Towards Spectral Sequences for Homology Floris van Doorn University of Pittsburgh November 22, 2018 Joint work with Jeremy Avigad, Steve Awodey, Ulrik Buchholtz, Egbert Rijke and Mike Shulman. Floris van Doorn (University of Pittsburgh) Towards Spectral Sequences for Homology November 22, 2018 1 / 27 Outline Spectra and cohomology Spectral sequences for cohomology Spectral sequences for homology Applications Floris van Doorn (University of Pittsburgh) Towards Spectral Sequences for Homology November 22, 2018 2 / 27 Spectra ∗ A prespectrum is a sequence of pointed types Y : Z ! Type with ∗ pointed maps ΣYn ! Yn+1 called structure maps. ∗ By the adjunction Σ a Ω, we can equivalently take maps Yn ! ΩYn+1. An Ω-spectrum or spectrum is a prespectrum Y where the maps ∗ Yn ! ΩYn+1 are equivalences. A spectrum Y is called n-truncated if Yk is (n + k)-truncated for all k : Z. The homotopy groups of an Ω-spectrum Y are πn(Y ) :≡ πn+k(Yk) (which is independent of k and also defined for negative n). Floris van Doorn (University of Pittsburgh) Towards Spectral Sequences for Homology November 22, 2018 3 / 27 Spectra Examples If A is an abelian group, the Eilenberg-MacLane spectrum HA : Ω-Spectrum where (HA)n = K(A; n) is a 0-truncated Ω-spectrum. If X and Y are prespectra, then X _ Y defined by (X _ Y )n :≡ Xn _ Yn is a prespectrum, since we have a pointed map ∗ ΩXn+1 _ ΩYn+1 ! Ω(Xn+1 _ Yn+1): If X is a pointed type and Y : X ! Ω-Spectrum is family of spectra parametrized over X we have a spectrum Π∗(x : X); Y x defined by ∗ ∗ (Π (x : X); Y x)n :≡ Π (x : X); (Y x)n Floris van Doorn (University of Pittsburgh) Towards Spectral Sequences for Homology November 22, 2018 4 / 27 Cohomology If X : Type∗ and Y : Ω-Spectrum, we have generalized reduced cohomology: n n ∗ ∗ Y (X) ≡ He (X; Y ) :≡ π−n(X ! Y ) ' kX ! Ynk0: If Y = HA, then we get the ordinary reduced cohomology He n(X; A). If X is any type, we get unreduced cohomology n n H (X; Y ) :≡ π−n(X ! Y ) ' He (X + 1; Y ): We get parametrized cohomology by replacing functions with dependent functions: n ∗ ∗ He (X; λx.Y x) :≡ π−n(Π (x : X); Y x) ' kΠ (x : X); (Y x)n)k0: Here Y : X ! Ω-Spectrum is a parametrized spectrum. Floris van Doorn (University of Pittsburgh) Towards Spectral Sequences for Homology November 22, 2018 5 / 27 Long Exact Sequence of Homotopy Groups Given a pointed map f : X !∗ Y with fiber F . Then we have the following long exact sequence. : : π2(f) π2(F ) π2(X) π2(Y ) π2(p1) π1(δ) π1(f) π1(F ) π1(X) π1(Y ) π1(p1) π0(δ) π0(f) π0(F ) π0(X) π0(Y ) π0(p1) Floris van Doorn (University of Pittsburgh) Towards Spectral Sequences for Homology November 22, 2018 6 / 27 Spectral Sequences p;q Definition.A spectral sequence consists of a family Er of abelian groups for p; q : Z and r ≥ 2. For a fixed r this gives the r-page of the spectral sequence. ::: q q • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • p;q p p;q p E2 E3 Floris van Doorn (University of Pittsburgh) Towards Spectral Sequences for Homology November 22, 2018 7 / 27 Spectral Sequences p;q p;q p+r;q−r+1 Definition. ::: with differentials dr : Er ! Er such that dr ◦ dr = 0 (this is cohomologically indexed) ::: q q • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • p;q p p;q p E2 E3 Floris van Doorn (University of Pittsburgh) Towards Spectral Sequences for Homology November 22, 2018 7 / 27 Spectral Sequences p;q p;q p;q Definition. ::: and with isomorphisms αr : H (Er) ' Er+1 p;q p;q p−r;q+r−1 where H (Er) = ker(dr )=im(dr ). q q • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • p;q p p;q p E2 E3 Floris van Doorn (University of Pittsburgh) Towards Spectral Sequences for Homology November 22, 2018 7 / 27 Spectral Sequences p;q p;q p;q Definition. ::: and with isomorphisms αr : H (Er) ' Er+1 p;q p;q p−r;q+r−1 where H (Er) = ker(dr )=im(dr ). The differentials of Er+1 are not determined by (Er; dr). q q • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • p;q p p;q p E2 E3 Floris van Doorn (University of Pittsburgh) Towards Spectral Sequences for Homology November 22, 2018 7 / 27 We can often compute the abutment, a \twisted sum" of the diagonals. Convergence of Spectral Sequences q • • • • • In many spectral sequences the p;q • • • • • pages converge to E1 . • • • • • • • • • • • • • • • D0 D1 D2 D3 D4 p p;q E1 Floris van Doorn (University of Pittsburgh) Towards Spectral Sequences for Homology November 22, 2018 8 / 27 Convergence of Spectral Sequences q • • • • • In many spectral sequences the p;q • • • • • pages converge to E1 . • • • • • We can often compute the abutment, a \twisted sum" of • • • • • the diagonals. • • • • • D0 D1 D2 D3 D4 p p;q E1 Floris van Doorn (University of Pittsburgh) Towards Spectral Sequences for Homology November 22, 2018 8 / 27 p;q p;q The second page is E2 = C ; p;q The spectral sequence converges to E1 ; n p;q The abutment D is a twisted sum of the E1 for n = p + q. n;q This means that there are groups (D )q and short exact sequences: n;0 n n;1 0 ! E1 !D ! D ! 0 : : p;q n;q n;q+1 0 ! E1 !D ! D ! 0 p−1;q+1 n;q+1 n;q+2 0 ! E1 !D ! D ! 0 : : 0;n n;n 0 ! E1 !D ! 0 p;q if there exists a spectral sequence Er such that Convergence of Spectral Sequences For a bigraded abelian group Cp;q and graded abelian group Dn we write p;q p;q p+q E2 = C ) D Floris van Doorn (University of Pittsburgh) Towards Spectral Sequences for Homology November 22, 2018 9 / 27 p;q p;q The second page is E2 = C ; p;q The spectral sequence converges to E1 ; n p;q The abutment D is a twisted sum of the E1 for n = p + q. n;q This means that there are groups (D )q and short exact sequences: n;0 n n;1 0 ! E1 !D ! D ! 0 : : p;q n;q n;q+1 0 ! E1 !D ! D ! 0 p−1;q+1 n;q+1 n;q+2 0 ! E1 !D ! D ! 0 : : 0;n n;n 0 ! E1 !D ! 0 Convergence of Spectral Sequences For a bigraded abelian group Cp;q and graded abelian group Dn we write p;q p;q p+q E2 = C ) D p;q if there exists a spectral sequence Er such that Floris van Doorn (University of Pittsburgh) Towards Spectral Sequences for Homology November 22, 2018 9 / 27 p;q The spectral sequence converges to E1 ; n p;q The abutment D is a twisted sum of the E1 for n = p + q. n;q This means that there are groups (D )q and short exact sequences: n;0 n n;1 0 ! E1 !D ! D ! 0 : : p;q n;q n;q+1 0 ! E1 !D ! D ! 0 p−1;q+1 n;q+1 n;q+2 0 ! E1 !D ! D ! 0 : : 0;n n;n 0 ! E1 !D ! 0 Convergence of Spectral Sequences For a bigraded abelian group Cp;q and graded abelian group Dn we write p;q p;q p+q E2 = C ) D p;q if there exists a spectral sequence Er such that p;q p;q The second page is E2 = C ; Floris van Doorn (University of Pittsburgh) Towards Spectral Sequences for Homology November 22, 2018 9 / 27 n p;q The abutment D is a twisted sum of the E1 for n = p + q. n;q This means that there are groups (D )q and short exact sequences: n;0 n n;1 0 ! E1 !D ! D ! 0 : : p;q n;q n;q+1 0 ! E1 !D ! D ! 0 p−1;q+1 n;q+1 n;q+2 0 ! E1 !D ! D ! 0 : : 0;n n;n 0 ! E1 !D ! 0 Convergence of Spectral Sequences For a bigraded abelian group Cp;q and graded abelian group Dn we write p;q p;q p+q E2 = C ) D p;q if there exists a spectral sequence Er such that p;q p;q The second page is E2 = C ; p;q The spectral sequence converges to E1 ; Floris van Doorn (University of Pittsburgh) Towards Spectral Sequences for Homology November 22, 2018 9 / 27 n;q This means that there are groups (D )q and short exact sequences: n;0 n n;1 0 ! E1 !D ! D ! 0 : : p;q n;q n;q+1 0 ! E1 !D ! D ! 0 p−1;q+1 n;q+1 n;q+2 0 ! E1 !D ! D ! 0 : : 0;n n;n 0 ! E1 !D ! 0 Convergence of Spectral Sequences For a bigraded abelian group Cp;q and graded abelian group Dn we write p;q p;q p+q E2 = C ) D p;q if there exists a spectral sequence Er such that p;q p;q The second page is E2 = C ; p;q The spectral sequence converges to E1 ; n p;q The abutment D is a twisted sum of the E1 for n = p + q. Floris van Doorn (University of Pittsburgh) Towards Spectral Sequences for Homology November 22, 2018 9 / 27 Convergence of Spectral Sequences For a bigraded abelian group Cp;q and graded abelian group Dn we write p;q p;q p+q E2 = C ) D p;q if there exists a spectral sequence Er such that p;q p;q The second page is E2 = C ; p;q The spectral sequence converges to E1 ; n p;q The abutment D is a twisted sum of the E1 for n = p + q.