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)

Adams-Novikov SS

Algebraic NSS

Adams SS

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 dened using combinatorics of partitions of n points,and is related to SP

n

(S

0

)=SP

n1

(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

Adams-Novikov SS

Algebraic NSS

Adams SS

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

)

Bouseld-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)

Adams-Novikov SS

Algebraic NSS

Adams SS

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 dened using combinatorics of partitions of n points,and is related to SP

n

(S

0

)=SP

n1

(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

Adams-Novikov SS

Algebraic NSS

Adams SS

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

)

Bouseld-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)

Adams-Novikov SS

Algebraic NSS

Adams SS

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 dened using combinatorics of partitions of n points,and is related to SP

n

(S

0

)=SP

n1

(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

Adams-Novikov SS

Algebraic NSS

Adams SS

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

)

Bouseld-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)

Adams-Novikov SS

Algebraic NSS

Adams SS

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 dened using combinatorics of partitions of n points,and is related to SP

n

(S

0

)=SP

n1

(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

Adams-Novikov SS

Algebraic NSS

Adams SS

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

)

Bouseld-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)

Adams-Novikov SS

Algebraic NSS

Adams SS

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 dened using combinatorics of partitions of n points,and is related to SP

n

(S

0

)=SP

n1

(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

Adams-Novikov SS

Algebraic NSS

Adams SS

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

)

Bouseld-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)

Adams-Novikov SS

Algebraic NSS

Adams SS

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 dened using combinatorics of partitions of n points,and is related to SP

n

(S

0

)=SP

n1

(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

Adams-Novikov SS

Algebraic NSS

Adams SS

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

)

Bouseld-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)

Adams-Novikov SS

Algebraic NSS

Adams SS

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 dened using combinatorics of partitions of n points,and is related to SP

n

(S

0

)=SP

n1

(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

Adams-Novikov SS

Algebraic NSS

Adams SS

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

)

Bouseld-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)

Adams-Novikov SS

Algebraic NSS

Adams SS

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 dened using combinatorics of partitions of n points,and is related to SP

n

(S

0

)=SP

n1

(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

Adams-Novikov SS

Algebraic NSS

Adams SS

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

)

Bouseld-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)

Adams-Novikov SS

Algebraic NSS

Adams SS

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 dened using combinatorics of partitions of n points,and is related to SP

n

(S

0

)=SP

n1

(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

Adams-Novikov SS

Algebraic NSS

Adams SS

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

)

Bouseld-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)

Adams-Novikov SS

Algebraic NSS

Adams SS

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 dened using combinatorics of partitions of n points,and is related to SP

n

(S

0

)=SP

n1

(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

Adams-Novikov SS

Algebraic NSS

Adams SS

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

)

Bouseld-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)

Adams-Novikov SS

Algebraic NSS

Adams SS

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 dened using combinatorics of partitions of n points,and is related to SP

n

(S

0

)=SP

n1

(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

Adams-Novikov SS

Algebraic NSS

Adams SS

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

)

Bouseld-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)

Adams-Novikov SS

Algebraic NSS

Adams SS

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 dened using combinatorics of partitions of n points,and is related to SP

n

(S

0

)=SP

n1

(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

Adams-Novikov SS

Algebraic NSS

Adams SS

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

)

Bouseld-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)

Adams-Novikov SS

Algebraic NSS

Adams SS

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 dened using combinatorics of partitions of n points,and is related to SP

n

(S

0

)=SP

n1

(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

Adams-Novikov SS

Algebraic NSS

Adams SS

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

)

Bouseld-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)

Adams-Novikov SS

Algebraic NSS

Adams SS

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 dened using combinatorics of partitions of n points,and is related to SP

n

(S

0

)=SP

n1

(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

Adams-Novikov SS

Algebraic NSS

Adams SS

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

)

Bouseld-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)

Adams-Novikov SS

Algebraic NSS

Adams SS

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 dened using combinatorics of partitions of n points,and is related to SP

n

(S

0

)=SP

n1

(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

Adams-Novikov SS

Algebraic NSS

Adams SS

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

)

Bouseld-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)

Adams-Novikov SS

Algebraic NSS

Adams SS

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 dened using combinatorics of partitions of n points,and is related to SP

n

(S

0

)=SP

n1

(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

Adams-Novikov SS

Algebraic NSS

Adams SS

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

)

Bouseld-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)

Adams-Novikov SS

Algebraic NSS

Adams SS

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 dened using combinatorics of partitions of n points,and is related to SP

n

(S

0

)=SP

n1

(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

Adams-Novikov SS

Algebraic NSS

Adams SS

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

)

Bouseld-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)

Adams-Novikov SS

Algebraic NSS

Adams SS

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 dened using combinatorics of partitions of n points,and is related to SP

n

(S

0

)=SP

n1

(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

Adams-Novikov SS

Algebraic NSS

Adams SS

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

)

Bouseld-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)

Adams-Novikov SS

Algebraic NSS

Adams SS

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 dened using combinatorics of partitions of n points,and is related to SP

n

(S

0

)=SP

n1

(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

Adams-Novikov SS

Algebraic NSS

Adams SS

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

)

Bouseld-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)

Adams-Novikov SS

Algebraic NSS

Adams SS

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 dened using combinatorics of partitions of n points,and is related to SP

n

(S

0

)=SP

n1

(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

Adams-Novikov SS

Algebraic NSS

Adams SS

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

)

Bouseld-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)

Adams-Novikov SS

Algebraic NSS

Adams SS

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 dened using combinatorics of partitions of n points,and is related to SP

n

(S

0

)=SP

n1

(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

Adams-Novikov SS

Algebraic NSS

Adams SS

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

)

Bouseld-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)

Adams-Novikov SS

Algebraic NSS

Adams SS

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 dened using combinatorics of partitions of n points,and is related to SP

n

(S

0

)=SP

n1

(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

Adams-Novikov SS

Algebraic NSS

Adams SS

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

)

Bouseld-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)

Adams-Novikov SS

Algebraic NSS

Adams SS

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 dened using combinatorics of partitions of n points,and is related to SP

n

(S

0

)=SP

n1

(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

Adams-Novikov SS

Algebraic NSS

Adams SS

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

)

Bouseld-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)

Adams-Novikov SS

Algebraic NSS

Adams SS

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 dened using combinatorics of partitions of n points,and is related to SP

n

(S

0

)=SP

n1

(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

Adams-Novikov SS

Algebraic NSS

Adams SS

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

)

Bouseld-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)

Adams-Novikov SS

Algebraic NSS

Adams SS

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 dened using combinatorics of partitions of n points,and is related to SP

n

(S

0

)=SP

n1

(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

Adams-Novikov SS

Algebraic NSS

Adams SS

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

)

Bouseld-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)

Adams-Novikov SS

Algebraic NSS

Adams SS

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 dened using combinatorics of partitions of n points,and is related to SP

n

(S

0

)=SP

n1

(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

Adams-Novikov SS

Algebraic NSS

Adams SS

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

)

Bouseld-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)

Adams-Novikov SS

Algebraic NSS

Adams SS

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 dened using combinatorics of partitions of n points,and is related to SP

n

(S

0

)=SP

n1

(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

Adams-Novikov SS

Algebraic NSS

Adams SS

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

)

Bouseld-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)

Adams-Novikov SS

Algebraic NSS

Adams SS

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 dened using combinatorics of partitions of n points,and is related to SP

n

(S

0

)=SP

n1

(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

Adams-Novikov SS

Algebraic NSS

Adams SS

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

)

Bouseld-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)

Adams-Novikov SS

Algebraic NSS

Adams SS

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)

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