[GAP Forum] Chain Complexes: The Moebius Strip

[Marc Bogaerts]

# The Moebius strip is obtained by gluing together a straight strip and 
and a twisted strip
# that has been rotated 180°. See figure in attachment where the former 
strip is colored in blue
# and the latter in red.
# The labeling of the vertices and the edges is as in following scheme:
#
#   *1---1---*2
#      | \          /|
#      |  \       /  |
#      4   \   /   2
#      |     \ /     |
#      |     / \     |
#      |   /     \   |
#      |  5     6  |
#      | /         \ |
#     *4---3--*3
#

# (vertex labels are preceded by an asterix, edge labels are written 
upon the edge)
# First we construct the 0-skeleton:
c0:=ChainComplex(0,[1..4]);
dim 0:[ 1 .. 4 ]

# Then we construct the 1-skeleton upon c0:
c1:=ChainComplex(1,c0,[[1,2],[2,3],[3,4],[4,1],[1,3],[2,4]]);
dim 0:[ 1 .. 4 ]
dim 1:[ [ 1, -1, 0, 0 ], [ 0, 1, -1, 0 ], [ 0, 0, 1, -1 ], [ -1, 0, 0, 1 
], [ 1, 0, -1, 0 ], [ 0, 1, 0, -1 ] ]
Basis of 2-cells: [ [ 1, 0, 0, 1, 0, 1 ], [ 0, 1, 0, -1, -1, -1 ], [ 0, 
0, 1, 1, 1, 0 ] ]

faces:=[];
[  ]
# let's first obtain the set of edges:
M:=GeneratorsOfBoundaries(0,c1);
[ [ 1, -1, 0, 0 ], [ 0, 1, -1, 0 ], [ 0, 0, 1, -1 ], [ -1, 0, 0, 1 ],
  [ 1, 0, -1, 0 ], [ 0, 1, 0, -1 ] ]
# we look for a a combination u of the edges 1,2,3 and 4 that form a 
cycle, the border of the straight strip:
u:=[1..6]*0;
[ 0, 0, 0, 0, 0, 0 ]
u{[1,2,3,4]}:=[a,b,c,d]; #a,b,c and d were predefined as indeterminates
[ a, b, c, d ]
u*M;
[ a-d, -a+b, -b+c, -c+d ]

u{[1,2,3,4]}:=[1,1,1,1];
[ 1, 1, 1, 1 ]
u*M;
[ 0, 0, 0, 0 ]

Add(faces,u);
#same for the edges 1,5,3 and 6 to form the border of "twisted" strip:
u:=[1..6]*0;
[ 0, 0, 0, 0, 0, 0 ]
u{[1,5,3,6]}:=[a,b,c,d];
[ a, b, c, d ]

u*M;
[ a+b, -a+d, -b+c, -c-d ]

u{[1,5,3,6]}:=[1,-1,-1,1];
[ 1, -1, -1, 1 ]
u*M;
[ 0, 0, 0, 0 ]

Add(faces,u);

"faces" now contains two combination of edges each forming a cycle each,
c2:=ChainComplex(2,c1,faces);
dim 0:[ 1 .. 4 ]
dim 1:[ [ 1, -1, 0, 0 ], [ 0, 1, -1, 0 ], [ 0, 0, 1, -1 ], [ -1, 0, 0, 1 
], [ 1, 0, -1, 0 ], [ 0, 1, 0, -1 ] ]
dim 2:[ [ 1, 1, 1, 1, 0, 0 ], [ 1, 0, -1, 0, -1, 1 ] ]
Basis of 3-cells: [  ]

HomologyGroups(c2);
[ [ 0 ], [ 0 ], [  ] ]

# The Moebius strip has the same homotopy type as a circle.

SingularFaces(c2);
[ [ 0, 1, -1, 0 ], [ -1, 0, 0, 1 ], [ 1, 0, -1, 0 ], [ 0, 1, 0, -1 ] ]

# And it has a border consisting of the edges 2, 4, 5 and 6.

Marc Bogaerts.