Computation of minimal model of GCDA’s

Miguel Angel Marco Buzunáriz

MEGA, Madrid, June 20th 2019

Motivation

Consider A_{DR}(M) the set of all forms in a manifold M

They form an algebra that is:

  • graded commutative: a\wedge b = (-1)^{|ab|}b\wedge a
  • differential: d(a\wedge b)=d(a)\wedge b + (-1)^{|a|}a\wedge d(b)

We call such an algebra a GCDA.

Free GCA’s

Consider a graded vector space

V=\bigoplus_{i=0}^\infty V_i

Take its even and odd parts:

V_e=\bigoplus_{i=0}^\infty V_{2i}, V_o=\bigoplus_{i=0}^\infty V_{2i+1}

The algebra \left(Sym V_e\right) \oplus \left(\bigwedge V_o\right) is a graded commutative algebra.

Free GCA’s

Theorem: Every GCA is the quotient of a free GCA by a bilateral ideal.

Finitely generated free GCA’s are given by:

  • A finite set of generators x_1,\ldots,x_n
  • The degree of each generator |x_1|,\ldots ,|x_n|

If we add a finite set of homogenous relations \left\{r_i\right\}, we determine every possible finitely generated GCA.

A noncommutative Gröbner basis provides a normal form for each element.

Example

SageMath has support for GCDA’s since version 6.4

Graded Commutative Algebra with generators ('x21', 'x22', 'x3', 'x5') in degrees (2, 2, 3, 5) over Rational Field
Defining x21, x22, x3, x5
x3*x5 - x21*x5 - 2*x21*x3 + x22*x3 - x21*x22

Example

[x22*x5, x21*x5, x22^2*x3, x21*x22*x3, x21^2*x3]

Example

Graded Commutative Algebra with generators ('x21', 'x22', 'x3', 'x5') in degrees (2, 2, 3, 5) with relations [-x21^2*x3 + x21*x5] over Rational Field
Defining x21, x22, x3, x5
x21^2*x5

Example

([x22*x5, x21*x5, x22^2*x3, x21*x22*x3, x21^2*x3],
 [x22*x5, x21*x5, x22^2*x3, x21*x22*x3])

Differentials

A differential in a GCA is a linear map of degree 1 d:A\rightarrow A such that d^2=0 and satisfying the Leibniz rule d(xy)=d(x)y+(-1)^{|x|}d(y)

Given the image of the generators d(x_1),\ldots,d(x_n), the image of every element is determined.

Note that not every choice of d(x_i) determines a differential.

Example

Commutative Differential Graded Algebra with generators ('x21', 'x22', 'x3', 'x5') in degrees (2, 2, 3, 5) over Rational Field with differential:
   x21 --> x3
   x22 --> 0
   x3 --> 0
   x5 --> x22^3

Cohomology

The differential induces a cochain complex structure on the GCDA:

A_0\xrightarrow{\quad d \quad}A_1\xrightarrow{\quad d \quad}A_2\xrightarrow{\quad d \quad}\cdots

So we can consider the corresponding cohomology

H^n(A)=Ker(d\mid_{A_n})/Im(d\mid_{A_{n-1}})

If there are no generators of degree 0, we have a finite basis of each graded part.

With that, a basis for the cohomology can be computed by gaussian eliminiation.

\quad

[x22^3, x21*x22^2, x21^2*x22, x21^3]
[x22*x5, x21*x5, x22^2*x3, x21*x22*x3, x21^2*x3]
[x3*x5, x22^4, x21*x22^3, x21^2*x22^2, x21^3*x22, x21^4]

Differential of Commutative Differential Graded Algebra with generators ('x21', 'x22', 'x3', 'x5') in degrees (2, 2, 3, 5) over Rational Field
  Defn: x21 --> x3
        x22 --> 0
        x3 --> 0
        x5 --> x22^3
[0 0 0 0 0]
[0 0 1 0 0]
[0 0 0 2 0]
[0 0 0 0 3]

[0 1 0 0 0 0]
[1 0 1 0 0 0]
[0 0 0 0 0 0]
[0 0 0 0 0 0]
[0 0 0 0 0 0]
Vector space of degree 5 and dimension 3 over Rational Field
Basis matrix:
[0 0 1 0 0]
[0 0 0 1 0]
[0 0 0 0 1]

Vector space of degree 5 and dimension 3 over Rational Field
Basis matrix:
[0 0 1 0 0]
[0 0 0 1 0]
[0 0 0 0 1]
Free module generated by {} over Rational Field

Cohomology algebras

The cohomology H^*(A) of a GCDA A is itself a GCA. Endowed with a trivial differential, it is a GCDA isomorphic to its own cohomology.

Note that even if A is finitely generated, H^*(A) might be infinitely generated (as an algebra).

However, we can compute it up to any given degree g. That is, we can compute a finitely GCDA that whose truncation to degree g isomorphic to that of H^*(A).

Example of cohomology algebras

Commutative Differential Graded Algebra with generators ('x', 'y', 'z') in degrees (1, 2, 2) over Rational Field with differential:
   x --> 0
   y --> x*z
   z --> 0

Example of cohomology algebras

Free module generated by {[x]} over Rational Field
Free module generated by {[z]} over Rational Field
Free module generated by {[x*y]} over Rational Field
Free module generated by {[z^2]} over Rational Field
Free module generated by {[x*y^2]} over Rational Field
Free module generated by {[z^3]} over Rational Field
Free module generated by {[x*y^3]} over Rational Field
Free module generated by {[z^4]} over Rational Field
Free module generated by {[x*y^4]} over Rational Field

Example of cohomology algebras

{1: [x], 2: [z], 3: [x*y], 5: [x*y^2], 7: [x*y^3], 9: [x*y^4]}
Commutative Differential Graded Algebra with generators ('x0', 'x1', 'x2', 'x3', 'x4', 'x5') in degrees (1, 2, 3, 5, 7, 9) with relations [x0*x1, x0*x2, x1*x2, x0*x1^2, x0*x3, x0*x1*x2, x1*x3, x1^2*x2, x0*x1^3, x2*x3, x0*x4, x0*x1*x3, x0*x1^2*x2, x0*x2*x3, x1*x4, x1^2*x3, x1^3*x2, x0*x1^4] over Rational Field with differential:
   x0 --> 0
   x1 --> 0
   x2 --> 0
   x3 --> 0
   x4 --> 0
   x5 --> 0

Minimal models

Definition: A GCDA A is said to be a Sullivan algebra if it admits an ordered free set of generators \left\{x_i\right\}_{i\in I}, such that d(x_i) can be expressed in terms of \left\{x_j\mid j<i\right\}.

Definition: A Sullivan algebra is said to be minimal if |x_i|\leq|x_j| for every i<j.

Definition: A morphism of GCDA’s \phi:M\longrightarrow A is said to be a quasi-isomorphism if the induced morphism \phi^*:H^*(M)\longrightarrow H^*(A) is an isomorphism.

Definition: A minimal model of a CDGA A is quasi-isomorphism \phi:M\longrightarrow A from a minimal algebra M to A.

Construction of minimal models

Theorem: Every GCDA A admits a unique (up to isomorphism) minimal model. Moreover, it is a complete invariant of the quasi-isomorphism type.

Theorem: For a topological space X, the rank of \pi_i(X,\mathbb{Q}) coincides with the number of generators of degree i in the minimal model of A_{PL}(X).

It can be constructed degree by degree.

The idea is to increase M at each step by adding new generators. For each generator we must give its differential and its image by the morphism.

Step zero

  • Let g_1 be the lowest degree with nontrivial cohomology.
    • Take a basis \left\{a^{g_1}_1,\ldots,a^{g_1}_{n_1}\right\} of H^{g_1}(A).
    • Consider one generator x_i^{g_1} for each a^{g_1}_i. We impose \phi(x_i^{g_1})=a^{g_1}_i.
    • These generators will have trivial differential.

\begin{array}{ccccc} a^{g_1}_1 & \xleftarrow{\quad \phi \quad} & x^{g_1}_1 & \xrightarrow{\quad d \quad} & 0 \\ a^{g_1}_2 & \xleftarrow{\quad \phi \quad} & x^{g_1}_2 & \xrightarrow{\quad d \quad} & 0 \\ \vdots & & \vdots & & \vdots \end{array}

After this step, we have a g_1-minimal model \phi_{g_1}:M_{g_1}\rightarrow A. That is, it induces isomorphisms \phi_{g}:H^g(M)\rightarrow H^g(A) for g\leq g_1.

Induction step: injectivity

Now assume we have a g_i-minimal model \phi_{g_i}:M_{g_i}\rightarrow A, we will extend it to get injectivity in \phi_{g_i+1}^{g_i+1}:H^{g_i+1}(M_{g_i+1})\rightarrow H^{g_i+1}(A).

Let [k_1],\ldots,[k_m] be a basis of Ker(\phi_{g_i}^{g_i+1}:M_{g_i}^{g_i+1}\rightarrow A^{g_i+1}). For each k_l we have:

  • d(k_l)=0 \quad\Rightarrow\quad d(\phi_{g_i}(k_l))=0
  • [\phi_{g_l}(k_l)]=0\quad\Rightarrow\quad \exists c_l\in M_{g_i} such that d(c_i)=\phi_{g_i}(k_l)

Induction step: injectivity

Extend the algebra M with new generators:

\begin{array}{ccccc} c_1 & \xleftarrow{\quad \phi \quad} & y_1^{g_i} & \xrightarrow{\quad d \quad} & k_1 \\ c_2 & \xleftarrow{\quad \phi \quad} & y_2^{g_i} & \xrightarrow{\quad d \quad} & k_2 \\ \vdots & & \vdots & & \vdots \end{array}

Repeat this process until we have no kernel.

Warning: If H^1(A)\neq 0, this process might not finish in a finite number of steps. In that case we cannot compute it effectively.

Induction step: surjectivity

assume we have a g_i-minimal model \phi_{g_i}:M_{g_i}\rightarrow A, such that \phi_{g_i+1}^{g_i+1}:H^{g_i+1}(M_{g_i+1})\rightarrow H^{g_i+1}(A) is injective. We will now extend the model to make it surjective.

Consider \phi_{g_i+1}^{g_i+1}(H^{g_i+1}(M_{g_i+1})) as subspace of H^{g_i+1}(A).

We can take a basis \left\{[b_1],\ldots,[b_p],[b_{p+1}],\ldots ,[b_s]\right\} of H^{g_i+1}(A) such that \left\{[b_{p+1}],\ldots ,[b_s]\right\} is a basis of \phi_{g_i+1}^{g_i+1}(H^{g_i+1}(M_{g_i+1})).

For each [b_l] , l\leq p, consider a new generator x^{g_i+1}_l, with d(x^{g_i+1}_l)=0. We impose \phi(x^{g_i+1}_l)=b_l.

Induction step: surjectivity

Extend the algebra M with new generators:

\begin{array}{ccccc} b_1 & \xleftarrow{\quad \phi \quad} & x_1^{g_i+1} & \xrightarrow{\quad d \quad} & 0 \\ b_2 & \xleftarrow{\quad \phi \quad} & x_2^{g_i+1} & \xrightarrow{\quad d \quad} & 0 \\ \vdots & & \vdots & & \vdots \end{array}

After one iteration of this step, we get a g_i+1-minimal model \phi_{g_i+1}:M_{g_i+1}\rightarrow A

Demo

We have implemented this method in SageMath (will be available in version 8.8)

Commutative Differential Graded Algebra with generators ('x', 'y', 'z') in degrees (1, 2, 2) over Rational Field with differential:
   x --> 0
   y --> x*z
   z --> 0

Demo

Commutative Differential Graded Algebra morphism:
  From: Commutative Differential Graded Algebra with generators ('x1_0', 'x2_0', 'y2_0') in degrees (1, 2, 2) over Rational Field with differential:
   x1_0 --> 0
   x2_0 --> 0
   y2_0 --> x1_0*x2_0
  To:   Commutative Differential Graded Algebra with generators ('x', 'y', 'z') in degrees (1, 2, 2) over Rational Field with differential:
   x --> 0
   y --> x*z
   z --> 0
  Defn: (x1_0, x2_0, y2_0) --> (x, z, y)

Non formal example

Commutative Differential Graded Algebra morphism:
  From: Commutative Differential Graded Algebra with generators ('x3_0',) in degrees (3,) over Rational Field with differential:
   x3_0 --> 0
  To:   Commutative Differential Graded Algebra with generators ('x', 'y', 'z') in degrees (1, 1, 1) over Rational Field with differential:
   x --> y*z
   y --> x*z
   z --> x*y
  Defn: (x3_0,) --> (x*y*z,)

Non formal example

Commutative Differential Graded Algebra morphism:
  From: Commutative Differential Graded Algebra with generators ('x1_0', 'x1_1', 'x2_0', 'x2_1', 'y2_0', 'y2_1', 'y3_0', 'y3_1', 'y3_2', 'y3_3', 'y3_4', 'y3_5', 'y3_6', 'y3_7', 'y3_8') in degrees (1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3) over Rational Field with differential:
   x1_0 --> 0
   x1_1 --> 0
   x2_0 --> 0
   x2_1 --> 0
   y2_0 --> x1_0*x2_0
   y2_1 --> -1/2*x1_0*x2_1 + x1_0*y2_0
   y3_0 --> -1/2*x2_1^2 + 1/2*x2_1*y2_0 - 1/2*y2_0^2 + x2_0*y2_1
   y3_1 --> x2_0*x2_1
   y3_2 --> x2_0^2
   y3_3 --> x2_0*y2_0 + x1_0*y3_2
   y3_4 --> 1/2*x2_1*y2_0 + 1/2*y2_0^2 - x2_0*y2_1 + x1_0*y3_1
   y3_5 --> x2_1*y2_1 + x1_0*y3_0 + x1_0*y3_4
   y3_6 --> 1/2*y2_0^2 + x1_0*y3_3
   y3_7 --> 1/3*x2_1*y2_1 + 1/3*y2_0*y2_1 + 1/3*x1_0*y3_0 + x1_0*y3_6
   y3_8 --> 1/6*y2_1^2 - 1/2*x1_0*y3_5 + x1_0*y3_7
  To:   Commutative Differential Graded Algebra with generators ('e1', 'e2', 'e3', 'e4', 'e5', 'e6') in degrees (1, 1, 1, 1, 1, 1) over Rational Field with differential:
   e1 --> 0
   e2 --> 0
   e3 --> -e1*e3 - e2*e5
   e4 --> e1*e4 - e2*e6
   e5 --> -e1*e5
   e6 --> e1*e6
  Defn: (x1_0, x1_1, x2_0, x2_1, y2_0, y2_1, y3_0, y3_1, y3_2, y3_3, y3_4, y3_5, y3_6, y3_7, y3_8) --> (e2, e1, e5*e6, e4*e5 + e3*e6, e4*e5, 1/2*e3*e4, 0, 0, 0, 0, 0, 0, 0, 0, 0)

Non formal example

Commutative Differential Graded Algebra with generators ('x0', 'x1', 'x2', 'x3', 'x4') in degrees (1, 1, 2, 2, 3) with relations [x0*x2] over Rational Field with differential:
   x0 --> 0
   x1 --> 0
   x2 --> 0
   x3 --> 0
   x4 --> 0

Non formal example

---------------------------------------------------------------------------

ValueError                                Traceback (most recent call last)

<ipython-input-28-ae993d1c5938> in <module>()
----> 1 H.minimal_model(Integer(3), max_iterations=Integer(6))


/home/mmarco/sage/local/lib/python2.7/site-packages/sage/algebras/commutative_dga.pyc in minimal_model(self, i, max_iterations)
   2528         for degree in range(degnzero+1, max_degree+1):
   2529             phi = extendx(phi, degree)
-> 2530             phi = extendy(phi, degree+1)
   2531             self._minimalmodels[degree] = phi
   2532


/home/mmarco/sage/local/lib/python2.7/site-packages/sage/algebras/commutative_dga.pyc in extendy(phi, degree)
   2489                     return phi
   2490                 if iteration == max_iterations-1:
-> 2491                     raise ValueError("could not cover all relations in max iterations in degree {}".format(degree))
   2492                 ndifs = [CB.lift(g) for g in K.basis()]
   2493                 basisdegree = B.basis(degree)


ValueError: could not cover all relations in max iterations in degree 3

Rational homotopy of wedges of spheres

Commutative Differential Graded Algebra morphism:
  From: Commutative Differential Graded Algebra with generators ('x5_0', 'x5_1', 'y9_0', 'y13_0', 'y13_1', 'y17_0', 'y17_1', 'y17_2', 'y21_0', 'y21_1', 'y21_2', 'y21_3', 'y21_4', 'y21_5', 'y25_0', 'y25_1', 'y25_2', 'y25_3', 'y25_4', 'y25_5', 'y25_6', 'y25_7', 'y25_8') in degrees (5, 5, 9, 13, 13, 17, 17, 17, 21, 21, 21, 21, 21, 21, 25, 25, 25, 25, 25, 25, 25, 25, 25) over Rational Field with differential:
   x5_0 --> 0
   x5_1 --> 0
   y9_0 --> x5_0*x5_1
   y13_0 --> x5_1*y9_0
   y13_1 --> x5_0*y9_0
   y17_0 --> x5_0*y13_0 + x5_1*y13_1
   y17_1 --> x5_1*y13_0
   y17_2 --> x5_0*y13_1
   y21_0 --> y9_0*y13_1 + x5_0*y17_0
   y21_1 --> y9_0*y13_0 + x5_0*y17_1
   y21_2 --> x5_0*y17_0 + x5_1*y17_2
   y21_3 --> x5_1*y17_1
   y21_4 --> x5_1*y17_0 + x5_0*y17_1
   y21_5 --> x5_0*y17_2
   y25_0 --> y13_0*y13_1 + x5_1*y21_0 - x5_0*y21_1 + x5_0*y21_4
   y25_1 --> y9_0*y17_2 + x5_0*y21_0 + x5_0*y21_2
   y25_2 --> y9_0*y17_1 + x5_1*y21_1 + 2*x5_0*y21_3
   y25_3 --> y9_0*y17_0 + x5_1*y21_0 + x5_0*y21_1 + 2*x5_0*y21_4
   y25_4 --> x5_0*y21_2 + x5_1*y21_5
   y25_5 --> x5_0*y21_3 + x5_1*y21_4
   y25_6 --> x5_1*y21_3
   y25_7 --> x5_1*y21_2 + x5_0*y21_4
   y25_8 --> x5_0*y21_5
  To:   Commutative Differential Graded Algebra with generators ('x', 'y') in degrees (5, 5) with relations [x*y] over Rational Field with differential:
   x --> 0
   y --> 0
  Defn: (x5_0, x5_1, y9_0, y13_0, y13_1, y17_0, y17_1, y17_2, y21_0, y21_1, y21_2, y21_3, y21_4, y21_5, y25_0, y25_1, y25_2, y25_3, y25_4, y25_5, y25_6, y25_7, y25_8) --> (y, x, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)

Rational homotopy of wedges of spheres

[2, 1, 2, 3, 6, 9]
Univariate Polynomial Ring in x over Finite Field of size 2 (using GF2X)

Rational homotopy of wedges of spheres

[x,
 x + 1,
 x^2 + x + 1,
 x^3 + x + 1,
 x^3 + x^2 + 1,
 x^4 + x + 1,
 x^4 + x^3 + 1,
 x^4 + x^3 + x^2 + x + 1,
 x^5 + x^2 + 1,
 x^5 + x^3 + 1,
 x^5 + x^3 + x^2 + x + 1,
 x^5 + x^4 + x^2 + x + 1,
 x^5 + x^4 + x^3 + x + 1,
 x^5 + x^4 + x^3 + x^2 + 1,
 x^6 + x + 1,
 x^6 + x^3 + 1,
 x^6 + x^4 + x^2 + x + 1,
 x^6 + x^4 + x^3 + x + 1,
 x^6 + x^5 + 1,
 x^6 + x^5 + x^2 + x + 1,
 x^6 + x^5 + x^3 + x^2 + 1,
 x^6 + x^5 + x^4 + x + 1,
 x^6 + x^5 + x^4 + x^2 + 1,
 x^7 + x + 1,
 x^7 + x^3 + 1,
 x^7 + x^3 + x^2 + x + 1,
 x^7 + x^4 + 1,
 x^7 + x^4 + x^3 + x^2 + 1,
 x^7 + x^5 + x^2 + x + 1,
 x^7 + x^5 + x^3 + x + 1,
 x^7 + x^5 + x^4 + x^3 + 1,
 x^7 + x^5 + x^4 + x^3 + x^2 + x + 1,
 x^7 + x^6 + 1,
 x^7 + x^6 + x^3 + x + 1,
 x^7 + x^6 + x^4 + x + 1,
 x^7 + x^6 + x^4 + x^2 + 1,
 x^7 + x^6 + x^5 + x^2 + 1,
 x^7 + x^6 + x^5 + x^3 + x^2 + x + 1,
 x^7 + x^6 + x^5 + x^4 + 1,
 x^7 + x^6 + x^5 + x^4 + x^2 + x + 1,
 x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + 1]

Rational homotopy of wedges of spheres

[2, 1, 2, 3, 6, 9, 18]

Conjecture: The rank of the 2i+1’th homotopy group of \mathbb{S}^3\vee \mathbb{S}^3 equals the number of irreducible polynomials of degree i in \mathbb{F}_2

Future work

  • Deal with (some) cases of infinite type.
    • In particular A_{PL}(X) for finite simplicial sets X.
  • Criterion for formality
    • Difficulty: checking if two minimal models are isomorphic or not.
    • Idea: Make sure that the algorithm gives identical presentations if they are.

Thank you