9 Towards the “Shape” of Cosmological Observables and the String …
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Fig. 9.3 Two 1-cycles are highlighted in green and blue. The green 1-cycle is not the boundary
of any 2-chain, and therefore represents a nontrivial 1-homology class. The blue 1-cycle is the
boundary of the 2-chain comprised of the interior triangle, and is therefore homologous to the
empty cycle. The 1-cycle that uses two of the green 1-cycle’s edges and two of the purple edges is
homologous to the green 1-cycle
[v 0 ] + [v 1 ] + [v 1 ] + · · · + [v n ] + [v n ] + [v 0 ] = 0. Since ∂ p is linear, the p-cycles
form a subgroup of C p , which we denote Z p . In other words, Z p = ker ∂ p . Analogously, we call a p-cycle σ a p-boundary if it is the boundary of a ( p + 1)-cycle,
i.e. σ = ∂ p τ for some ( p + 1)-cycle τ . The p-boundaries also form a group, which
we denote B p . A crucial observation is that the boundary of a boundary is always
empty, ∂ p ∂ p+1 σ = 0 for all σ . This is called the Fundamental Lemma of Homology.
In other words B p is a subgroup of Z p . This observation leads us to define the p-th
homology group as
H p = Z p /B p
(9.2)
In other words, elements of the p-th homology group are p-cycles modulo the equivalence relation σ σ + ∂ p+1 τ . Nontrivial homology elements are cycles that are not
boundaries of higher dimensional cycles. These cycles are homologous to the empty
p-cycle. See Fig. 9.3 for examples.
Recall that we are interested in identifying “holes” of various dimensions in a
simplicial complex. The reason we have gone through the preceding mathematical
definitions is that a hole of dimension p corresponds to a nontrivial element of the
p-th homology group.
2 Moreover, since the p-th homology group is defined in terms
of subspaces of the linear boundary operator ∂ p , its rank (counting the number of
independent p-holes) can be calculated via simple linear algebra. These ranks are
called the Betti numbers, which we denote b p . They count the number of “holes” of
dimension p.
9.2.2 Persistent Simplicial Homology
The machinery of simplicial homology allows us to compute topological invariants of
simplicial complexes. These invariants are preserved by simplicial homeomorphisms.
As in the continuous case, we might hope that in the context of data analysis we
2 Strictly speaking, the 0-th homology group counts the independent components of the simplicial
complex, i.e. disconnected clusters.
223
Fig. 9.3 Two 1-cycles are highlighted in green and blue. The green 1-cycle is not the boundary
of any 2-chain, and therefore represents a nontrivial 1-homology class. The blue 1-cycle is the
boundary of the 2-chain comprised of the interior triangle, and is therefore homologous to the
empty cycle. The 1-cycle that uses two of the green 1-cycle’s edges and two of the purple edges is
homologous to the green 1-cycle
[v 0 ] + [v 1 ] + [v 1 ] + · · · + [v n ] + [v n ] + [v 0 ] = 0. Since ∂ p is linear, the p-cycles
form a subgroup of C p , which we denote Z p . In other words, Z p = ker ∂ p . Analogously, we call a p-cycle σ a p-boundary if it is the boundary of a ( p + 1)-cycle,
i.e. σ = ∂ p τ for some ( p + 1)-cycle τ . The p-boundaries also form a group, which
we denote B p . A crucial observation is that the boundary of a boundary is always
empty, ∂ p ∂ p+1 σ = 0 for all σ . This is called the Fundamental Lemma of Homology.
In other words B p is a subgroup of Z p . This observation leads us to define the p-th
homology group as
H p = Z p /B p
(9.2)
In other words, elements of the p-th homology group are p-cycles modulo the equivalence relation σ σ + ∂ p+1 τ . Nontrivial homology elements are cycles that are not
boundaries of higher dimensional cycles. These cycles are homologous to the empty
p-cycle. See Fig. 9.3 for examples.
Recall that we are interested in identifying “holes” of various dimensions in a
simplicial complex. The reason we have gone through the preceding mathematical
definitions is that a hole of dimension p corresponds to a nontrivial element of the
p-th homology group.
2 Moreover, since the p-th homology group is defined in terms
of subspaces of the linear boundary operator ∂ p , its rank (counting the number of
independent p-holes) can be calculated via simple linear algebra. These ranks are
called the Betti numbers, which we denote b p . They count the number of “holes” of
dimension p.
9.2.2 Persistent Simplicial Homology
The machinery of simplicial homology allows us to compute topological invariants of
simplicial complexes. These invariants are preserved by simplicial homeomorphisms.
As in the continuous case, we might hope that in the context of data analysis we
2 Strictly speaking, the 0-th homology group counts the independent components of the simplicial
complex, i.e. disconnected clusters.