The periodicity theorem in the solution to the Arf-Kervaire - PowerPoint PPT Presentation

The periodicity The spectrum M theorem Mike Hill Mike Hopkins As explained previously, there is an action of the cyclic group Doug Ravenel C 8 on the 4-fold smash product MU ( 4 ) . It is derived using a norm induction from the action of C 2 on MU by complex conjugation. We show that its homotopy fixed point set ( MU ( 4 ) ) hC 8 and its actual fixed point set ( MU ( 4 ) ) C 8 are equivalent. It is an E ∞ -ring Our strategy spectrum, and M is obtained from it by inverting an element The spectrum M D ∈ π 256 which we will identify below. The slice spectral sequence S m ρ 8 ∧ H Z The homotopy of ( MU ( 4 ) ) hC 8 can be computed using the Implications homotopy fixed point spectral sequence, for which Geometric fixed points Some slice differentials E 2 = H ∗ ( C 8 ; π ∗ ( MU ( 4 ) )) The proof In this case it concides with the Adams-Novikov spectral sequence for π ∗ (( MU ( 4 ) ) hC 8 ) 1.4

The periodicity The spectrum M theorem Mike Hill Mike Hopkins As explained previously, there is an action of the cyclic group Doug Ravenel C 8 on the 4-fold smash product MU ( 4 ) . It is derived using a norm induction from the action of C 2 on MU by complex conjugation. We show that its homotopy fixed point set ( MU ( 4 ) ) hC 8 and its actual fixed point set ( MU ( 4 ) ) C 8 are equivalent. It is an E ∞ -ring Our strategy spectrum, and M is obtained from it by inverting an element The spectrum M D ∈ π 256 which we will identify below. The slice spectral sequence S m ρ 8 ∧ H Z The homotopy of ( MU ( 4 ) ) hC 8 can be computed using the Implications homotopy fixed point spectral sequence, for which Geometric fixed points Some slice differentials E 2 = H ∗ ( C 8 ; π ∗ ( MU ( 4 ) )) The proof In this case it concides with the Adams-Novikov spectral sequence for π ∗ (( MU ( 4 ) ) hC 8 ) The algebraic methods described by Hopkins can be used to show that it detects the θ j s. 1.4

The periodicity The spectrum M theorem Mike Hill Mike Hopkins As explained previously, there is an action of the cyclic group Doug Ravenel C 8 on the 4-fold smash product MU ( 4 ) . It is derived using a norm induction from the action of C 2 on MU by complex conjugation. We show that its homotopy fixed point set ( MU ( 4 ) ) hC 8 and its actual fixed point set ( MU ( 4 ) ) C 8 are equivalent. It is an E ∞ -ring Our strategy spectrum, and M is obtained from it by inverting an element The spectrum M D ∈ π 256 which we will identify below. The slice spectral sequence S m ρ 8 ∧ H Z The homotopy of ( MU ( 4 ) ) hC 8 can be computed using the Implications homotopy fixed point spectral sequence, for which Geometric fixed points Some slice differentials E 2 = H ∗ ( C 8 ; π ∗ ( MU ( 4 ) )) The proof In this case it concides with the Adams-Novikov spectral sequence for π ∗ (( MU ( 4 ) ) hC 8 ) The algebraic methods described by Hopkins can be used to show that it detects the θ j s. D has to be chosen so that this is still true after we invert it. 1.4

The periodicity The spectrum M (continued) theorem Mike Hill Mike Hopkins Doug Ravenel Our strategy The spectrum M The slice spectral sequence S m ρ 8 ∧ H Z Implications Geometric fixed points Some slice differentials The proof 1.5

The periodicity The spectrum M (continued) theorem Mike Hill Mike Hopkins Doug Ravenel The homotopy of ( MU ( 4 ) ) C 8 and M = D − 1 ( MU ( 4 ) ) C 8 can be also computed using the slice spectral sequence described by Hill. Our strategy The spectrum M The slice spectral sequence S m ρ 8 ∧ H Z Implications Geometric fixed points Some slice differentials The proof 1.5

The periodicity The spectrum M (continued) theorem Mike Hill Mike Hopkins Doug Ravenel The homotopy of ( MU ( 4 ) ) C 8 and M = D − 1 ( MU ( 4 ) ) C 8 can be also computed using the slice spectral sequence described by Hill. It has the convenient property that π − 2 vanishes in the E 2 -term. Our strategy The spectrum M The slice spectral sequence S m ρ 8 ∧ H Z Implications Geometric fixed points Some slice differentials The proof 1.5

The periodicity The spectrum M (continued) theorem Mike Hill Mike Hopkins Doug Ravenel The homotopy of ( MU ( 4 ) ) C 8 and M = D − 1 ( MU ( 4 ) ) C 8 can be also computed using the slice spectral sequence described by Hill. It has the convenient property that π − 2 vanishes in the E 2 -term. In fact π k vanishes for − 4 < k < 0. Our strategy The spectrum M The slice spectral sequence S m ρ 8 ∧ H Z Implications Geometric fixed points Some slice differentials The proof 1.5

The periodicity The spectrum M (continued) theorem Mike Hill Mike Hopkins Doug Ravenel The homotopy of ( MU ( 4 ) ) C 8 and M = D − 1 ( MU ( 4 ) ) C 8 can be also computed using the slice spectral sequence described by Hill. It has the convenient property that π − 2 vanishes in the E 2 -term. In fact π k vanishes for − 4 < k < 0. Our strategy The spectrum M This is our main motivation for developing the slice spectral The slice spectral sequence. sequence S m ρ 8 ∧ H Z Implications Geometric fixed points Some slice differentials The proof 1.5

The periodicity The spectrum M (continued) theorem Mike Hill Mike Hopkins Doug Ravenel The homotopy of ( MU ( 4 ) ) C 8 and M = D − 1 ( MU ( 4 ) ) C 8 can be also computed using the slice spectral sequence described by Hill. It has the convenient property that π − 2 vanishes in the E 2 -term. In fact π k vanishes for − 4 < k < 0. Our strategy The spectrum M This is our main motivation for developing the slice spectral The slice spectral sequence. We do not know how to show this vanishing using sequence S m ρ 8 ∧ H Z the other spectral sequence. Implications Geometric fixed points Some slice differentials The proof 1.5

The periodicity The spectrum M (continued) theorem Mike Hill Mike Hopkins Doug Ravenel The homotopy of ( MU ( 4 ) ) C 8 and M = D − 1 ( MU ( 4 ) ) C 8 can be also computed using the slice spectral sequence described by Hill. It has the convenient property that π − 2 vanishes in the E 2 -term. In fact π k vanishes for − 4 < k < 0. Our strategy The spectrum M This is our main motivation for developing the slice spectral The slice spectral sequence. We do not know how to show this vanishing using sequence S m ρ 8 ∧ H Z the other spectral sequence. Implications Geometric fixed points In order to identify D we need to study the slice spectral Some slice differentials sequence in more detail. The proof 1.5

� � � The periodicity The slice spectral sequence theorem Mike Hill Mike Hopkins Doug Ravenel Recall that for G = C 8 we have a slice tower . . . � P n + 1 � P n � P n − 1 � . . . MU ( 4 ) G MU ( 4 ) MU ( 4 ) G G G P n + 1 n + 1 MU ( 4 ) G P n − 1 n − 1 MU ( 4 ) G P n n MU ( 4 ) in which Our strategy The spectrum M The slice spectral sequence S m ρ 8 ∧ H Z Implications Geometric fixed points Some slice differentials The proof 1.6

� � � The periodicity The slice spectral sequence theorem Mike Hill Mike Hopkins Doug Ravenel Recall that for G = C 8 we have a slice tower . . . � P n + 1 � P n � P n − 1 � . . . MU ( 4 ) G MU ( 4 ) MU ( 4 ) G G G P n + 1 n + 1 MU ( 4 ) G P n − 1 n − 1 MU ( 4 ) G P n n MU ( 4 ) in which Our strategy The spectrum M • the inverse limit is MU ( 4 ) , The slice spectral sequence S m ρ 8 ∧ H Z Implications Geometric fixed points Some slice differentials The proof 1.6

� � � The periodicity The slice spectral sequence theorem Mike Hill Mike Hopkins Doug Ravenel Recall that for G = C 8 we have a slice tower . . . � P n + 1 � P n � P n − 1 � . . . MU ( 4 ) G MU ( 4 ) MU ( 4 ) G G G P n + 1 n + 1 MU ( 4 ) G P n − 1 n − 1 MU ( 4 ) G P n n MU ( 4 ) in which Our strategy The spectrum M • the inverse limit is MU ( 4 ) , The slice spectral sequence • the direct limit is contractible and S m ρ 8 ∧ H Z Implications Geometric fixed points Some slice differentials The proof 1.6

� � � The periodicity The slice spectral sequence theorem Mike Hill Mike Hopkins Doug Ravenel Recall that for G = C 8 we have a slice tower . . . � P n + 1 � P n � P n − 1 � . . . MU ( 4 ) G MU ( 4 ) MU ( 4 ) G G G P n + 1 n + 1 MU ( 4 ) G P n − 1 n − 1 MU ( 4 ) G P n n MU ( 4 ) in which Our strategy The spectrum M • the inverse limit is MU ( 4 ) , The slice spectral sequence • the direct limit is contractible and S m ρ 8 ∧ H Z n MU ( 4 ) is the fiber of the map P n G MU ( 4 ) → P n − 1 • G P n MU ( 4 ) . Implications G Geometric fixed points Some slice differentials The proof 1.6

� � � The periodicity The slice spectral sequence theorem Mike Hill Mike Hopkins Doug Ravenel Recall that for G = C 8 we have a slice tower . . . � P n + 1 � P n � P n − 1 � . . . MU ( 4 ) G MU ( 4 ) MU ( 4 ) G G G P n + 1 n + 1 MU ( 4 ) G P n − 1 n − 1 MU ( 4 ) G P n n MU ( 4 ) in which Our strategy The spectrum M • the inverse limit is MU ( 4 ) , The slice spectral sequence • the direct limit is contractible and S m ρ 8 ∧ H Z n MU ( 4 ) is the fiber of the map P n G MU ( 4 ) → P n − 1 • G P n MU ( 4 ) . Implications G Geometric fixed points Some slice differentials n MU ( 4 ) is the nth slice and the decreasing sequence of G P n The proof subgroups of π ∗ ( MU ( 4 ) ) is the slice filtration. 1.6

� � � The periodicity The slice spectral sequence theorem Mike Hill Mike Hopkins Doug Ravenel Recall that for G = C 8 we have a slice tower . . . � P n + 1 � P n � P n − 1 � . . . MU ( 4 ) G MU ( 4 ) MU ( 4 ) G G G P n + 1 n + 1 MU ( 4 ) G P n − 1 n − 1 MU ( 4 ) G P n n MU ( 4 ) in which Our strategy The spectrum M • the inverse limit is MU ( 4 ) , The slice spectral sequence • the direct limit is contractible and S m ρ 8 ∧ H Z n MU ( 4 ) is the fiber of the map P n G MU ( 4 ) → P n − 1 • G P n MU ( 4 ) . Implications G Geometric fixed points Some slice differentials n MU ( 4 ) is the nth slice and the decreasing sequence of G P n The proof subgroups of π ∗ ( MU ( 4 ) ) is the slice filtration. We also get slice filtrations of the RO ( G ) -graded homotopy π ⋆ ( MU ( 4 ) ) and the homotopy groups of fixed point sets π ∗ (( MU ( 4 ) ) H ) for each subgroup H . 1.6

The periodicity The slice spectral sequence (continued) theorem This means the slice filtration leads to a slice spectral Mike Hill Mike Hopkins sequence converging to π ∗ ( MU ( 4 ) ) and its variants. Doug Ravenel Our strategy The spectrum M The slice spectral sequence S m ρ 8 ∧ H Z Implications Geometric fixed points Some slice differentials The proof 1.7

The periodicity The slice spectral sequence (continued) theorem This means the slice filtration leads to a slice spectral Mike Hill Mike Hopkins sequence converging to π ∗ ( MU ( 4 ) ) and its variants. Doug Ravenel One variant has the form E s , t = π G t − s ( G P t t MU ( 4 ) ) = ⇒ π G t − s ( MU ( 4 ) ) . 2 ∗ ( MU ( 4 ) ) is by definition π ∗ (( MU ( 4 ) ) G ) , the Recall that π G homotopy of the fixed point set. Our strategy The spectrum M The slice spectral sequence S m ρ 8 ∧ H Z Implications Geometric fixed points Some slice differentials The proof 1.7

The periodicity The slice spectral sequence (continued) theorem This means the slice filtration leads to a slice spectral Mike Hill Mike Hopkins sequence converging to π ∗ ( MU ( 4 ) ) and its variants. Doug Ravenel One variant has the form E s , t = π G t − s ( G P t t MU ( 4 ) ) = ⇒ π G t − s ( MU ( 4 ) ) . 2 ∗ ( MU ( 4 ) ) is by definition π ∗ (( MU ( 4 ) ) G ) , the Recall that π G homotopy of the fixed point set. Our strategy The spectrum M Slice Theorem The slice spectral In the slice tower for MU ( 4 ) , every odd slice is contractible and sequence S m ρ 8 ∧ H Z 2 n = ˆ P 2 n W n ∧ H Z , where H Z is the integer Eilenberg-Mac Lane Implications spectrum and ˆ W n is a certain wedge of the following three Geometric fixed points Some slice differentials types of finite G-spectra: The proof 1.7

The periodicity The slice spectral sequence (continued) theorem This means the slice filtration leads to a slice spectral Mike Hill Mike Hopkins sequence converging to π ∗ ( MU ( 4 ) ) and its variants. Doug Ravenel One variant has the form E s , t = π G t − s ( G P t t MU ( 4 ) ) = ⇒ π G t − s ( MU ( 4 ) ) . 2 ∗ ( MU ( 4 ) ) is by definition π ∗ (( MU ( 4 ) ) G ) , the Recall that π G homotopy of the fixed point set. Our strategy The spectrum M Slice Theorem The slice spectral In the slice tower for MU ( 4 ) , every odd slice is contractible and sequence S m ρ 8 ∧ H Z 2 n = ˆ P 2 n W n ∧ H Z , where H Z is the integer Eilenberg-Mac Lane Implications spectrum and ˆ W n is a certain wedge of the following three Geometric fixed points Some slice differentials types of finite G-spectra: The proof • S ( n / 4 ) ρ 8 , where ρ g denotes the regular real representation of C g , 1.7

The periodicity The slice spectral sequence (continued) theorem This means the slice filtration leads to a slice spectral Mike Hill Mike Hopkins sequence converging to π ∗ ( MU ( 4 ) ) and its variants. Doug Ravenel One variant has the form E s , t = π G t − s ( G P t t MU ( 4 ) ) = ⇒ π G t − s ( MU ( 4 ) ) . 2 ∗ ( MU ( 4 ) ) is by definition π ∗ (( MU ( 4 ) ) G ) , the Recall that π G homotopy of the fixed point set. Our strategy The spectrum M Slice Theorem The slice spectral In the slice tower for MU ( 4 ) , every odd slice is contractible and sequence S m ρ 8 ∧ H Z 2 n = ˆ P 2 n W n ∧ H Z , where H Z is the integer Eilenberg-Mac Lane Implications spectrum and ˆ W n is a certain wedge of the following three Geometric fixed points Some slice differentials types of finite G-spectra: The proof • S ( n / 4 ) ρ 8 , where ρ g denotes the regular real representation of C g , • C 8 ∧ C 4 S ( n / 2 ) ρ 4 and 1.7

The periodicity The slice spectral sequence (continued) theorem This means the slice filtration leads to a slice spectral Mike Hill Mike Hopkins sequence converging to π ∗ ( MU ( 4 ) ) and its variants. Doug Ravenel One variant has the form E s , t = π G t − s ( G P t t MU ( 4 ) ) = ⇒ π G t − s ( MU ( 4 ) ) . 2 ∗ ( MU ( 4 ) ) is by definition π ∗ (( MU ( 4 ) ) G ) , the Recall that π G homotopy of the fixed point set. Our strategy The spectrum M Slice Theorem The slice spectral In the slice tower for MU ( 4 ) , every odd slice is contractible and sequence S m ρ 8 ∧ H Z 2 n = ˆ P 2 n W n ∧ H Z , where H Z is the integer Eilenberg-Mac Lane Implications spectrum and ˆ W n is a certain wedge of the following three Geometric fixed points Some slice differentials types of finite G-spectra: The proof • S ( n / 4 ) ρ 8 , where ρ g denotes the regular real representation of C g , • C 8 ∧ C 4 S ( n / 2 ) ρ 4 and • C 8 ∧ C 2 S n ρ 2 . 1.7

The periodicity The slice spectral sequence (continued) theorem This means the slice filtration leads to a slice spectral Mike Hill Mike Hopkins sequence converging to π ∗ ( MU ( 4 ) ) and its variants. Doug Ravenel One variant has the form E s , t = π G t − s ( G P t t MU ( 4 ) ) = ⇒ π G t − s ( MU ( 4 ) ) . 2 ∗ ( MU ( 4 ) ) is by definition π ∗ (( MU ( 4 ) ) G ) , the Recall that π G homotopy of the fixed point set. Our strategy The spectrum M Slice Theorem The slice spectral In the slice tower for MU ( 4 ) , every odd slice is contractible and sequence S m ρ 8 ∧ H Z 2 n = ˆ P 2 n W n ∧ H Z , where H Z is the integer Eilenberg-Mac Lane Implications spectrum and ˆ W n is a certain wedge of the following three Geometric fixed points Some slice differentials types of finite G-spectra: The proof • S ( n / 4 ) ρ 8 , where ρ g denotes the regular real representation of C g , • C 8 ∧ C 4 S ( n / 2 ) ρ 4 and • C 8 ∧ C 2 S n ρ 2 . The same holds after we invert D, in which case negative 1.7 values of n can occur.

Slices of the form S m ρ 8 ∧ H Z The periodicity theorem Mike Hill Here is a picture of some slices S m ρ 8 ∧ H Z . Mike Hopkins Doug Ravenel Our strategy The spectrum M The slice spectral sequence S m ρ 8 ∧ H Z Implications Geometric fixed points Some slice differentials The proof 1.8

Slices of the form S m ρ 8 ∧ H Z The periodicity theorem Mike Hill Here is a picture of some slices S m ρ 8 ∧ H Z . Mike Hopkins Doug Ravenel ◦ 28 ◦ ◦ ⋄ ◦ ⋄ ◦ ⋄ ◦ ⋄ ⋄ ⋄ ⋄ 14 ◦ ⋄ ◦ ⋄ ⋄ ⋄ Our strategy ⋄ ⋄ ⋄ ⋄ The spectrum M − 2 ⋄ ⋄ − 4 0 The slice spectral � � � � � ⋄⋄⋄⋄⋄⋄⋄⋄◦◦◦◦ ⋄⋄⋄⋄◦◦ sequence 0 2 4 S m ρ 8 ∧ H Z Implications Geometric fixed points Some slice differentials − 14 The proof ⋄⋄⋄⋄◦◦ − 28 − 32 − 16 0 16 32 1.8

Slices of the form S m ρ 8 ∧ H Z (continued) The periodicity theorem Mike Hill Mike Hopkins Doug Ravenel • Note that all elements are in the first and third quadrants between certain black lines with slopes 0 and orchid lines with slope 7, Our strategy The spectrum M The slice spectral sequence S m ρ 8 ∧ H Z Implications Geometric fixed points Some slice differentials The proof 1.9

Slices of the form S m ρ 8 ∧ H Z (continued) The periodicity theorem Mike Hill Mike Hopkins Doug Ravenel • Note that all elements are in the first and third quadrants between certain black lines with slopes 0 and orchid lines with slope 7, and are concentrated on diagonals where t is divisible by 8. Our strategy The spectrum M The slice spectral sequence S m ρ 8 ∧ H Z Implications Geometric fixed points Some slice differentials The proof 1.9

Slices of the form S m ρ 8 ∧ H Z (continued) The periodicity theorem Mike Hill Mike Hopkins Doug Ravenel • Note that all elements are in the first and third quadrants between certain black lines with slopes 0 and orchid lines with slope 7, and are concentrated on diagonals where t is divisible by 8. • Bullets, circles and diamonds indicate cyclic groups of Our strategy order 2, 4 and 8, and boxes indicate copies of the integers. The spectrum M The slice spectral sequence S m ρ 8 ∧ H Z Implications Geometric fixed points Some slice differentials The proof 1.9

Slices of the form S m ρ 8 ∧ H Z (continued) The periodicity theorem Mike Hill Mike Hopkins Doug Ravenel • Note that all elements are in the first and third quadrants between certain black lines with slopes 0 and orchid lines with slope 7, and are concentrated on diagonals where t is divisible by 8. • Bullets, circles and diamonds indicate cyclic groups of Our strategy order 2, 4 and 8, and boxes indicate copies of the integers. The spectrum M • A similar picture for S m ρ 4 ∧ H Z would be confined to the The slice spectral sequence S m ρ 8 ∧ H Z regions between the black lines and blue lines with slope 3 Implications Geometric fixed points Some slice differentials The proof 1.9

Slices of the form S m ρ 8 ∧ H Z (continued) The periodicity theorem Mike Hill Mike Hopkins Doug Ravenel • Note that all elements are in the first and third quadrants between certain black lines with slopes 0 and orchid lines with slope 7, and are concentrated on diagonals where t is divisible by 8. • Bullets, circles and diamonds indicate cyclic groups of Our strategy order 2, 4 and 8, and boxes indicate copies of the integers. The spectrum M • A similar picture for S m ρ 4 ∧ H Z would be confined to the The slice spectral sequence S m ρ 8 ∧ H Z regions between the black lines and blue lines with slope 3 Implications and concentrated on diagonals where t is divisible by 4. Geometric fixed points Some slice differentials The proof 1.9

Slices of the form S m ρ 8 ∧ H Z (continued) The periodicity theorem Mike Hill Mike Hopkins Doug Ravenel • Note that all elements are in the first and third quadrants between certain black lines with slopes 0 and orchid lines with slope 7, and are concentrated on diagonals where t is divisible by 8. • Bullets, circles and diamonds indicate cyclic groups of Our strategy order 2, 4 and 8, and boxes indicate copies of the integers. The spectrum M • A similar picture for S m ρ 4 ∧ H Z would be confined to the The slice spectral sequence S m ρ 8 ∧ H Z regions between the black lines and blue lines with slope 3 Implications and concentrated on diagonals where t is divisible by 4. Geometric fixed points • A similar picture for S m ρ 2 ∧ H Z would be confined to the Some slice differentials The proof regions between the black lines and green lines with slope 1 1.9

Slices of the form S m ρ 8 ∧ H Z (continued) The periodicity theorem Mike Hill Mike Hopkins Doug Ravenel • Note that all elements are in the first and third quadrants between certain black lines with slopes 0 and orchid lines with slope 7, and are concentrated on diagonals where t is divisible by 8. • Bullets, circles and diamonds indicate cyclic groups of Our strategy order 2, 4 and 8, and boxes indicate copies of the integers. The spectrum M • A similar picture for S m ρ 4 ∧ H Z would be confined to the The slice spectral sequence S m ρ 8 ∧ H Z regions between the black lines and blue lines with slope 3 Implications and concentrated on diagonals where t is divisible by 4. Geometric fixed points • A similar picture for S m ρ 2 ∧ H Z would be confined to the Some slice differentials The proof regions between the black lines and green lines with slope 1 and concentrated on diagonals where t is divisible by 2. 1.9

The periodicity Implications for the slice spectral sequence theorem Mike Hill Mike Hopkins Doug Ravenel These calculations imply the following. Our strategy The spectrum M The slice spectral sequence S m ρ 8 ∧ H Z Implications Geometric fixed points Some slice differentials The proof 1.10

The periodicity Implications for the slice spectral sequence theorem Mike Hill Mike Hopkins Doug Ravenel These calculations imply the following. • The slice spectral sequence for MU ( 4 ) is concentrated in the first quadrant and confined by the same vanishing lines. Our strategy The spectrum M The slice spectral sequence S m ρ 8 ∧ H Z Implications Geometric fixed points Some slice differentials The proof 1.10

The periodicity Implications for the slice spectral sequence theorem Mike Hill Mike Hopkins Doug Ravenel These calculations imply the following. • The slice spectral sequence for MU ( 4 ) is concentrated in the first quadrant and confined by the same vanishing lines. Our strategy • Later we will invert elements in π m ρ 8 ( MU ( 4 ) ) . The spectrum M The slice spectral sequence S m ρ 8 ∧ H Z Implications Geometric fixed points Some slice differentials The proof 1.10

The periodicity Implications for the slice spectral sequence theorem Mike Hill Mike Hopkins Doug Ravenel These calculations imply the following. • The slice spectral sequence for MU ( 4 ) is concentrated in the first quadrant and confined by the same vanishing lines. Our strategy • Later we will invert elements in π m ρ 8 ( MU ( 4 ) ) . The fact that The spectrum M The slice spectral S − ρ 8 ∧ ( C 8 ∧ H S m ρ h ) = C 8 ∧ H S ( m − 8 / h ) ρ h sequence S m ρ 8 ∧ H Z Implications Geometric fixed points Some slice differentials The proof 1.10

The periodicity Implications for the slice spectral sequence theorem Mike Hill Mike Hopkins Doug Ravenel These calculations imply the following. • The slice spectral sequence for MU ( 4 ) is concentrated in the first quadrant and confined by the same vanishing lines. Our strategy • Later we will invert elements in π m ρ 8 ( MU ( 4 ) ) . The fact that The spectrum M The slice spectral S − ρ 8 ∧ ( C 8 ∧ H S m ρ h ) = C 8 ∧ H S ( m − 8 / h ) ρ h sequence S m ρ 8 ∧ H Z Implications means that the resulting slice spectral sequence is Geometric fixed points confined to the regions of the first and third quadrants Some slice differentials The proof shown in the picture. 1.10

The periodicity Geometric fixed points theorem Mike Hill Mike Hopkins Doug Ravenel In order to proceed further, we need another concept from equivariant stable homotopy theory. Our strategy The spectrum M The slice spectral sequence S m ρ 8 ∧ H Z Implications Geometric fixed points Some slice differentials The proof 1.11

The periodicity Geometric fixed points theorem Mike Hill Mike Hopkins Doug Ravenel In order to proceed further, we need another concept from equivariant stable homotopy theory. Unstably a G -space X has a fixed point set , X G = { x ∈ X : γ ( x ) = x ∀ γ ∈ G } . Our strategy The spectrum M The slice spectral sequence S m ρ 8 ∧ H Z Implications Geometric fixed points Some slice differentials The proof 1.11

The periodicity Geometric fixed points theorem Mike Hill Mike Hopkins Doug Ravenel In order to proceed further, we need another concept from equivariant stable homotopy theory. Unstably a G -space X has a fixed point set , X G = { x ∈ X : γ ( x ) = x ∀ γ ∈ G } . Our strategy The spectrum M This is the same as F ( S 0 , X + ) G , the space of based The slice spectral equivariant maps S 0 → X + , sequence S m ρ 8 ∧ H Z Implications Geometric fixed points Some slice differentials The proof 1.11

The periodicity Geometric fixed points theorem Mike Hill Mike Hopkins Doug Ravenel In order to proceed further, we need another concept from equivariant stable homotopy theory. Unstably a G -space X has a fixed point set , X G = { x ∈ X : γ ( x ) = x ∀ γ ∈ G } . Our strategy The spectrum M This is the same as F ( S 0 , X + ) G , the space of based The slice spectral equivariant maps S 0 → X + , which is the same as the space of sequence S m ρ 8 ∧ H Z unbased equivariant maps ∗ → X . Implications Geometric fixed points Some slice differentials The proof 1.11

The periodicity Geometric fixed points theorem Mike Hill Mike Hopkins Doug Ravenel In order to proceed further, we need another concept from equivariant stable homotopy theory. Unstably a G -space X has a fixed point set , X G = { x ∈ X : γ ( x ) = x ∀ γ ∈ G } . Our strategy The spectrum M This is the same as F ( S 0 , X + ) G , the space of based The slice spectral equivariant maps S 0 → X + , which is the same as the space of sequence S m ρ 8 ∧ H Z unbased equivariant maps ∗ → X . Implications Geometric fixed points The homotopy fixed point set X hG is the space of based Some slice differentials The proof equivariant maps EG + → X + , where EG is a contractible free G -space. 1.11

The periodicity Geometric fixed points theorem Mike Hill Mike Hopkins Doug Ravenel In order to proceed further, we need another concept from equivariant stable homotopy theory. Unstably a G -space X has a fixed point set , X G = { x ∈ X : γ ( x ) = x ∀ γ ∈ G } . Our strategy The spectrum M This is the same as F ( S 0 , X + ) G , the space of based The slice spectral equivariant maps S 0 → X + , which is the same as the space of sequence S m ρ 8 ∧ H Z unbased equivariant maps ∗ → X . Implications Geometric fixed points The homotopy fixed point set X hG is the space of based Some slice differentials The proof equivariant maps EG + → X + , where EG is a contractible free G -space. The equivariant homotopy type of X hG is independent of the choice of EG . 1.11

The periodicity Geometric fixed points (continued) theorem Mike Hill Mike Hopkins Doug Ravenel Both of these definitions have stable analogs, but the fixed point functor is awkward for two reasons: Our strategy The spectrum M The slice spectral sequence S m ρ 8 ∧ H Z Implications Geometric fixed points Some slice differentials The proof 1.12

The periodicity Geometric fixed points (continued) theorem Mike Hill Mike Hopkins Doug Ravenel Both of these definitions have stable analogs, but the fixed point functor is awkward for two reasons: • it fails to commute with smash products and Our strategy The spectrum M The slice spectral sequence S m ρ 8 ∧ H Z Implications Geometric fixed points Some slice differentials The proof 1.12

The periodicity Geometric fixed points (continued) theorem Mike Hill Mike Hopkins Doug Ravenel Both of these definitions have stable analogs, but the fixed point functor is awkward for two reasons: • it fails to commute with smash products and • it fails to commute with infinite suspensions. Our strategy The spectrum M The slice spectral sequence S m ρ 8 ∧ H Z Implications Geometric fixed points Some slice differentials The proof 1.12

The periodicity Geometric fixed points (continued) theorem Mike Hill Mike Hopkins Doug Ravenel Both of these definitions have stable analogs, but the fixed point functor is awkward for two reasons: • it fails to commute with smash products and • it fails to commute with infinite suspensions. Our strategy The geometric fixed set Φ G X is a convenient substitute that The spectrum M avoids these difficulties. The slice spectral sequence S m ρ 8 ∧ H Z Implications Geometric fixed points Some slice differentials The proof 1.12

The periodicity Geometric fixed points (continued) theorem Mike Hill Mike Hopkins Doug Ravenel Both of these definitions have stable analogs, but the fixed point functor is awkward for two reasons: • it fails to commute with smash products and • it fails to commute with infinite suspensions. Our strategy The geometric fixed set Φ G X is a convenient substitute that The spectrum M avoids these difficulties. In order to define it we need the The slice spectral sequence isotropy separation sequence , which in the case of a finite S m ρ 8 ∧ H Z cyclic 2-group G is Implications Geometric fixed points EC 2 + → S 0 → ˜ Some slice differentials EC 2 . The proof 1.12

The periodicity Geometric fixed points (continued) theorem Mike Hill Mike Hopkins Doug Ravenel Both of these definitions have stable analogs, but the fixed point functor is awkward for two reasons: • it fails to commute with smash products and • it fails to commute with infinite suspensions. Our strategy The geometric fixed set Φ G X is a convenient substitute that The spectrum M avoids these difficulties. In order to define it we need the The slice spectral sequence isotropy separation sequence , which in the case of a finite S m ρ 8 ∧ H Z cyclic 2-group G is Implications Geometric fixed points EC 2 + → S 0 → ˜ Some slice differentials EC 2 . The proof Here E Z / 2 is a G -space via the projection G → Z / 2 and S 0 has the trivial action, so ˜ EC 2 is also a G -space. 1.12

The periodicity Geometric fixed points (continued) theorem Mike Hill Mike Hopkins Doug Ravenel Under this action EC G 2 is empty while for any proper subgroup H of G , EC H 2 = EC 2 , which is contractible. For an arbitrary finite group G it is possible to construct a G -space with the similar properties. Our strategy The spectrum M The slice spectral sequence S m ρ 8 ∧ H Z Implications Geometric fixed points Some slice differentials The proof 1.13

The periodicity Geometric fixed points (continued) theorem Mike Hill Mike Hopkins Doug Ravenel Under this action EC G 2 is empty while for any proper subgroup H of G , EC H 2 = EC 2 , which is contractible. For an arbitrary finite group G it is possible to construct a G -space with the similar properties. Our strategy Definition The spectrum M The slice spectral For a finite cyclic 2-group G and G-spectrum X, the geometric sequence S m ρ 8 ∧ H Z fixed point spectrum is Implications Geometric fixed points Φ G X = ( X ∧ ˜ EC 2 ) G . Some slice differentials The proof 1.13

The periodicity Geometric fixed points (continued) theorem Mike Hill Mike Hopkins Doug Ravenel This functor has the following properties: Our strategy The spectrum M The slice spectral sequence S m ρ 8 ∧ H Z Implications Geometric fixed points Some slice differentials The proof 1.14

The periodicity Geometric fixed points (continued) theorem Mike Hill Mike Hopkins Doug Ravenel This functor has the following properties: • For G -spectra X and Y , Φ G ( X ∧ Y ) = Φ G X ∧ Φ G Y . Our strategy The spectrum M The slice spectral sequence S m ρ 8 ∧ H Z Implications Geometric fixed points Some slice differentials The proof 1.14

The periodicity Geometric fixed points (continued) theorem Mike Hill Mike Hopkins Doug Ravenel This functor has the following properties: • For G -spectra X and Y , Φ G ( X ∧ Y ) = Φ G X ∧ Φ G Y . • For a G -space X , Φ G Σ ∞ X = Σ ∞ ( X G ) . Our strategy The spectrum M The slice spectral sequence S m ρ 8 ∧ H Z Implications Geometric fixed points Some slice differentials The proof 1.14

The periodicity Geometric fixed points (continued) theorem Mike Hill Mike Hopkins Doug Ravenel This functor has the following properties: • For G -spectra X and Y , Φ G ( X ∧ Y ) = Φ G X ∧ Φ G Y . • For a G -space X , Φ G Σ ∞ X = Σ ∞ ( X G ) . • A map f : X → Y is a G -equivalence iff Φ H f is an ordinary equivalence for each subgroup H ⊂ G . Our strategy The spectrum M The slice spectral sequence S m ρ 8 ∧ H Z Implications Geometric fixed points Some slice differentials The proof 1.14

The periodicity Geometric fixed points (continued) theorem Mike Hill Mike Hopkins Doug Ravenel This functor has the following properties: • For G -spectra X and Y , Φ G ( X ∧ Y ) = Φ G X ∧ Φ G Y . • For a G -space X , Φ G Σ ∞ X = Σ ∞ ( X G ) . • A map f : X → Y is a G -equivalence iff Φ H f is an ordinary equivalence for each subgroup H ⊂ G . Our strategy From the suspension property we can deduce that The spectrum M Φ C 8 MU ( 4 ) = MO , The slice spectral sequence S m ρ 8 ∧ H Z Implications the unoriented cobordism spectrum. Geometric fixed points Some slice differentials Geometric Fixed Point Theorem The proof Let σ denote the sign representation. Then for any G-spectrum σ π ⋆ ( X ) , where a σ : S 0 → S σ is the X, π ⋆ (˜ EC 2 ∧ X ) = a − 1 element defined in Hill’s lecture. 1.14

The periodicity Geometric fixed points (continued) theorem Mike Hill Recall that π ∗ ( MO ) = Z / 2 [ y i : i > 0 , i � = 2 k − 1 ] where | y i | = i . Mike Hopkins Doug Ravenel Our strategy The spectrum M The slice spectral sequence S m ρ 8 ∧ H Z Implications Geometric fixed points Some slice differentials The proof 1.15

The periodicity Geometric fixed points (continued) theorem Mike Hill Recall that π ∗ ( MO ) = Z / 2 [ y i : i > 0 , i � = 2 k − 1 ] where | y i | = i . Mike Hopkins Doug Ravenel It is not hard to show that π ∗ ( MU ( 4 ) ) = Z [ r i , γ ( r i ) , γ 2 ( r i ) , γ 3 ( r i ) : i > 0 ] where | r i | = 2 i , γ is a generator of G and γ 4 ( r i ) = ( − 1 ) i r i . Our strategy The spectrum M The slice spectral sequence S m ρ 8 ∧ H Z Implications Geometric fixed points Some slice differentials The proof 1.15

The periodicity Geometric fixed points (continued) theorem Mike Hill Recall that π ∗ ( MO ) = Z / 2 [ y i : i > 0 , i � = 2 k − 1 ] where | y i | = i . Mike Hopkins Doug Ravenel It is not hard to show that π ∗ ( MU ( 4 ) ) = Z [ r i , γ ( r i ) , γ 2 ( r i ) , γ 3 ( r i ) : i > 0 ] where | r i | = 2 i , γ is a generator of G and γ 4 ( r i ) = ( − 1 ) i r i . In π i ρ 8 ( MU ( 4 ) ) we have the element Our strategy Nr i = r i γ ( r i ) γ 2 ( r i ) γ 3 ( r i ) . The spectrum M The slice spectral sequence S m ρ 8 ∧ H Z Implications Geometric fixed points Some slice differentials The proof 1.15

The periodicity Geometric fixed points (continued) theorem Mike Hill Recall that π ∗ ( MO ) = Z / 2 [ y i : i > 0 , i � = 2 k − 1 ] where | y i | = i . Mike Hopkins Doug Ravenel It is not hard to show that π ∗ ( MU ( 4 ) ) = Z [ r i , γ ( r i ) , γ 2 ( r i ) , γ 3 ( r i ) : i > 0 ] where | r i | = 2 i , γ is a generator of G and γ 4 ( r i ) = ( − 1 ) i r i . In π i ρ 8 ( MU ( 4 ) ) we have the element Our strategy Nr i = r i γ ( r i ) γ 2 ( r i ) γ 3 ( r i ) . The spectrum M The slice spectral Applying the functor Φ G to the map Nr i : S i ρ 8 → MU ( 4 ) gives a sequence S m ρ 8 ∧ H Z map S i → MO . Implications Geometric fixed points Some slice differentials The proof 1.15

The periodicity Geometric fixed points (continued) theorem Mike Hill Recall that π ∗ ( MO ) = Z / 2 [ y i : i > 0 , i � = 2 k − 1 ] where | y i | = i . Mike Hopkins Doug Ravenel It is not hard to show that π ∗ ( MU ( 4 ) ) = Z [ r i , γ ( r i ) , γ 2 ( r i ) , γ 3 ( r i ) : i > 0 ] where | r i | = 2 i , γ is a generator of G and γ 4 ( r i ) = ( − 1 ) i r i . In π i ρ 8 ( MU ( 4 ) ) we have the element Our strategy Nr i = r i γ ( r i ) γ 2 ( r i ) γ 3 ( r i ) . The spectrum M The slice spectral Applying the functor Φ G to the map Nr i : S i ρ 8 → MU ( 4 ) gives a sequence S m ρ 8 ∧ H Z map S i → MO . Implications Geometric fixed points Lemma Some slice differentials The proof The generators r i and y i can be chosen so that for i = 2 k − 1 � 0 Φ G Nr i = y i otherwise. 1.15

The periodicity Some slice differentials theorem Mike Hill Mike Hopkins Doug Ravenel It follows from the above that the slice spectral sequence for MU ( 4 ) has a vanishing line of slope 7. Our strategy The spectrum M The slice spectral sequence S m ρ 8 ∧ H Z Implications Geometric fixed points Some slice differentials The proof 1.16

The periodicity Some slice differentials theorem Mike Hill Mike Hopkins Doug Ravenel It follows from the above that the slice spectral sequence for MU ( 4 ) has a vanishing line of slope 7. We will describe the subring of elements lying on it. Our strategy The spectrum M The slice spectral sequence S m ρ 8 ∧ H Z Implications Geometric fixed points Some slice differentials The proof 1.16

The periodicity Some slice differentials theorem Mike Hill Mike Hopkins Doug Ravenel It follows from the above that the slice spectral sequence for MU ( 4 ) has a vanishing line of slope 7. We will describe the subring of elements lying on it. Let f i ∈ π i ( MU ( 4 ) ) be the composite Our strategy a i ρ 8 � S i ρ 8 Nr i � MU ( 4 ) . The spectrum M S i The slice spectral sequence S m ρ 8 ∧ H Z Implications Geometric fixed points Some slice differentials The proof 1.16

The periodicity Some slice differentials theorem Mike Hill Mike Hopkins Doug Ravenel It follows from the above that the slice spectral sequence for MU ( 4 ) has a vanishing line of slope 7. We will describe the subring of elements lying on it. Let f i ∈ π i ( MU ( 4 ) ) be the composite Our strategy a i ρ 8 � S i ρ 8 Nr i � MU ( 4 ) . The spectrum M S i The slice spectral sequence S m ρ 8 ∧ H Z The following facts about f i are easy to prove. Implications Geometric fixed points Some slice differentials The proof 1.16

The periodicity Some slice differentials theorem Mike Hill Mike Hopkins Doug Ravenel It follows from the above that the slice spectral sequence for MU ( 4 ) has a vanishing line of slope 7. We will describe the subring of elements lying on it. Let f i ∈ π i ( MU ( 4 ) ) be the composite Our strategy a i ρ 8 � S i ρ 8 Nr i � MU ( 4 ) . The spectrum M S i The slice spectral sequence S m ρ 8 ∧ H Z The following facts about f i are easy to prove. Implications • It appears in the slice spectral sequence in E 7 i , 8 i Geometric fixed points , which is 2 Some slice differentials on the vanishing line. The proof 1.16

The periodicity Some slice differentials theorem Mike Hill Mike Hopkins Doug Ravenel It follows from the above that the slice spectral sequence for MU ( 4 ) has a vanishing line of slope 7. We will describe the subring of elements lying on it. Let f i ∈ π i ( MU ( 4 ) ) be the composite Our strategy a i ρ 8 � S i ρ 8 Nr i � MU ( 4 ) . The spectrum M S i The slice spectral sequence S m ρ 8 ∧ H Z The following facts about f i are easy to prove. Implications • It appears in the slice spectral sequence in E 7 i , 8 i Geometric fixed points , which is 2 Some slice differentials on the vanishing line. The proof • The subring of elements on the vanishing line is the polynomial algebra on the f i . 1.16

The periodicity Some slice differentials (continued) theorem Mike Hill Mike Hopkins Doug Ravenel • Under the map π ∗ ( MU ( g / 2 ) ) → π ∗ (Φ G MU ( g / 2 ) ) = π ∗ ( MO ) Our strategy we have for i = 2 k − 1 The spectrum M � 0 f i �→ The slice spectral y i otherwise sequence S m ρ 8 ∧ H Z Implications Geometric fixed points Some slice differentials The proof 1.17

The periodicity Some slice differentials (continued) theorem Mike Hill Mike Hopkins Doug Ravenel • Under the map π ∗ ( MU ( g / 2 ) ) → π ∗ (Φ G MU ( g / 2 ) ) = π ∗ ( MO ) Our strategy we have for i = 2 k − 1 The spectrum M � 0 f i �→ The slice spectral y i otherwise sequence S m ρ 8 ∧ H Z • Any differential landing on the vanishing line must have a Implications Geometric fixed points target in the ideal ( f 1 , f 3 , f 7 , . . . ) . Some slice differentials The proof 1.17

The periodicity Some slice differentials (continued) theorem Mike Hill Mike Hopkins Doug Ravenel • Under the map π ∗ ( MU ( g / 2 ) ) → π ∗ (Φ G MU ( g / 2 ) ) = π ∗ ( MO ) Our strategy we have for i = 2 k − 1 The spectrum M � 0 f i �→ The slice spectral y i otherwise sequence S m ρ 8 ∧ H Z • Any differential landing on the vanishing line must have a Implications Geometric fixed points target in the ideal ( f 1 , f 3 , f 7 , . . . ) . A similar statement can be Some slice differentials made after smashing with S 2 k σ . The proof 1.17

The periodicity Some slice differentials (continued) theorem Mike Hill Mike Hopkins Doug Ravenel Recall that for an oriented representation V there is a map u V : S | V | → Σ V H Z , which lies in π V −| V | ( H Z ) . Our strategy The spectrum M The slice spectral sequence S m ρ 8 ∧ H Z Implications Geometric fixed points Some slice differentials The proof 1.18

The periodicity Some slice differentials (continued) theorem Mike Hill Mike Hopkins Doug Ravenel Recall that for an oriented representation V there is a map u V : S | V | → Σ V H Z , which lies in π V −| V | ( H Z ) . Slice Differentials Theorem In the slice spectral sequence for Σ 2 k σ MU ( 4 ) (for k > 0 ) we have d r ( u 2 k σ ) = 0 for r < 1 + 8 ( 2 k − 1 ) , and Our strategy The spectrum M The slice spectral d 1 + 8 ( 2 k − 1 ) ( u 2 k σ ) = a 2 k σ f 2 k − 1 . sequence S m ρ 8 ∧ H Z Implications Geometric fixed points Some slice differentials The proof 1.18

The periodicity Some slice differentials (continued) theorem Mike Hill Mike Hopkins Doug Ravenel Recall that for an oriented representation V there is a map u V : S | V | → Σ V H Z , which lies in π V −| V | ( H Z ) . Slice Differentials Theorem In the slice spectral sequence for Σ 2 k σ MU ( 4 ) (for k > 0 ) we have d r ( u 2 k σ ) = 0 for r < 1 + 8 ( 2 k − 1 ) , and Our strategy The spectrum M The slice spectral d 1 + 8 ( 2 k − 1 ) ( u 2 k σ ) = a 2 k σ f 2 k − 1 . sequence S m ρ 8 ∧ H Z Implications Geometric fixed points Some slice differentials Inverting a σ in the slice spectral sequence will make it The proof converge to π ∗ ( MO ) . 1.18

The periodicity Some slice differentials (continued) theorem Mike Hill Mike Hopkins Doug Ravenel Recall that for an oriented representation V there is a map u V : S | V | → Σ V H Z , which lies in π V −| V | ( H Z ) . Slice Differentials Theorem In the slice spectral sequence for Σ 2 k σ MU ( 4 ) (for k > 0 ) we have d r ( u 2 k σ ) = 0 for r < 1 + 8 ( 2 k − 1 ) , and Our strategy The spectrum M The slice spectral d 1 + 8 ( 2 k − 1 ) ( u 2 k σ ) = a 2 k σ f 2 k − 1 . sequence S m ρ 8 ∧ H Z Implications Geometric fixed points Some slice differentials Inverting a σ in the slice spectral sequence will make it The proof converge to π ∗ ( MO ) . This means each f 2 k − 1 must be killed by some power of a σ . 1.18

The periodicity Some slice differentials (continued) theorem Mike Hill Mike Hopkins Doug Ravenel Recall that for an oriented representation V there is a map u V : S | V | → Σ V H Z , which lies in π V −| V | ( H Z ) . Slice Differentials Theorem In the slice spectral sequence for Σ 2 k σ MU ( 4 ) (for k > 0 ) we have d r ( u 2 k σ ) = 0 for r < 1 + 8 ( 2 k − 1 ) , and Our strategy The spectrum M The slice spectral d 1 + 8 ( 2 k − 1 ) ( u 2 k σ ) = a 2 k σ f 2 k − 1 . sequence S m ρ 8 ∧ H Z Implications Geometric fixed points Some slice differentials Inverting a σ in the slice spectral sequence will make it The proof converge to π ∗ ( MO ) . This means each f 2 k − 1 must be killed by some power of a σ . The only way this can happen is as indicated in the theorem. 1.18

The periodicity Some slice differentials (continued) theorem Mike Hill Mike Hopkins Doug Ravenel Let ( 8 ) = Nr 2 k − 1 ∈ π ( 2 k − 1 ) ρ 8 ( MU ( 4 ) ) . ∆ k Our strategy The spectrum M The slice spectral sequence S m ρ 8 ∧ H Z Implications Geometric fixed points Some slice differentials The proof 1.19

The periodicity Some slice differentials (continued) theorem Mike Hill Mike Hopkins Doug Ravenel Let ( 8 ) = Nr 2 k − 1 ∈ π ( 2 k − 1 ) ρ 8 ( MU ( 4 ) ) . ∆ k We want to invert this element and study the resulting slice spectral sequence. Our strategy The spectrum M The slice spectral sequence S m ρ 8 ∧ H Z Implications Geometric fixed points Some slice differentials The proof 1.19

The periodicity Some slice differentials (continued) theorem Mike Hill Mike Hopkins Doug Ravenel Let ( 8 ) = Nr 2 k − 1 ∈ π ( 2 k − 1 ) ρ 8 ( MU ( 4 ) ) . ∆ k We want to invert this element and study the resulting slice spectral sequence. As explained previously, it is confined to the Our strategy first and third quadrants with vanishing lines of slopes 0 and 7. The spectrum M The slice spectral sequence S m ρ 8 ∧ H Z Implications Geometric fixed points Some slice differentials The proof 1.19

The periodicity Some slice differentials (continued) theorem Mike Hill Mike Hopkins Doug Ravenel Let ( 8 ) = Nr 2 k − 1 ∈ π ( 2 k − 1 ) ρ 8 ( MU ( 4 ) ) . ∆ k We want to invert this element and study the resulting slice spectral sequence. As explained previously, it is confined to the Our strategy first and third quadrants with vanishing lines of slopes 0 and 7. The spectrum M The slice spectral The differential d r on u 2 k + 1 σ described in the theorem is the last sequence S m ρ 8 ∧ H Z one possible since its target, a 2 k + 1 f 2 k + 1 − 1 , lies on the vanishing Implications σ Geometric fixed points line. Some slice differentials The proof 1.19

The periodicity Some slice differentials (continued) theorem Mike Hill Mike Hopkins Doug Ravenel Let ( 8 ) = Nr 2 k − 1 ∈ π ( 2 k − 1 ) ρ 8 ( MU ( 4 ) ) . ∆ k We want to invert this element and study the resulting slice spectral sequence. As explained previously, it is confined to the Our strategy first and third quadrants with vanishing lines of slopes 0 and 7. The spectrum M The slice spectral The differential d r on u 2 k + 1 σ described in the theorem is the last sequence S m ρ 8 ∧ H Z one possible since its target, a 2 k + 1 f 2 k + 1 − 1 , lies on the vanishing Implications σ Geometric fixed points line. If we can show that this target is killed by an earlier ( 8 ) Some slice differentials differential after inverting ∆ k , then u 2 k + 1 σ will be a permanent The proof cycle. 1.19

The periodicity Some slice differentials (continued) theorem Mike Hill Mike Hopkins Doug Ravenel We have ( 8 ) f 2 k + 1 − 1 ∆ = a ( 2 k + 1 − 1 ) ρ 8 Nr 2 k + 1 − 1 Nr 2 k − 1 k Our strategy The spectrum M The slice spectral sequence S m ρ 8 ∧ H Z Implications Geometric fixed points Some slice differentials The proof 1.20

The periodicity Some slice differentials (continued) theorem Mike Hill Mike Hopkins Doug Ravenel We have ( 8 ) f 2 k + 1 − 1 ∆ = a ( 2 k + 1 − 1 ) ρ 8 Nr 2 k + 1 − 1 Nr 2 k − 1 k ( 8 ) = a 2 k ρ 8 ∆ k + 1 f 2 k − 1 Our strategy The spectrum M The slice spectral sequence S m ρ 8 ∧ H Z Implications Geometric fixed points Some slice differentials The proof 1.20

The periodicity Some slice differentials (continued) theorem Mike Hill Mike Hopkins Doug Ravenel We have ( 8 ) f 2 k + 1 − 1 ∆ = a ( 2 k + 1 − 1 ) ρ 8 Nr 2 k + 1 − 1 Nr 2 k − 1 k ( 8 ) = a 2 k ρ 8 ∆ k + 1 f 2 k − 1 Our strategy ( 8 ) k + 1 d r ′ ( u 2 k σ ) for r ′ < r . = ∆ The spectrum M The slice spectral sequence S m ρ 8 ∧ H Z Implications Geometric fixed points Some slice differentials The proof 1.20