CoCalc -- palmieri-sd74.ipynb

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Project: Sage Days 74

Path: Talks/palmieri-sd74.ipynb

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Kernel: SageMath 7.3.beta2

Nerves

The nerve of a category (or group or monoid) is naturally constructed as a simplicial set.

The nerve (or classifying space) of a group:

one vertex
one edge for each element of the group
one $n$ -simplex for each $n$ -tuple $(a_1, a_2, ..., a_n)$ , non-degenerate if $a_i \neq 1$ for all $i$
face maps: multiply consecutive elements:
- $d_{0} (a_1 a_2 \cdots a_n) = a_{2} \cdots a_{n}$
- $d_{n} (a_{1} a_{2} \cdots a_{n}) = a_{1} \cdots a_{n-1}$
- $d_{i} (a_{1} a_{2} \cdots a_{n}) = a_{1} \cdots (a_{i} a_{i+1}) \cdots a_{n-1}$ , $1 \leq i \leq n-1$
degeneracy maps: insert $1$ in the $j$ th spot:
- $s_{j} (a_{1} \cdots a_{n}) = a_{1} \cdots a_{j} 1 a_{j+1} \cdots a_{n}$ , $0 \leq j \leq n$

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Similar for monoids or categories: one vertex for each object, one 1-simplex for each morphism, one $n$ -simplex for each collection of $n$ composable morphisms.

Also, given the nerve of a category, you can recover the category.

Are categories in good enough shape in Sage to be able to define the nerve of a (finite) category?

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Example: real projective space

Infinite dimensional real projective space $\mathbb{R} P^{\infty}$ is the classifying space of the group $\mathbb{Z}/2\mathbb{Z}$ . There is one non-degenerate simplex in each dimension. $\mathbb{R} P^{n}$ is its $n$ -skeleton.

Look at the $f$ -vectors for simplicial complex versions of $\mathbb{R} P^{n}$ for small values of $n$ :

In [1]:

simplicial_complexes.RealProjectiveSpace(2).f_vector()

[1, 6, 15, 10]

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simplicial_complexes.RealProjectiveSpace(3).f_vector()

[1, 11, 51, 80, 40]

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simplicial_complexes.RealProjectiveSpace(4).f_vector()

[1, 16, 120, 330, 375, 150]

In [4]:

simplicial_complexes.RealProjectiveSpace?

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For $\mathbb{R}P^5$ , the $f$ -vector is (1, 63, 903, 4200, 8400, 7560, 2520).

In comparison, as a simplicial set:

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simplicial_sets.RealProjectiveSpace(10).f_vector()

[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]

Some calculations...

In [6]:

%time simplicial_complexes.RealProjectiveSpace(4).cohomology_ring(GF(2))

CPU times: user 938 ms, sys: 101 ms, total: 1.04 s
Wall time: 4.23 s

Cohomology ring of Minimal triangulation of RP^4 over Finite Field of size 2

In [7]:

%time simplicial_sets.RealProjectiveSpace(36).cohomology_ring(GF(2))

CPU times: user 340 ms, sys: 17.5 ms, total: 358 ms
Wall time: 347 ms

Cohomology ring of RP^36 over Finite Field of size 2

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H = simplicial_sets.RealProjectiveSpace(36).cohomology_ring(GF(2))

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H.basis()

Finite family {(18, 0): h^{18,0}, (3, 0): h^{3,0}, (8, 0): h^{8,0}, (29, 0): h^{29,0}, (34, 0): h^{34,0}, (14, 0): h^{14,0}, (19, 0): h^{19,0}, (24, 0): h^{24,0}, (4, 0): h^{4,0}, (9, 0): h^{9,0}, (30, 0): h^{30,0}, (35, 0): h^{35,0}, (15, 0): h^{15,0}, (20, 0): h^{20,0}, (25, 0): h^{25,0}, (5, 0): h^{5,0}, (10, 0): h^{10,0}, (31, 0): h^{31,0}, (36, 0): h^{36,0}, (0, 0): h^{0,0}, (21, 0): h^{21,0}, (26, 0): h^{26,0}, (6, 0): h^{6,0}, (11, 0): h^{11,0}, (16, 0): h^{16,0}, (1, 0): h^{1,0}, (22, 0): h^{22,0}, (27, 0): h^{27,0}, (32, 0): h^{32,0}, (7, 0): h^{7,0}, (12, 0): h^{12,0}, (17, 0): h^{17,0}, (2, 0): h^{2,0}, (23, 0): h^{23,0}, (28, 0): h^{28,0}, (33, 0): h^{33,0}, (13, 0): h^{13,0}}

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x = list(H.basis(1))[0]
x

h^{1,0}

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x**25

h^{25,0}

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x.Sq(1).Sq(2).Sq(4)

h^{8,0}

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Classifying space of $C_{3}$

In Sage, the cyclic group $C_{3}$ has three elements, $1$ , $f$ , $f^{2}$ .

So its classifying space has two nondegenerate 1-simplices, four non-degenerate 2-simplices ( $f*f$ , $f^{2}*f$ , $f*f^{2}$ , $f^{2}*f^{2}$ ), eight non-degenerate 3-simplices, etc.

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C3 = groups.misc.MultiplicativeAbelian([3])
BC3 = C3.nerve_n_skeleton(5)
BC3

Simplicial set with 63 non-degenerate simplices

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BC3.homology()

{0: 0, 1: C3, 2: 0, 3: C3, 4: 0, 5: Z^22}

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BC3.fundamental_group()

Finitely presented group < e0 | e0^3 >

Classifying space of $\Sigma_{3}$

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Sigma3 = groups.permutation.Symmetric(3)
BSigma3 = Sigma3.nerve_n_skeleton(4)

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BSigma3

Simplicial set with 781 non-degenerate simplices

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BSigma3.homology()

{0: 0, 1: C2, 2: 0, 3: C6, 4: Z^520}

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BSigma3.fundamental_group()

Finitely presented group < e0, e1 | e0^2, e1^3, (e0*e1^-1)^2 >

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BSigma4 = groups.permutation.Symmetric(4).nerve_n_skeleton(3)
BSigma4

Simplicial set with 12720 non-degenerate simplices

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BSigma4.fundamental_group()

Finitely presented group < e3, e19 | e19^3, (e3*e19^-1)^2, e3^4 >

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In [24]:

BD17 = groups.permutation.Dihedral(17).nerve_n_skeleton(2)
BD17.f_vector()

[1, 33, 1089]

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BD17.fundamental_group()

Finitely presented group < e1, e4 | e4^2, (e4*e1^-1)^2, e1^17 >

Example: complex projective space

$\mathbb{C} P^{n}$ is the $2n$ -skeleton of the classifying space of the Lie group $S^{1}$ . Sage can't construct it that way, but work of Sergeraert (Kenzo, CAT) leads to constructions we can use in Sage.

$f$ -vectors as simplicial complexes:

In [26]:

simplicial_complexes.ComplexProjectivePlane().f_vector()

[1, 9, 36, 84, 90, 36]

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simplicial_complexes.ComplexProjectiveSpace(3).f_vector()

---------------------------------------------------------------------------
AttributeError                            Traceback (most recent call last)
<ipython-input-27-4e74e8a803a4> in <module>()
----> 1 simplicial_complexes.ComplexProjectiveSpace(Integer(3)).f_vector()

AttributeError: 'module' object has no attribute 'ComplexProjectiveSpace'

As simplicial sets:

In [28]:

simplicial_sets.ComplexProjectiveSpace(2).f_vector()

[1, 0, 2, 3, 3]

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simplicial_sets.ComplexProjectiveSpace(3).f_vector()

[1, 0, 3, 10, 25, 30, 15]

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simplicial_sets.ComplexProjectiveSpace(4).f_vector()

[1, 0, 4, 22, 97, 255, 390, 315, 105]

In [31]:

CP3 = simplicial_sets.ComplexProjectiveSpace(3)

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CP3.homology()

{0: 0, 1: 0, 2: Z, 3: 0, 4: Z, 5: 0, 6: Z}

In [35]:

CP3.fundamental_group()

Group( <identity> of ... )

To do:

good conversions from simplicial complexes (and other objects) to simplicial sets
simplicial abelian groups, $k$ -skeleton of $K(\pi,n)$
infinite simplicial sets
general framework for simplicial objects in a category
higher homotopy groups (?)

In [38]:

from sage.homology.simplicial_set import *

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v = NonDegenerateSimplex(0)
w = NonDegenerateSimplex(0)

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e = NonDegenerateSimplex(1)

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SimplicialSet({e: (v, w)})

Simplicial set with 3 non-degenerate simplices

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