Unstable Homotopy Theory from the Chromatic Point of guozhen/K(2) آ  Bounded torsion

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Transcript of Unstable Homotopy Theory from the Chromatic Point of guozhen/K(2) آ  Bounded torsion

  • The EHP Sequence Periodic Unstable Homotopy Theory

    The K(2)-local Goodwillie Tower of Spheres Computation of π∗(ΦK(2)S

    3)

    Unstable Homotopy Theory from the Chromatic Point of View

    Guozhen Wang

    MIT

    April 13, 2015

    Guozhen Wang Unstable Homotopy Theory from the Chromatic Point of View

  • The EHP Sequence Periodic Unstable Homotopy Theory

    The K(2)-local Goodwillie Tower of Spheres Computation of π∗(ΦK(2)S

    3)

    Outline

    1 The EHP Sequence Definition of EHP sequence EHP sequence for p = 3 Exponent of unstable homotopy groups

    2 Periodic Unstable Homotopy Theory Periodic unstable homotopy groups Bousfield-Kuhn functor v1-periodic unstable homotopy groups Bounded torsion phenomenon in chromatic level 2

    3 The K (2)-local Goodwillie Tower of Spheres The Goodwillie tower of the identity The Goodwillie derivatives of spheres Goodwillie differentials on En-cohomology

    4 Computation of π∗(ΦK(2)S 3)

    Guozhen Wang Unstable Homotopy Theory from the Chromatic Point of View

  • The EHP Sequence Periodic Unstable Homotopy Theory

    The K(2)-local Goodwillie Tower of Spheres Computation of π∗(ΦK(2)S

    3)

    Definition of EHP sequence EHP sequence for p = 3 Exponent of unstable homotopy groups

    Section 1

    The EHP Sequence

    Guozhen Wang Unstable Homotopy Theory from the Chromatic Point of View

  • The EHP Sequence Periodic Unstable Homotopy Theory

    The K(2)-local Goodwillie Tower of Spheres Computation of π∗(ΦK(2)S

    3)

    Definition of EHP sequence EHP sequence for p = 3 Exponent of unstable homotopy groups

    The Hopf invariant

    Theorem (James Splitting)

    Let X be a connected space. Then there is a homotopy equivalence ΣΩΣX = ∨ΣX∧i .

    Definition (Hopf invariant)

    The Hopf map H : ΩΣX → ΩΣX∧p at prime p is defined to be the adjoint of the projection map ΣΩΣX+ ∼= ∨ΣX∧i → ΣX∧p.

    Guozhen Wang Unstable Homotopy Theory from the Chromatic Point of View

  • The EHP Sequence Periodic Unstable Homotopy Theory

    The K(2)-local Goodwillie Tower of Spheres Computation of π∗(ΦK(2)S

    3)

    Definition of EHP sequence EHP sequence for p = 3 Exponent of unstable homotopy groups

    The EHP sequence

    Theorem (James)

    We have a 2-local fiber sequence:

    Sk E−→ ΩSk+1 H−→ ΩS2k+1

    Theorem (Toda)

    At an odd prime p, we have fiber sequences:

    ˆS2k E−→ ΩS2k+1 H−→ ΩS2pk+1

    S2k−1 E−→ Ω ˆS2k H−→ ΩS2kp−1

    where ˆS2k is the (2kp − 1)-skeleton of ΩS2k+1.

    Guozhen Wang Unstable Homotopy Theory from the Chromatic Point of View

  • The EHP Sequence Periodic Unstable Homotopy Theory

    The K(2)-local Goodwillie Tower of Spheres Computation of π∗(ΦK(2)S

    3)

    Definition of EHP sequence EHP sequence for p = 3 Exponent of unstable homotopy groups

    EHP sequence for p = 3

    0 1 2 3 4 5 6 7 8 9 10 11 12 1 α1 α2 β1 α3/2

    1 * * * * * * * * * * * * 1 * * α1 * * * α2 * *

    1 * * α1 * * * α2 * 1 * * α1 * *

    1 * * α1 * 1 *

    1

    Guozhen Wang Unstable Homotopy Theory from the Chromatic Point of View

  • The EHP Sequence Periodic Unstable Homotopy Theory

    The K(2)-local Goodwillie Tower of Spheres Computation of π∗(ΦK(2)S

    3)

    Definition of EHP sequence EHP sequence for p = 3 Exponent of unstable homotopy groups

    EHP sequence for p = 3

    13 14 15 16 17 18 19 20 21 22 23 α1β1 α4 α5 β1^2 α1β1^2 ; α6/2

    * * * * * * * * * * * β1~ α3/2 * α1β1 μ[α2] α4 * * μ[α3/2] α5 β1^2

    * β1 α3/2 * α1β1 μ[α1] α4 * * μ[α2] α5 * α2 * * β1 α3/2 * α1β1 * α4 * * * α2 * * β1 α3/2 * α1β1 * α4 * α1 * * * α2 * * β1 α3/2 * * * α1 * * * α2 * * β1 α3/2

    1 * * α1 * * * α2 * 1 * * α1 * * * α2

    1 * * α1 * 1 * * α1

    1

    Guozhen Wang Unstable Homotopy Theory from the Chromatic Point of View

  • The EHP Sequence Periodic Unstable Homotopy Theory

    The K(2)-local Goodwillie Tower of Spheres Computation of π∗(ΦK(2)S

    3)

    Definition of EHP sequence EHP sequence for p = 3 Exponent of unstable homotopy groups

    p-exponent of unstable homotopy groups

    Theorem (James, Toda)

    1 The d1-differential on odd rows of the EHP spectral sequence is the multiplication by p map.

    2 The p-component of π∗S 2k+1 is annialated by p2k .

    Theorem (Cohen-Moore-Neisendorfer)

    At an odd prime p,

    1 The multiplication by p map on the fiber of double suspension S2k−1 → Ω2S2k+1 is zero.

    2 The p-component of π∗S 2k+1 is annihilated by pk .

    Guozhen Wang Unstable Homotopy Theory from the Chromatic Point of View

  • The EHP Sequence Periodic Unstable Homotopy Theory

    The K(2)-local Goodwillie Tower of Spheres Computation of π∗(ΦK(2)S

    3)

    Periodic unstable homotopy groups Bousfield-Kuhn functor v1-periodic unstable homotopy groups Bounded torsion phenomenon in chromatic level 2

    Section 2

    Periodic Unstable Homotopy Theory

    Guozhen Wang Unstable Homotopy Theory from the Chromatic Point of View

  • The EHP Sequence Periodic Unstable Homotopy Theory

    The K(2)-local Goodwillie Tower of Spheres Computation of π∗(ΦK(2)S

    3)

    Periodic unstable homotopy groups Bousfield-Kuhn functor v1-periodic unstable homotopy groups Bounded torsion phenomenon in chromatic level 2

    Type n complex

    Definition

    A finite CW -complex W is type n if K̃ (h)∗W = 0 for h < n, and

    K̃ (n)∗W is nontrivial.

    Theorem (Hopkins-Smith)

    For a type n complex W , there exist positive integers t,N and map

    v tn : Σ N+t|vn|W → ΣNW

    such that v tn induces multiplication by v t n on K (n)-homology.

    Guozhen Wang Unstable Homotopy Theory from the Chromatic Point of View

  • The EHP Sequence Periodic Unstable Homotopy Theory

    The K(2)-local Goodwillie Tower of Spheres Computation of π∗(ΦK(2)S

    3)

    Periodic unstable homotopy groups Bousfield-Kuhn functor v1-periodic unstable homotopy groups Bounded torsion phenomenon in chromatic level 2

    Periodic homotopy groups

    Definition

    Let X be a space. The homotopy groups of X with coefficients in W is defined by

    πi (X ; W ) = [Σ iW ,X ]

    When W is type n, the map v tn on W induces a map v tn : πi (X ; W )→ πi+t|vn|(X ; W ) for i ≥ N.

    Definition

    The vn-periodic homotopy groups of X with coefficients in W is defined by

    v−1n π∗(X ; W ) = (v t n) −1π∗(X ; W )

    Guozhen Wang Unstable Homotopy Theory from the Chromatic Point of View

  • The EHP Sequence Periodic Unstable Homotopy Theory

    The K(2)-local Goodwillie Tower of Spheres Computation of π∗(ΦK(2)S

    3)

    Periodic unstable homotopy groups Bousfield-Kuhn functor v1-periodic unstable homotopy groups Bounded torsion phenomenon in chromatic level 2

    The Bousfield-Kuhn functor

    Let T (n) be the Bousfield class (in the sense of localization) of v−1n Σ

    ∞W for any type n complex W .

    Theorem (Bousfield, Kuhn)

    There exists a functor Φn from the category of based spaces to spectrum, such that:

    1 If Y is a spectrum, then Φn(Ω ∞Y ) ∼= LT (n)Y .

    2 For any space X , we have v−1n π∗(X ; W ) = π∗(ΦnX ; W ), for any type n complex W .

    We have the variations ΦK(n) = LK(n)Φn.

    Guozhen Wang Unstable Homotopy Theory from the Chromatic Point of View

  • The EHP Sequence Periodic Unstable Homotopy Theory

    The K(2)-local Goodwillie Tower of Spheres Computation of π∗(ΦK(2)S

    3)

    Periodic unstable homotopy groups Bousfield-Kuhn functor v1-periodic unstable homotopy groups Bounded torsion phenomenon in chromatic level 2

    v1-periodic homotopy type of unstable spheres

    Let P∞1 = Σ∞BΣp. We can make P∞1 into a CW complex with cells in dimension q − 1, q, 2q − 1, 2q, . . . , where q = 2(p − 1). Define P2k1 to be the kq-skeleton of P∞1 , which has cells in dimension q − 1, q, . . . , kq − 1, kq.

    Theorem (Mahowald-Thompson)

    ΦK(1)S 2k+1 is homotopy equivalent to LK(1)P2k1 .

    Remark

    At an odd prime, we have LK(1)P∞1 ∼= LK(1)S, and LK(1)P2k1 ∼= Σ−1S/pk .

    Guozhen Wang Unstable Homotopy Theory from the Chromatic Point of View

  • The EHP Sequence Periodic Unstable Homotopy Theory

    The K(2)-local Goodwillie Tower of Spheres Computation of π∗(ΦK(2)S

    3)

    Periodic unstable homotopy groups Bousfield-Kuhn functor v1-periodic unstable homotopy groups Bounded torsion phenomenon in chromatic level 2

    vn-torsion in unstable homotopy groups

    Theorem (W.)

    The group π∗(ΦK(2)S 3) is annihilated by v 21 for p ≥ 5.

    Remark

    The map v 21 : Σ 2|v1|ΦK(2)S

    3 → ΦK(2)S3 is non-trivial because it is not zero on E2-homology.

    Theorem (W.)

    The group π∗(ΦK(2)S 2k+1) has bounded v1-torsion for p ≥ 5.

    Conjecture (generalization of Cohen-Moore-Neisendorfer)

    The vn-torsion part of π∗(S 2k+1) is annihilated by a fixed power

    (which depends on k) of vn.

    Guozhen Wang Unstable Homotopy Theory from the Chromatic Point of View

  • The EHP Sequence Periodic Unsta