homology.sage In Sage Math, implementation of simplicial complexes for computing chain complexes and homology cf. https://www.math.hmc.edu/~su/pcmi/projects/simplicial_module/simplicial_pcmi.pdf and J.M. Lee's Introduction to Topological Manifolds

homology.sage

## homology.sage
## This is my implementation of Algebraic Topology, Homology, Simplicial Complexes 
##   utilizing
## Sage Math
##  from 
## 
## The main reference that I'll liberally copy from is from is 
##   Introduction to Topological Manifolds
##   by J.M. Lee
################################################################################
## Copyleft 2015, Ernest Yeung <ernestyalumni@gmail.com>
## 
## 20150515
## 
## This program, along with all its code, is free software; you can redistribute 
## it and/or modify it under the terms of the GNU General Public License as 
## published by the Free Software Foundation; either version 2 of the License, 
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## 
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## but WITHOUT ANY WARRANTY; without even the implied warranty of 
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 
## GNU General Public License for more details.
## 
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# cf. Sage Reference Manual: Cell complexes and their homology, Release 6.6 
# homology.pdf
# Ch. 4 Finite Simplicial Complexes

closedinterval = SimplicialComplex((('a','b'), ['a','b']))

from itertools import combinations

def create_n_simplex_all(n=1,n_fill=1,n_0 = 0):
    """
    create_n_simplex_all = create_n_simplex_all(n=1,n_fill,n_0=0)
    Creates a n-simplex that has all possible combinations of the vertices for the faces

    e.g. Example of usage
    cinqsimpl = create_n_simplex_all(5,5)
    cinqsimpl.cells()
    cinqsimpl.dimension() # 5
    cinqsimpl.euler_characteristic() # 1
    
    sage: len(create_n_simplex_all(5,5).cells()[0])
    6
    sage: len(create_n_simplex_all(5,5).cells()[1])
    15
    sage: len(create_n_simplex_all(5,5).cells()[2])
    20
    sage: len(create_n_simplex_all(5,5).cells()[3])
    15
    sage: len(create_n_simplex_all(5,5).cells()[4])
    6
    sage: len(create_n_simplex_all(5,5).cells()[5])
    1
    

    create_n_simplex_all(2,2,1)
    """
    assert n>=n_0
    assert n>=n_fill

    vertices = [ (i,) for i in range(n_0,n+1+n_0)]
    arg = vertices+[combo for i in range(2,n_fill+2) for combo in [I for I in combinations(range(n_0,n+1+n_0),i)]]

    return SimplicialComplex(arg)

def create_thin_n_polygon(n=1,n_0 = 0):
    """
    create_thin_n_polygon = create_thin_n_polygon(n=1,n_0=0)
    Creates a 1-simplex that's path connected and is really just a n-polygon without any filling
    
    e.g. Example of usage
    create_thin_n_polygon(2).homology() # {0: 0, 1: Z}
    sage: create_thin_n_polygon(3).homology() # {0: 0, 1: Z}
    sage: create_thin_n_polygon(5).homology() # {0: 0, 1: Z}
    # as expected
    """
    assert n>=n_0
    
    vertices = [ (i,) for i in range(n_0,n+1)]
    
    arg = vertices + [(i,i+1) if i < n else (i,n_0) for i in range(n_0,n+1) ]
    return SimplicialComplex(arg)


# cf. https://www.math.hmc.edu/~su/pcmi/projects/simplicial_module/simplicial_pcmi.pdf
# Demonstrating Simplicial Homology, (augmented) chain complexes, and ith reduced homology pp. 2 of the pdf, the closed interval and hollow triangle

# closed interval

create_thin_n_polygon(1)
create_thin_n_polygon(1).faces()  # {-1: {()}, 0: {(0,), (1,)}, 1: {(0, 1)}}
create_thin_n_polygon(1).chain_complex()  
create_thin_n_polygon(1).chain_complex().differential()  
# {0: [], 1: [-1]
# [ 1], 2: []}

create_thin_n_polygon(2).chain_complex().differential()
create_thin_n_polygon(2).chain_complex().differential()[1]

ascii_art( create_thin_n_polygon(2).chain_complex() )

Matrix(create_thin_n_polygon(2).chain_complex().differential()[1]).rank()  # 2 
Matrix(create_thin_n_polygon(2).chain_complex().differential()[1])





# cf. https://www.math.hmc.edu/~su/pcmi/projects/simplicial_module/simplicial_pcmi.pdf
# Example on [5] which is 5 vertices

hmcExample5 = create_n_simplex_all(2,2,1)
for face in create_thin_n_polygon(4,2).faces()[1]:
    hmcExample5.add_face(face)

hmcExample5.add_face((5,))

hmcExample5.homology()  # sage: X.homology() # {0: Z, 1: Z, 2: 0}
hmcExample5.chain_complex().differential() # {0: [], 1: [ 1  1  0  0  0]
# [ 0  0  0  0  0]
# [ 0 -1  1  1  0]
# [ 0  0  0 -1 -1]
# [-1  0 -1  0  1], 2: [ 0]
# [ 0]
# [ 1]
# [-1]
# [ 1], 3: []}

hmcExample5.chain_complex().differential()[1].rank() # 3

##
# 2-Sphere
# from built-in 2-Sphere simplicial complex for Sage Math
##

simplicial_complexes.Sphere(2).faces()
simplicial_complexes.Sphere(2).homology()
scS2 = simplicial_complexes.Sphere(2)
scS2.chain_complex().differential()
scS2.chain_complex().differential()[2].rank() # 3
scS2.chain_complex().differential()[3].rank() # 3

# Exercise 5.
# cf. https://www.math.hmc.edu/~su/pcmi/projects/simplicial_module/simplicial_pcmi.pdf

hmcExercise5 = SimplicialComplex( [(i,) for i in range(1,6)] + [(1,3),(1,4),(3,4),
	     (1,5),(4,5),(1,6),(3,6),(2,3),(2,4),
	     (2,5),(2,6),(5,6)] + 
	     [ 
	     (1,4,5),(1,3,6),(2,3,4),
	     (2,4,5),(1,5,6),(2,3,6),
	     (2,5,6),
	     (1,3,4)] )

hmcExercise5.homology() # {0: 0, 1: 0, 2: Z}
hmcExercise5.homology() == scS2.homology() # True

# Exercise 6
# cf. https://www.math.hmc.edu/~su/pcmi/projects/simplicial_module/simplicial_pcmi.pdf

# Using the built-in Torus simplicial complex minimal triangulation
simplicial_complexes.Torus().homology() # {0: 0, 1: Z x Z, 2: Z}

hmcExercise6 = SimplicialComplex( [(i,) for i in range(1,10)] + 
	     [
	     (1,2),(1,3),(2,3),
	     (1,4),(1,5),(4,5),
	     (1,6),(1,7),(2,6),(2,7),(3,7),
	     (1,9),(4,6),(4,9),(5,9),	     
	     (1,8),(2,9),(3,8),(3,9),
	     (4,7),(5,7),(5,8),
	     (6,7),(6,9),(7,8),(8,9),
	     (6,8)] +
	     [
	     (1,4,7),(1,3,7),(2,3,7),(2,6,7),(1,2,6),(1,4,6),
	     (4,6,9),(4,5,9),(1,2,9),(1,5,9),
	     (2,3,9),(3,8,9),(1,3,8),(1,5,8),	
	     (4,5,7),(5,7,8),
	     (6,7,8),(6,8,9)
]
)

hmcExercise6.homology()  # {0: 0, 1: Z x Z, 2: Z}

# Exercise 7

simplicial_complexes.KleinBottle().homology()  # {0: 0, 1: Z x C2, 2: 0}

simplicial_complexes.RealProjectiveSpace(2)
simplicial_complexes.RealProjectiveSpace(2).homology()

simplicial_complexes.RealProjectivePlane()
simplicial_complexes.RealProjectivePlane().homology()

simplicial_complexes.ProjectivePlane()
simplicial_complexes.ProjectivePlane().homology()


hmcExercise7_Kleinbottle = SimplicialComplex( [(i,) for i in range(1,10)] + 
	     [
	     (1,2),(1,3),(2,3),
	     (1,4),(1,5),(4,5),
	     (1,6),(1,7),(2,6),(2,7),(3,7),
	     (1,9),(4,6),(4,9),(5,9),	     
	     (1,8),(2,9),(2,8),(3,9),
	     (4,7),(5,7),(5,8),
	     (6,7),(6,9),(7,8),(8,9),
	     (6,8)] +
	     [
	     (1,4,7),(1,3,7),(2,3,7),(2,6,7),(1,2,6),(1,4,6),
	     (4,6,9),(4,5,9),(1,3,9),(1,5,9),
	     (2,3,9),(2,8,9),(1,2,8),(1,5,8),	
	     (4,5,7),(5,7,8),
	     (6,7,8),(6,8,9)
]
)

hmcExercise7_Kleinbottle.homology()
hmcExercise7_Kleinbottle.chain_complex().differential()

hmcExercise7_RP2 = SimplicialComplex( [(i,) for i in range(1,10)] + 
	     [
	     (1,2),(1,3),(2,3),
	     (1,4),(1,5),(4,5),
	     (1,6),(1,7),(2,6),(2,7),(3,7),
	     (1,9),(4,6),(4,9),(5,9),	     
	     (1,8),(2,9),(2,8),(3,9),
	     (4,7),(5,7),(4,8),
	     (6,7),(6,9),(7,8),(8,9),
	     (6,8)] +
	     [
	     (1,5,7),(1,3,7),(2,3,7),(2,6,7),(1,2,6),(1,4,6),
	     (4,6,9),(4,5,9),(1,3,9),(1,5,9),
	     (2,3,9),(2,8,9),(1,2,8),(1,4,8),	
	     (4,5,7),(4,7,8),
	     (6,7,8),(6,8,9)
]
)

hmcExercise7_RP2.homology()
hmcExercise7_RP2.chain_complex().differential()