)

2
L(2)
SP
p
3
(S
0
)

3
L(3)
SP
1
(S
0
)=H

1
L() is a DGA up to homotopy,chain equivalent to Z (Whitehead,Kuhn,Priddy)
Symmetric power ltration
S
MU
MU
MU^
MU
MU
(2)
MU^
MU
(2)
MU
(3)
MU^
MU
(3)
MU
(4)
Algebraic NSS
Unstable Adams SS,Lambda algebra,central series for simplicial groups
X
1
=
S
1
=Z
QS
0
X
2
X
3

1
Q(2)
h
2
X
4

1
Q(3)
h
3
X
5

1
Q(4)
h
4
Q(n) is a certain nite 
n
-spectrum,with H

Q(n) = H
0
Q(n) = Lie(n).
It is dened using combinatorics of partitions of n points,and is related to SP
n
(S
0
)=SP
n1
(S
0
)
There is a similar tower for
S
k+1
,with bres

1
(S
nk
^ Q(n))
h
n
(Goodwillie,Johnson,Arone,Mahowald)
EHPSS
Goodwillie tower
Algebraic NSS
Unstable Adams SS,Lambda algebra,central series for simplicial groups
KU

(S
0
)
Ell

(S
0
)
K(n)

(S
0
)

(L
K(n)
S
0
)
v
1
n

(S

)
Bouseld-Kuhn

n
:Spaces!Spectra 
n
(

1
X) = L
K(n)
(X)
Overview of homotopy theory

(S

)
H

(S
0
)

S

(S
0
)
MU

(S
0
)
Formal groups

k+1
S
1
E

k+2
S
2
E

k+3
S
3
E

k+4
S
4

k
(QS
0
)=
S
k
(S
0
)

k+2
S
3
H

k+3
S
5
H

k+4
S
7
H
EHPSS
S
0
=X(1)
X(2)
X(3)
X(4)
X(1)=MU
X(n)=Thom(
SU(n)!
SU=BU)
X(n)=X(n;0)
X(n;1)
X(n;2)
X(n;2)
X(n;1)=X(n+1)
X(n;k) from the James ltration on
(SU(n+1)=SU(n))=
S
2n+1
=JS
2n
Nilpotence ltration
Koszul ltration
S
0
=SP
1
(S
0
)
SP
2
(S
0
)
SP
3
(S
0
)
SP
4
(S
0
)
SP
1
(S
0
)=H
SP
n
(S
0
)= prespectrum with k'th space (S
k
)
n
=
n
S
0
=SP
1
(S
0
)
SP
p
(S
0
)
SP
p
2
(S
0
)
SP
p
3
(S
0
)
SP
1
(S
0
)=H
SP
n
(S
0
)= prespectrum with k'th space (S
k
)
n
=
n
S
0
=SP
1
(S
0
)
L(0)
SP
p
(S
0
)
L(1)
SP
p
2
(S
0
)

2
L(2)
SP
p
3
(S
0
)

3
L(3)
SP
1
(S
0
)=H

1
L() is a DGA up to homotopy,chain equivalent to Z (Whitehead,Kuhn,Priddy)
Symmetric power ltration
S
MU
MU
MU^
MU
MU
(2)
MU^
MU
(2)
MU
(3)
MU^
MU
(3)
MU
(4)
Algebraic NSS
Unstable Adams SS,Lambda algebra,central series for simplicial groups
X
1
=
S
1
=Z
QS
0
X
2
X
3

1
Q(2)
h
2
X
4

1
Q(3)
h
3
X
5

1
Q(4)
h
4
Q(n) is a certain nite 
n
-spectrum,with H

Q(n) = H
0
Q(n) = Lie(n).
It is dened using combinatorics of partitions of n points,and is related to SP
n
(S
0
)=SP
n1
(S
0
)
There is a similar tower for
S
k+1
,with bres

1
(S
nk
^ Q(n))
h
n
(Goodwillie,Johnson,Arone,Mahowald)
EHPSS
Goodwillie tower
Algebraic NSS
Unstable Adams SS,Lambda algebra,central series for simplicial groups
KU

(S
0
)
Ell

(S
0
)
K(n)

(S
0
)

(L
K(n)
S
0
)
v
1
n

(S

)
Bouseld-Kuhn

n
:Spaces!Spectra 
n
(

1
X) = L
K(n)
(X)
Overview of homotopy theory

(S

)
H

(S
0
)

S

(S
0
)
MU

(S
0
)
Formal groups

k+1
S
1
E

k+2
S
2
E

k+3
S
3
E

k+4
S
4

k
(QS
0
)=
S
k
(S
0
)

k+2
S
3
H

k+3
S
5
H

k+4
S
7
H
EHPSS
S
0
=X(1)
X(2)
X(3)
X(4)
X(1)=MU
X(n)=Thom(
SU(n)!
SU=BU)
X(n)=X(n;0)
X(n;1)
X(n;2)
X(n;2)
X(n;1)=X(n+1)
X(n;k) from the James ltration on
(SU(n+1)=SU(n))=
S
2n+1
=JS
2n
Nilpotence ltration
Koszul ltration
S
0
=SP
1
(S
0
)
SP
2
(S
0
)
SP
3
(S
0
)
SP
4
(S
0
)
SP
1
(S
0
)=H
SP
n
(S
0
)= prespectrum with k'th space (S
k
)
n
=
n
S
0
=SP
1
(S
0
)
SP
p
(S
0
)
SP
p
2
(S
0
)
SP
p
3
(S
0
)
SP
1
(S
0
)=H
SP
n
(S
0
)= prespectrum with k'th space (S
k
)
n
=
n
S
0
=SP
1
(S
0
)
L(0)
SP
p
(S
0
)
L(1)
SP
p
2
(S
0
)

2
L(2)
SP
p
3
(S
0
)

3
L(3)
SP
1
(S
0
)=H

1
L() is a DGA up to homotopy,chain equivalent to Z (Whitehead,Kuhn,Priddy)
Symmetric power ltration
S
MU
MU
MU^
MU
MU
(2)
MU^
MU
(2)
MU
(3)
MU^
MU
(3)
MU
(4)
Algebraic NSS
Unstable Adams SS,Lambda algebra,central series for simplicial groups
X
1
=
S
1
=Z
QS
0
X
2
X
3

1
Q(2)
h
2
X
4

1
Q(3)
h
3
X
5

1
Q(4)
h
4
Q(n) is a certain nite 
n
-spectrum,with H

Q(n) = H
0
Q(n) = Lie(n).
It is dened using combinatorics of partitions of n points,and is related to SP
n
(S
0
)=SP
n1
(S
0
)
There is a similar tower for
S
k+1
,with bres

1
(S
nk
^ Q(n))
h
n
(Goodwillie,Johnson,Arone,Mahowald)
EHPSS
Goodwillie tower
Algebraic NSS
Unstable Adams SS,Lambda algebra,central series for simplicial groups
KU

(S
0
)
Ell

(S
0
)
K(n)

(S
0
)

(L
K(n)
S
0
)
v
1
n

(S

)
Bouseld-Kuhn

n
:Spaces!Spectra 
n
(

1
X) = L
K(n)
(X)
Overview of homotopy theory

(S

)
H

(S
0
)

S

(S
0
)
MU

(S
0
)
Formal groups

k+1
S
1
E

k+2
S
2
E

k+3
S
3
E

k+4
S
4

k
(QS
0
)=
S
k
(S
0
)

k+2
S
3
H

k+3
S
5
H

k+4
S
7
H
EHPSS
S
0
=X(1)
X(2)
X(3)
X(4)
X(1)=MU
X(n)=Thom(
SU(n)!
SU=BU)
X(n)=X(n;0)
X(n;1)
X(n;2)
X(n;2)
X(n;1)=X(n+1)
X(n;k) from the James ltration on
(SU(n+1)=SU(n))=
S
2n+1
=JS
2n
Nilpotence ltration
Koszul ltration
S
0
=SP
1
(S
0
)
SP
2
(S
0
)
SP
3
(S
0
)
SP
4
(S
0
)
SP
1
(S
0
)=H
SP
n
(S
0
)= prespectrum with k'th space (S
k
)
n
=
n
S
0
=SP
1
(S
0
)
SP
p
(S
0
)
SP
p
2
(S
0
)
SP
p
3
(S
0
)
SP
1
(S
0
)=H
SP
n
(S
0
)= prespectrum with k'th space (S
k
)
n
=
n
S
0
=SP
1
(S
0
)
L(0)
SP
p
(S
0
)
L(1)
SP
p
2
(S
0
)

2
L(2)
SP
p
3
(S
0
)

3
L(3)
SP
1
(S
0
)=H

1
L() is a DGA up to homotopy,chain equivalent to Z (Whitehead,Kuhn,Priddy)
Symmetric power ltration
S
MU
MU
MU^
MU
MU
(2)
MU^
MU
(2)
MU
(3)
MU^
MU
(3)
MU
(4)
Algebraic NSS
Unstable Adams SS,Lambda algebra,central series for simplicial groups
X
1
=
S
1
=Z
QS
0
X
2
X
3

1
Q(2)
h
2
X
4

1
Q(3)
h
3
X
5

1
Q(4)
h
4
Q(n) is a certain nite 
n
-spectrum,with H

Q(n) = H
0
Q(n) = Lie(n).
It is dened using combinatorics of partitions of n points,and is related to SP
n
(S
0
)=SP
n1
(S
0
)
There is a similar tower for
S
k+1
,with bres

1
(S
nk
^ Q(n))
h
n
(Goodwillie,Johnson,Arone,Mahowald)
EHPSS
Goodwillie tower
Algebraic NSS
Unstable Adams SS,Lambda algebra,central series for simplicial groups
KU

(S
0
)
Ell

(S
0
)
K(n)

(S
0
)

(L
K(n)
S
0
)
v
1
n

(S

)
Bouseld-Kuhn

n
:Spaces!Spectra 
n
(

1
X) = L
K(n)
(X)
Overview of homotopy theory

(S

)
H

(S
0
)

S

(S
0
)
MU

(S
0
)
Formal groups

k+1
S
1
E

k+2
S
2
E

k+3
S
3
E

k+4
S
4

k
(QS
0
)=
S
k
(S
0
)

k+2
S
3
H

k+3
S
5
H

k+4
S
7
H
EHPSS
S
0
=X(1)
X(2)
X(3)
X(4)
X(1)=MU
X(n)=Thom(
SU(n)!
SU=BU)
X(n)=X(n;0)
X(n;1)
X(n;2)
X(n;2)
X(n;1)=X(n+1)
X(n;k) from the James ltration on
(SU(n+1)=SU(n))=
S
2n+1
=JS
2n
Nilpotence ltration
Koszul ltration
S
0
=SP
1
(S
0
)
SP
2
(S
0
)
SP
3
(S
0
)
SP
4
(S
0
)
SP
1
(S
0
)=H
SP
n
(S
0
)= prespectrum with k'th space (S
k
)
n
=
n
S
0
=SP
1
(S
0
)
SP
p
(S
0
)
SP
p
2
(S
0
)
SP
p
3
(S
0
)
SP
1
(S
0
)=H
SP
n
(S
0
)= prespectrum with k'th space (S
k
)
n
=
n
S
0
=SP
1
(S
0
)
L(0)
SP
p
(S
0
)
L(1)
SP
p
2
(S
0
)

2
L(2)
SP
p
3
(S
0
)

3
L(3)
SP
1
(S
0
)=H

1
L() is a DGA up to homotopy,chain equivalent to Z (Whitehead,Kuhn,Priddy)
Symmetric power ltration
S
MU
MU
MU^
MU
MU
(2)
MU^
MU
(2)
MU
(3)
MU^
MU
(3)
MU
(4)
Algebraic NSS
Unstable Adams SS,Lambda algebra,central series for simplicial groups
X
1
=
S
1
=Z
QS
0
X
2
X
3

1
Q(2)
h
2
X
4

1
Q(3)
h
3
X
5

1
Q(4)
h
4
Q(n) is a certain nite 
n
-spectrum,with H

Q(n) = H
0
Q(n) = Lie(n).
It is dened using combinatorics of partitions of n points,and is related to SP
n
(S
0
)=SP
n1
(S
0
)
There is a similar tower for
S
k+1
,with bres

1
(S
nk
^ Q(n))
h
n
(Goodwillie,Johnson,Arone,Mahowald)
EHPSS
Goodwillie tower
Algebraic NSS
Unstable Adams SS,Lambda algebra,central series for simplicial groups
KU

(S
0
)
Ell

(S
0
)
K(n)

(S
0
)

(L
K(n)
S
0
)
v
1
n

(S

)
Bouseld-Kuhn

n
:Spaces!Spectra 
n
(

1
X) = L
K(n)
(X)
Overview of homotopy theory

(S

)
H

(S
0
)

S

(S
0
)
MU

(S
0
)
Formal groups

k+1
S
1
E

k+2
S
2
E

k+3
S
3
E

k+4
S
4

k
(QS
0
)=
S
k
(S
0
)

k+2
S
3
H

k+3
S
5
H

k+4
S
7
H
EHPSS
S
0
=X(1)
X(2)
X(3)
X(4)
X(1)=MU
X(n)=Thom(
SU(n)!
SU=BU)
X(n)=X(n;0)
X(n;1)
X(n;2)
X(n;2)
X(n;1)=X(n+1)
X(n;k) from the James ltration on
(SU(n+1)=SU(n))=
S
2n+1
=JS
2n
Nilpotence ltration
Koszul ltration
S
0
=SP
1
(S
0
)
SP
2
(S
0
)
SP
3
(S
0
)
SP
4
(S
0
)
SP
1
(S
0
)=H
SP
n
(S
0
)= prespectrum with k'th space (S
k
)
n
=
n
S
0
=SP
1
(S
0
)
SP
p
(S
0
)
SP
p
2
(S
0
)
SP
p
3
(S
0
)
SP
1
(S
0
)=H
SP
n
(S
0
)= prespectrum with k'th space (S
k
)
n
=
n
S
0
=SP
1
(S
0
)
L(0)
SP
p
(S
0
)
L(1)
SP
p
2
(S
0
)

2
L(2)
SP
p
3
(S
0
)

3
L(3)
SP
1
(S
0
)=H

1
L() is a DGA up to homotopy,chain equivalent to Z (Whitehead,Kuhn,Priddy)
Symmetric power ltration
S
MU
MU
MU^
MU
MU
(2)
MU^
MU
(2)
MU
(3)
MU^
MU
(3)
MU
(4)
Algebraic NSS
Unstable Adams SS,Lambda algebra,central series for simplicial groups
X
1
=
S
1
=Z
QS
0
X
2
X
3

1
Q(2)
h
2
X
4

1
Q(3)
h
3
X
5

1
Q(4)
h
4
Q(n) is a certain nite 
n
-spectrum,with H

Q(n) = H
0
Q(n) = Lie(n).
It is dened using combinatorics of partitions of n points,and is related to SP
n
(S
0
)=SP
n1
(S
0
)
There is a similar tower for
S
k+1
,with bres

1
(S
nk
^ Q(n))
h
n
(Goodwillie,Johnson,Arone,Mahowald)
EHPSS
Goodwillie tower
Algebraic NSS
Unstable Adams SS,Lambda algebra,central series for simplicial groups
KU

(S
0
)
Ell

(S
0
)
K(n)

(S
0
)

(L
K(n)
S
0
)
v
1
n

(S

)
Bouseld-Kuhn

n
:Spaces!Spectra 
n
(

1
X) = L
K(n)
(X)
Overview of homotopy theory

(S

)
H

(S
0
)

S

(S
0
)
MU

(S
0
)
Formal groups

k+1
S
1
E

k+2
S
2
E

k+3
S
3
E

k+4
S
4

k
(QS
0
)=
S
k
(S
0
)

k+2
S
3
H

k+3
S
5
H

k+4
S
7
H
EHPSS
S
0
=X(1)
X(2)
X(3)
X(4)
X(1)=MU
X(n)=Thom(
SU(n)!
SU=BU)
X(n)=X(n;0)
X(n;1)
X(n;2)
X(n;2)
X(n;1)=X(n+1)
X(n;k) from the James ltration on
(SU(n+1)=SU(n))=
S
2n+1
=JS
2n
Nilpotence ltration
Koszul ltration
S
0
=SP
1
(S
0
)
SP
2
(S
0
)
SP
3
(S
0
)
SP
4
(S
0
)
SP
1
(S
0
)=H
SP
n
(S
0
)= prespectrum with k'th space (S
k
)
n
=
n
S
0
=SP
1
(S
0
)
SP
p
(S
0
)
SP
p
2
(S
0
)
SP
p
3
(S
0
)
SP
1
(S
0
)=H
SP
n
(S
0
)= prespectrum with k'th space (S
k
)
n
=
n
S
0
=SP
1
(S
0
)
L(0)
SP
p
(S
0
)
L(1)
SP
p
2
(S
0
)

2
L(2)
SP
p
3
(S
0
)

3
L(3)
SP
1
(S
0
)=H

1
L() is a DGA up to homotopy,chain equivalent to Z (Whitehead,Kuhn,Priddy)
Symmetric power ltration
S
MU
MU
MU^
MU
MU
(2)
MU^
MU
(2)
MU
(3)
MU^
MU
(3)
MU
(4)
Algebraic NSS
Unstable Adams SS,Lambda algebra,central series for simplicial groups
X
1
=
S
1
=Z
QS
0
X
2
X
3

1
Q(2)
h
2
X
4

1
Q(3)
h
3
X
5

1
Q(4)
h
4
Q(n) is a certain nite 
n
-spectrum,with H

Q(n) = H
0
Q(n) = Lie(n).
It is dened using combinatorics of partitions of n points,and is related to SP
n
(S
0
)=SP
n1
(S
0
)
There is a similar tower for
S
k+1
,with bres

1
(S
nk
^ Q(n))
h
n
(Goodwillie,Johnson,Arone,Mahowald)
EHPSS
Goodwillie tower
Algebraic NSS
Unstable Adams SS,Lambda algebra,central series for simplicial groups
KU

(S
0
)
Ell

(S
0
)
K(n)

(S
0
)

(L
K(n)
S
0
)
v
1
n

(S

)
Bouseld-Kuhn

n
:Spaces!Spectra 
n
(

1
X) = L
K(n)
(X)
Overview of homotopy theory

(S

)
H

(S
0
)

S

(S
0
)
MU

(S
0
)
Formal groups

k+1
S
1
E

k+2
S
2
E

k+3
S
3
E

k+4
S
4

k
(QS
0
)=
S
k
(S
0
)

k+2
S
3
H

k+3
S
5
H

k+4
S
7
H
EHPSS
S
0
=X(1)
X(2)
X(3)
X(4)
X(1)=MU
X(n)=Thom(
SU(n)!
SU=BU)
X(n)=X(n;0)
X(n;1)
X(n;2)
X(n;2)
X(n;1)=X(n+1)
X(n;k) from the James ltration on
(SU(n+1)=SU(n))=
S
2n+1
=JS
2n
Nilpotence ltration
Koszul ltration
S
0
=SP
1
(S
0
)
SP
2
(S
0
)
SP
3
(S
0
)
SP
4
(S
0
)
SP
1
(S
0
)=H
SP
n
(S
0
)= prespectrum with k'th space (S
k
)
n
=
n
S
0
=SP
1
(S
0
)
SP
p
(S
0
)
SP
p
2
(S
0
)
SP
p
3
(S
0
)
SP
1
(S
0
)=H
SP
n
(S
0
)= prespectrum with k'th space (S
k
)
n
=
n
S
0
=SP
1
(S
0
)
L(0)
SP
p
(S
0
)
L(1)
SP
p
2
(S
0
)

2
L(2)
SP
p
3
(S
0
)

3
L(3)
SP
1
(S
0
)=H

1
L() is a DGA up to homotopy,chain equivalent to Z (Whitehead,Kuhn,Priddy)
Symmetric power ltration
S
MU
MU
MU^
MU
MU
(2)
MU^
MU
(2)
MU
(3)
MU^
MU
(3)
MU
(4)
Algebraic NSS
Unstable Adams SS,Lambda algebra,central series for simplicial groups
X
1
=
S
1
=Z
QS
0
X
2
X
3

1
Q(2)
h
2
X
4

1
Q(3)
h
3
X
5

1
Q(4)
h
4
Q(n) is a certain nite 
n
-spectrum,with H

Q(n) = H
0
Q(n) = Lie(n).
It is dened using combinatorics of partitions of n points,and is related to SP
n
(S
0
)=SP
n1
(S
0
)
There is a similar tower for
S
k+1
,with bres

1
(S
nk
^ Q(n))
h
n
(Goodwillie,Johnson,Arone,Mahowald)
EHPSS
Goodwillie tower
Algebraic NSS
Unstable Adams SS,Lambda algebra,central series for simplicial groups
KU

(S
0
)
Ell

(S
0
)
K(n)

(S
0
)

(L
K(n)
S
0
)
v
1
n

(S

)
Bouseld-Kuhn

n
:Spaces!Spectra 
n
(

1
X) = L
K(n)
(X)
Overview of homotopy theory

(S

)
H

(S
0
)

S

(S
0
)
MU

(S
0
)
Formal groups

k+1
S
1
E

k+2
S
2
E

k+3
S
3
E

k+4
S
4

k
(QS
0
)=
S
k
(S
0
)

k+2
S
3
H

k+3
S
5
H

k+4
S
7
H
EHPSS
S
0
=X(1)
X(2)
X(3)
X(4)
X(1)=MU
X(n)=Thom(
SU(n)!
SU=BU)
X(n)=X(n;0)
X(n;1)
X(n;2)
X(n;2)
X(n;1)=X(n+1)
X(n;k) from the James ltration on
(SU(n+1)=SU(n))=
S
2n+1
=JS
2n
Nilpotence ltration
Koszul ltration
S
0
=SP
1
(S
0
)
SP
2
(S
0
)
SP
3
(S
0
)
SP
4
(S
0
)
SP
1
(S
0
)=H
SP
n
(S
0
)= prespectrum with k'th space (S
k
)
n
=
n
S
0
=SP
1
(S
0
)
SP
p
(S
0
)
SP
p
2
(S
0
)
SP
p
3
(S
0
)
SP
1
(S
0
)=H
SP
n
(S
0
)= prespectrum with k'th space (S
k
)
n
=
n
S
0
=SP
1
(S
0
)
L(0)
SP
p
(S
0
)
L(1)
SP
p
2
(S
0
)

2
L(2)
SP
p
3
(S
0
)

3
L(3)
SP
1
(S
0
)=H

1
L() is a DGA up to homotopy,chain equivalent to Z (Whitehead,Kuhn,Priddy)
Symmetric power ltration
S
MU
MU
MU^
MU
MU
(2)
MU^
MU
(2)
MU
(3)
MU^
MU
(3)
MU
(4)
Algebraic NSS
Unstable Adams SS,Lambda algebra,central series for simplicial groups
X
1
=
S
1
=Z
QS
0
X
2
X
3

1
Q(2)
h
2
X
4

1
Q(3)
h
3
X
5

1
Q(4)
h
4
Q(n) is a certain nite 
n
-spectrum,with H

Q(n) = H
0
Q(n) = Lie(n).
It is dened using combinatorics of partitions of n points,and is related to SP
n
(S
0
)=SP
n1
(S
0
)
There is a similar tower for
S
k+1
,with bres

1
(S
nk
^ Q(n))
h
n
(Goodwillie,Johnson,Arone,Mahowald)
EHPSS
Goodwillie tower
Algebraic NSS
Unstable Adams SS,Lambda algebra,central series for simplicial groups
KU

(S
0
)
Ell

(S
0
)
K(n)

(S
0
)

(L
K(n)
S
0
)
v
1
n

(S

)
Bouseld-Kuhn

n
:Spaces!Spectra 
n
(

1
X) = L
K(n)
(X)
Overview of homotopy theory

(S

)
H

(S
0
)

S

(S
0
)
MU

(S
0
)
Formal groups

k+1
S
1
E

k+2
S
2
E

k+3
S
3
E

k+4
S
4

k
(QS
0
)=
S
k
(S
0
)

k+2
S
3
H

k+3
S
5
H

k+4
S
7
H
EHPSS
S
0
=X(1)
X(2)
X(3)
X(4)
X(1)=MU
X(n)=Thom(
SU(n)!
SU=BU)
X(n)=X(n;0)
X(n;1)
X(n;2)
X(n;2)
X(n;1)=X(n+1)
X(n;k) from the James ltration on
(SU(n+1)=SU(n))=
S
2n+1
=JS
2n
Nilpotence ltration
Koszul ltration
S
0
=SP
1
(S
0
)
SP
2
(S
0
)
SP
3
(S
0
)
SP
4
(S
0
)
SP
1
(S
0
)=H
SP
n
(S
0
)= prespectrum with k'th space (S
k
)
n
=
n
S
0
=SP
1
(S
0
)
SP
p
(S
0
)
SP
p
2
(S
0
)
SP
p
3
(S
0
)
SP
1
(S
0
)=H
SP
n
(S
0
)= prespectrum with k'th space (S
k
)
n
=
n
S
0
=SP
1
(S
0
)
L(0)
SP
p
(S
0
)
L(1)
SP
p
2
(S
0
)

2
L(2)
SP
p
3
(S
0
)

3
L(3)
SP
1
(S
0
)=H

1
L() is a DGA up to homotopy,chain equivalent to Z (Whitehead,Kuhn,Priddy)
Symmetric power ltration
S
MU
MU
MU^
MU
MU
(2)
MU^
MU
(2)
MU
(3)
MU^
MU
(3)
MU
(4)
Algebraic NSS
Unstable Adams SS,Lambda algebra,central series for simplicial groups
X
1
=
S
1
=Z
QS
0
X
2
X
3

1
Q(2)
h
2
X
4

1
Q(3)
h
3
X
5

1
Q(4)
h
4
Q(n) is a certain nite 
n
-spectrum,with H

Q(n) = H
0
Q(n) = Lie(n).
It is dened using combinatorics of partitions of n points,and is related to SP
n
(S
0
)=SP
n1
(S
0
)
There is a similar tower for
S
k+1
,with bres

1
(S
nk
^ Q(n))
h
n
(Goodwillie,Johnson,Arone,Mahowald)
EHPSS
Goodwillie tower
Algebraic NSS
Unstable Adams SS,Lambda algebra,central series for simplicial groups
KU

(S
0
)
Ell

(S
0
)
K(n)

(S
0
)

(L
K(n)
S
0
)
v
1
n

(S

)
Bouseld-Kuhn

n
:Spaces!Spectra 
n
(

1
X) = L
K(n)
(X)
Overview of homotopy theory

(S

)
H

(S
0
)

S

(S
0
)
MU

(S
0
)
Formal groups

k+1
S
1
E

k+2
S
2
E

k+3
S
3
E

k+4
S
4

k
(QS
0
)=
S
k
(S
0
)

k+2
S
3
H

k+3
S
5
H

k+4
S
7
H
EHPSS
S
0
=X(1)
X(2)
X(3)
X(4)
X(1)=MU
X(n)=Thom(
SU(n)!
SU=BU)
X(n)=X(n;0)
X(n;1)
X(n;2)
X(n;2)
X(n;1)=X(n+1)
X(n;k) from the James ltration on
(SU(n+1)=SU(n))=
S
2n+1
=JS
2n
Nilpotence ltration
Koszul ltration
S
0
=SP
1
(S
0
)
SP
2
(S
0
)
SP
3
(S
0
)
SP
4
(S
0
)
SP
1
(S
0
)=H
SP
n
(S
0
)= prespectrum with k'th space (S
k
)
n
=
n
S
0
=SP
1
(S
0
)
SP
p
(S
0
)
SP
p
2
(S
0
)
SP
p
3
(S
0
)
SP
1
(S
0
)=H
SP
n
(S
0
)= prespectrum with k'th space (S
k
)
n
=
n
S
0
=SP
1
(S
0
)
L(0)
SP
p
(S
0
)
L(1)
SP
p
2
(S
0
)

2
L(2)
SP
p
3
(S
0
)

3
L(3)
SP
1
(S
0
)=H

1
L() is a DGA up to homotopy,chain equivalent to Z (Whitehead,Kuhn,Priddy)
Symmetric power ltration
S
MU
MU
MU^
MU
MU
(2)
MU^
MU
(2)
MU
(3)
MU^
MU
(3)
MU
(4)
Algebraic NSS
Unstable Adams SS,Lambda algebra,central series for simplicial groups
X
1
=
S
1
=Z
QS
0
X
2
X
3

1
Q(2)
h
2
X
4

1
Q(3)
h
3
X
5

1
Q(4)
h
4
Q(n) is a certain nite 
n
-spectrum,with H

Q(n) = H
0
Q(n) = Lie(n)