Miguel Angel Marco Buzunáriz
MEGA, Madrid, June 20th 2019
Consider A_{DR}(M) the set of all forms in a manifold M
They form an algebra that is:
We call such an algebra a GCDA.
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.
Theorem: Every GCA is the quotient of a free GCA by a bilateral ideal.
Finitely generated free GCA’s are given by:
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.
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
[x22*x5, x21*x5, x22^2*x3, x21*x22*x3, x21^2*x3]
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
([x22*x5, x21*x5, x22^2*x3, x21*x22*x3, x21^2*x3],
[x22*x5, x21*x5, x22^2*x3, x21*x22*x3])
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.
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
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.
[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
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).
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
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
{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
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.
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.
\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.
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:
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.
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.
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
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
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)
A.<e1,e2,e3,e4,e5,e6> = GradedCommutativeAlgebra(QQ)
B = A.cdg_algebra({e3:-e1*e3-e2*e5, e4:e1*e4-e2*e6, e5:-e1*e5, e6:e1*e6})
N.<x,y,z> = GradedCommutativeAlgebra(QQ)
NN = N.cdg_algebra({x:y*z,y:x*z,z:x*y})
NN.minimal_model(4)
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,)
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)
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
---------------------------------------------------------------------------
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
A.<x,y> = GradedCommutativeAlgebra(QQ,degrees=(5,5))
I = A.ideal([x*y])
B = A.quotient(I).cdg_algebra({})
phi = B.minimal_model(25)
phi
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)
[2, 1, 2, 3, 6, 9]
Univariate Polynomial Ring in x over Finite Field of size 2 (using GF2X)
[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]
[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