Spectral Sequences in Type Theory

Floris van Doorn

Carnegie Mellon University

June 4, 2018

Joint work with Jeremy Avigad, Steve Awodey, Ulrik Buchholtz, Egbert Rijke and Mike Shulman.

Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 1 / 27 Outline

Cohomology in HoTT Spectral sequences Atiyah-Hirzebruch and Serre spectral sequences for Future work: Spectral sequences for homology

Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 2 / 27 Classical theorem: The Eilenberg-MacLane spaces K(A, n) classify cohomology. Recall. The Eilenberg-MacLane space K(A, n) is the unique pointed type X with one nontrivial πn(X) ' A.

Cohomology

How do we define cohomology? The usual constructions of singular cohomology is not homotopy invariant.

Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 3 / 27 Cohomology

How do we define cohomology? The usual constructions of singular cohomology is not homotopy invariant. Classical theorem: The Eilenberg-MacLane spaces K(A, n) classify cohomology. Recall. The Eilenberg-MacLane space K(A, n) is the unique pointed type X with one nontrivial homotopy group πn(X) ' A.

Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 3 / 27 Cohomology

We define the reduced cohomology of a pointed type X with coefficients in an abelian group A to be

n ∗ He (X,A) :≡ kX → K(A, n)k0.

The unreduced cohomology can be defined similarly for any (not necessarily pointed) type X:

n n H (X,A) :≡ kX → K(A, n)k0 = He (X + 1,A).

The group structure comes from the equivalence K(A, n) '∗ ΩK(A, n + 1).

Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 4 / 27 Long 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 (CMU) Spectral Sequences in HoTT June 4, 2018 5 / 27 Spectral Sequences

p,q DefinitionA 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 (CMU) Spectral Sequences in HoTT June 4, 2018 6 / 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 (CMU) Spectral Sequences in HoTT June 4, 2018 6 / 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 (CMU) Spectral Sequences in HoTT June 4, 2018 6 / 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. q q • • • • • • • • • •

• • • • • • • • • •

• • • • • • • • • •

• • • • • • • • • •

• • • • • • • • • •

p,q p p,q p E2 E3

Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 6 / 27 And we can get information about the diagonals on the infinity page.

Convergence of Spectral Sequences

q • • • • •

In many spectral sequences the p,q • • • • • pages converge to E∞ . • • • • •

• • • • •

• • • • • D0 D1 D2 D3 D4 p p,q E∞

Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 7 / 27 Convergence of Spectral Sequences

q • • • • •

In many spectral sequences the p,q • • • • • pages converge to E∞ . • • • • • And we can get information about the diagonals on the • • • • • infinity page.

• • • • • D0 D1 D2 D3 D4 p p,q E∞

Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 7 / 27 p,q p,q The second page is E2 = C ; p,q The spectral sequence converges to E∞ ; n p,q D is built up from E∞ for n = p + q in the following way: We have short exact sequences: 0,n n n,1 0 → E∞ →D → D → 0 . . p,q n,p n,p+1 0 → E∞ →D → D → 0 p+1,q−1 n,p+1 n,p+2 0 → E∞ →D → D → 0 . . n,0 n,n 0 → E∞ →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 (CMU) Spectral Sequences in HoTT June 4, 2018 8 / 27 p,q p,q The second page is E2 = C ; p,q The spectral sequence converges to E∞ ; n p,q D is built up from E∞ for n = p + q in the following way: We have short exact sequences: 0,n n n,1 0 → E∞ →D → D → 0 . . p,q n,p n,p+1 0 → E∞ →D → D → 0 p+1,q−1 n,p+1 n,p+2 0 → E∞ →D → D → 0 . . n,0 n,n 0 → E∞ →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 (CMU) Spectral Sequences in HoTT June 4, 2018 8 / 27 p,q The spectral sequence converges to E∞ ; n p,q D is built up from E∞ for n = p + q in the following way: We have short exact sequences: 0,n n n,1 0 → E∞ →D → D → 0 . . p,q n,p n,p+1 0 → E∞ →D → D → 0 p+1,q−1 n,p+1 n,p+2 0 → E∞ →D → D → 0 . . n,0 n,n 0 → E∞ →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 (CMU) Spectral Sequences in HoTT June 4, 2018 8 / 27 n p,q D is built up from E∞ for n = p + q in the following way: We have short exact sequences: 0,n n n,1 0 → E∞ →D → D → 0 . . p,q n,p n,p+1 0 → E∞ →D → D → 0 p+1,q−1 n,p+1 n,p+2 0 → E∞ →D → D → 0 . . n,0 n,n 0 → E∞ →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 E∞ ;

Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 8 / 27 We have short exact sequences: 0,n n n,1 0 → E∞ →D → D → 0 . . p,q n,p n,p+1 0 → E∞ →D → D → 0 p+1,q−1 n,p+1 n,p+2 0 → E∞ →D → D → 0 . . n,0 n,n 0 → E∞ →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 E∞ ; n p,q D is built up from E∞ for n = p + q in the following way:

Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 8 / 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 E∞ ; n p,q D is built up from E∞ for n = p + q in the following way: We have short exact sequences: 0,n n n,1 0 → E∞ →D → D → 0 . . p,q n,p n,p+1 0 → E∞ →D → D → 0 p+1,q−1 n,p+1 n,p+2 0 → E∞ →D → D → 0 . . n,0 n,n 0 → E∞ →D → 0

Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 8 / 27 Serre Spectral Sequence (Special Case)

Theorem. Suppose f : X → B and b0 : B and let F :≡ fibf (b0). Suppose that B is simply connected and A is an abelian group. Then

p,q p q p+q E2 = H (B,H (F,A)) ⇒ H (X,A).

This is only true for unreduced cohomology.

Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 9 / 27 Example: Cohomology of K(Z, 2)

∞ We will compute the cohomology groups of B = K(Z, 2) (which is CP ). f We define the map 1 −→ K(Z, 2) determined by the basepoint ? : K(Z, 2). It has fiber

1 fibf (?) = ΩK(Z, 2) = K(Z, 1) = S .

The spectral sequence for A = Z gives

p,q p q 1 p+q E2 = H (B,H (S )) ⇒ H (1).

Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 10 / 27 q q

p p p,q E2 p,q E∞

Example: Cohomology of K(Z, 2)

p,q p q 1 p+q E2 = H (B,H (S )) ⇒ H (1).

( ( n 1 Z if n = 0, 1 n Z if n = 0 H (S ) = H (1) = 0 otherwise 0 otherwise

Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 11 / 27 Example: Cohomology of K(Z, 2)

p,q p q 1 p+q E2 = H (B,H (S )) ⇒ H (1).

( ( n 1 Z if n = 0, 1 n Z if n = 0 H (S ) = H (1) = 0 otherwise 0 otherwise

q 0 1 2 3 4 q H (B) H (B) H (B) H (B) H (B) • • • • •

0 1 2 3 4 H (B) H (B) H (B) H (B) H (B) • • • • • Z 0 0 0 0 p p p,q E2 p,q E∞

Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 11 / 27 Example: Cohomology of K(Z, 2)

p,q p q 1 p+q E2 = H (B,H (S )) ⇒ H (1).

( ( n 1 Z if n = 0, 1 n Z if n = 0 H (S ) = H (1) = 0 otherwise 0 otherwise

q 0 1 2 3 4 q H (B) H (B) H (B) H (B) H (B) 0 0 0 0 0

H0(B) H1(B) H2(B) H3(B) H4(B) Z 0 0 0 0 p p p,q E2 p,q E∞

Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 11 / 27 Example: Cohomology of K(Z, 2)

p,q p q 1 p+q E2 = H (B,H (S )) ⇒ H (1).

( ( n 1 Z if n = 0, 1 n Z if n = 0 H (S ) = H (1) = 0 otherwise 0 otherwise

q 1 2 3 4 q Z H (B) H (B) H (B) H (B) 0 0 0 0 0

H1(B) H2(B) H3(B) H4(B) Z Z 0 0 0 0 p p p,q E2 p,q E∞

Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 11 / 27 Example: Cohomology of K(Z, 2)

p,q p q 1 p+q E2 = H (B,H (S )) ⇒ H (1).

( ( n 1 Z if n = 0, 1 n Z if n = 0 H (S ) = H (1) = 0 otherwise 0 otherwise

q 2 3 4 q Z 0 H (B) H (B) H (B) 0 0 0 0 0

H2(B) H3(B) H4(B) Z 0 Z 0 0 0 0 p p p,q E2 p,q E∞

Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 11 / 27 Example: Cohomology of K(Z, 2)

p,q p q 1 p+q E2 = H (B,H (S )) ⇒ H (1).

( ( n 1 Z if n = 0, 1 n Z if n = 0 H (S ) = H (1) = 0 otherwise 0 otherwise

q 3 4 q Z 0 Z H (B) H (B) 0 0 0 0 0

H3(B) H4(B) Z 0 Z Z 0 0 0 0 p p p,q E2 p,q E∞

Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 11 / 27 Example: Cohomology of K(Z, 2)

p,q p q 1 p+q E2 = H (B,H (S )) ⇒ H (1).

( ( n 1 Z if n = 0, 1 n Z if n = 0 H (S ) = H (1) = 0 otherwise 0 otherwise

q q Z 0 Z 0 Z 0 0 0 0 0

Z 0 Z 0 Z Z 0 0 0 0 p p p,q E2 p,q E∞

Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 11 / 27 Spectra

For the general Serre spectral sequence, we need generalized and parametrized cohomology.

∗ An (Ω-)spectrum is a sequence of pointed types Y : Z → Type such that ΩYn+1 = Yn.

Example. If A is an abelian group, HA : Spectrum where (HA)n = K(A, n).

A spectrum Y is called n-truncated if Yk is (n + k)-truncated for all k : Z.

The homotopy groups are πn(Y ) :≡ πn+k(Yk) (which is independent of k and also defined for negative n).

Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 12 / 27 Generalized Cohomology

If X is a type and Y is a spectrum, we have generalized cohomology:

n H (X,Y ) :≡ kX → Ynk0 ' π−n(X → Y ).

We get generalized and parametrized cohomology by replacing functions with dependent functions:

n H (X, λx.Y x) :≡ kΠ(x : X),Yn(x)k0. ' π−n(Π(x : X), Y x)

Here X is a type and Y : X → Spectrum.

Reduced cohomology is defined similar with basepoint-preserving sections.

Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 13 / 27 Serre Spectral Sequence

Theorem. (Serre Spectral Sequence) If f : X → B is any map and Y is a truncated spectrum, then

p,q p q p+q E2 = H (B, λb.H (fibf (b),Y )) ⇒ H (X,Y ).

If Y = HA and B is pointed simply connected, then this reduces to the previous case

p,q p q p+q E2 = H (B,H (fibf (b0),A)) ⇒ H (X,A).

Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 14 / 27 The Atiyah-Hirzebruch spectral sequence is also true if we replace all cohomologies by reduced cohomologies:

p,q p p+q E2 = He (X, λx.π−q(Y x)) ⇒ He (X, λx.Y x).

Atiyah-Hirzebruch Spectral Sequence

Theorem. (Atiyah-Hirzebruch Spectral Sequence) If X is any type and Y : X → Spectrum is a family of k-truncated spectra over X, then

p,q p p+q E2 = H (X, λx.π−q(Y x)) ⇒ H (X, λx.Y x).

Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 15 / 27 Atiyah-Hirzebruch Spectral Sequence

Theorem. (Atiyah-Hirzebruch Spectral Sequence) If X is any type and Y : X → Spectrum is a family of k-truncated spectra over X, then

p,q p p+q E2 = H (X, λx.π−q(Y x)) ⇒ H (X, λx.Y x).

The Atiyah-Hirzebruch spectral sequence is also true if we replace all cohomologies by reduced cohomologies:

p,q p p+q E2 = He (X, λx.π−q(Y x)) ⇒ He (X, λx.Y x).

Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 15 / 27 ∗ Given X : Type and Y : X → Spectrum which are s0-truncated. For x : X and s : Z we have the following fiber sequence of spectra:

K(πs(Y x), s) −−→kY xks −−→kY xks−1. The functor Y 7→ Π∗(x : X), Y x preserves fiber sequences:

∗ ∗ Π (x : X),K(πs(Y x), s) −−→ Π(x : X), kY xks −−→ Π (x : X), kY xks−1.

Let’s call these types Bs and As:

Bs −−→ As −−→ As−1.

Construction (1)

Based on the construction (sketch) by Shulman [ncatlab.org/homotopytypetheory/show/spectral+sequences] For pointed types we have the fiber sequence

K(πk(Z), k) −−→kZkk −−→kZkk−1.

Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 16 / 27 The functor Y 7→ Π∗(x : X), Y x preserves fiber sequences:

∗ ∗ Π (x : X),K(πs(Y x), s) −−→ Π(x : X), kY xks −−→ Π (x : X), kY xks−1.

Let’s call these types Bs and As:

Bs −−→ As −−→ As−1.

Construction (1)

Based on the construction (sketch) by Shulman [ncatlab.org/homotopytypetheory/show/spectral+sequences] For pointed types we have the fiber sequence

K(πk(Z), k) −−→kZkk −−→kZkk−1. ∗ Given X : Type and Y : X → Spectrum which are s0-truncated. For x : X and s : Z we have the following fiber sequence of spectra:

K(πs(Y x), s) −−→kY xks −−→kY xks−1.

Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 16 / 27 Construction (1)

Based on the construction (sketch) by Shulman [ncatlab.org/homotopytypetheory/show/spectral+sequences] For pointed types we have the fiber sequence

K(πk(Z), k) −−→kZkk −−→kZkk−1. ∗ Given X : Type and Y : X → Spectrum which are s0-truncated. For x : X and s : Z we have the following fiber sequence of spectra:

K(πs(Y x), s) −−→kY xks −−→kY xks−1. The functor Y 7→ Π∗(x : X), Y x preserves fiber sequences:

∗ ∗ Π (x : X),K(πs(Y x), s) −−→ Π(x : X), kY xks −−→ Π (x : X), kY xks−1.

Let’s call these types Bs and As:

Bs −−→ As −−→ As−1.

Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 16 / 27 Construction (2)

Bs −−→ As −−→ As−1. The long exact sequence of this fiber sequence gives: . . i πn+1(Bs) πn+1(As) πn+1(As−1) k j

i πn(Bs) πn(As) πn(As−1) k j

i πn−1(Bs) πn−1(As) πn−1(As−1) k . .

Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 17 / 27 Construction (3)

Define M M E = πn(Bs) and D = πn(As). n,s n,s These long exact sequences give an exact couple between bigraded abelian groups.

i D D

k j E

Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 18 / 27 Construction (4)

From an exact couple

i D D

k j E

we build a derived exact couple

i0 D0 D0

k0 j0 E0

where E0 is the (co)homology of E with differential d :≡ j ◦ k : E → E.

Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 19 / 27 Construction (5)

We iterate this process, so that we get a sequence of exact couples (Er,Dr, ir, jr, kr).

Now (Er, dr)r forms the Atiyah-Hirzebruch spectral sequence:

p,q p p+q E2 = He (X, λx.π−q(Y x)) ⇒ He (X, λx.Y x).

(We have applied the reindexing (p, q) = (s − n, −s).)

Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 20 / 27 For the Serre spectral sequence, we’re given a map f : X → B and a truncated spectrum Z. We define

Y = λ(b : B).fibf (b) → Z : B → Spectrum . Then q π−q(Y b) = π−q(fibf (b) → Z) = H (fibf (b),Z)

p+q H (B, λb.Y b) = π−(p+q)(Π(b : B), fibf (b) → Z)

= π−(p+q)((Σ(b : B), fibf (b)) → Z)

= π−(p+q)(X → Z) = Hp+q(X,Z) This gives the Serre spectral sequence

Construction (6)

p,q p p+q E2 = H (X, λx.π−q(Y x)) ⇒ H (X, λx.Y x).

p,q p q p+q E2 = H (B, λb.H (fibf (b),Z)) ⇒ H (X,Z).

Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 21 / 27 Then q π−q(Y b) = π−q(fibf (b) → Z) = H (fibf (b),Z)

p+q H (B, λb.Y b) = π−(p+q)(Π(b : B), fibf (b) → Z)

= π−(p+q)((Σ(b : B), fibf (b)) → Z)

= π−(p+q)(X → Z) = Hp+q(X,Z) This gives the Serre spectral sequence

Construction (6)

p,q p p+q E2 = H (X, λx.π−q(Y x)) ⇒ H (X, λx.Y x). For the Serre spectral sequence, we’re given a map f : X → B and a truncated spectrum Z. We define

Y = λ(b : B).fibf (b) → Z : B → Spectrum .

p,q p q p+q E2 = H (B, λb.H (fibf (b),Z)) ⇒ H (X,Z).

Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 21 / 27 Construction (6)

p,q p p+q E2 = H (X, λx.π−q(Y x)) ⇒ H (X, λx.Y x). For the Serre spectral sequence, we’re given a map f : X → B and a truncated spectrum Z. We define

Y = λ(b : B).fibf (b) → Z : B → Spectrum . Then q π−q(Y b) = π−q(fibf (b) → Z) = H (fibf (b),Z)

p+q H (B, λb.Y b) = π−(p+q)(Π(b : B), fibf (b) → Z)

= π−(p+q)((Σ(b : B), fibf (b)) → Z)

= π−(p+q)(X → Z) = Hp+q(X,Z) This gives the Serre spectral sequence p,q p q p+q E2 = H (B, λb.H (fibf (b),Z)) ⇒ H (X,Z).

Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 21 / 27 Formalization

Construction formalized in the Lean proof assistant. Available at github.com/cmu-phil/Spectral. The formalization took almost 2 years: November 2015 – July 2017. Formalized by vD, Jeremy Avigad, Steve Awodey, Ulrik Buchholtz, Egbert Rijke and Mike Shulman. Formalization is ∼10k-20k LoC (the total size of Lean-HoTT is 53k LoC).

Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 22 / 27 Applications

The remainder of the slides is mostly future work.

We can compute cohomology groups of certain spaces (such as n K(Z, n) and ΩS ). We can compute cohomology groups of generalized cohomology theories (K-theory). We can construct the Gysin and Wang sequences. To compute more homotopy groups of spheres, we need the Serre spectral sequence for homology.

Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 23 / 27 Homology with coefficients in a spectrum Y can be defined as

Hen(X,Y ) = πn(X ∧ Y ) = colimk(πn+k(X ∧ Yk)).

Smash Product

For pointed types A and B, the smash product A ∧ B is the following homotopy pushout.

A ∨ B 1 • b0

A × B A ∧ B B • A a0

Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 24 / 27 Smash Product

For pointed types A and B, the smash product A ∧ B is the following homotopy pushout.

A ∨ B 1 • b0

A × B A ∧ B B • A a0 Homology with coefficients in a spectrum Y can be defined as

Hen(X,Y ) = πn(X ∧ Y ) = colimk(πn+k(X ∧ Yk)).

Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 24 / 27 Parametrized Homology

We will also need parametrized homology.

Hen(X, λx.Y x) :≡ πn((x : X) ∧ Y x)

(x : A) ∧ B(x) is a parametrized version of the smash product, the following homotopy pushout:

B(a ) A ∨ B(a0) 1 0 B b0

Σ(x : A),B(x) (x : A) ∧ B(x) • A a0

Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 25 / 27 Spectral Sequences for Homology

Some challenges: Smashing doesn’t preserve spectra: we need to apply spectrification. We need to prove that smashing preserves fiber sequences. We should get the corresponding spectral sequences for homology:

2 Ep,q = Hep(X, λx.πq(Y x)) ⇒ Hep+q(X, λx.Y x).

2 Ep,q = Hp(B, λb.Hq(fibf (b),Y )) ⇒ Hp+q(X,Y ).

Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 26 / 27 Applications

Applications of the homology Serre spectral sequence: Serre class theorem (constructively?) n Computation of πn+k(S ) for k ≤ 3.

Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 27 / 27