<s>
In	O
algebraic	O
topology	O
,	O
simplicial	B-Algorithm
homology	I-Algorithm
is	O
the	O
sequence	O
of	O
homology	O
groups	O
of	O
a	O
simplicial	O
complex	O
.	O
</s>
<s>
Simplicial	B-Algorithm
homology	I-Algorithm
arose	O
as	O
a	O
way	O
to	O
study	O
topological	O
spaces	O
whose	O
building	O
blocks	O
are	O
n-simplices	O
,	O
the	O
n-dimensional	O
analogs	O
of	O
triangles	O
.	O
</s>
<s>
Many	O
topological	O
spaces	O
of	O
interest	O
can	O
be	O
triangulated	O
,	O
including	O
every	O
smooth	O
manifold	B-Architecture
(	O
Cairns	O
and	O
Whitehead	O
)	O
.	O
</s>
<s>
Simplicial	B-Algorithm
homology	I-Algorithm
is	O
defined	O
by	O
a	O
simple	O
recipe	O
for	O
any	O
abstract	O
simplicial	O
complex	O
.	O
</s>
<s>
It	O
is	O
a	O
remarkable	O
fact	O
that	O
simplicial	B-Algorithm
homology	I-Algorithm
only	O
depends	O
on	O
the	O
associated	O
topological	O
space	O
.	O
</s>
<s>
A	O
key	O
concept	O
in	O
defining	O
simplicial	B-Algorithm
homology	I-Algorithm
is	O
the	O
notion	O
of	O
an	O
orientation	O
of	O
a	O
simplex	O
.	O
</s>
<s>
This	O
construction	O
makes	O
simplicial	B-Algorithm
homology	I-Algorithm
a	O
functor	B-Language
from	O
simplicial	O
complexes	O
to	O
abelian	O
groups	O
.	O
</s>
<s>
This	O
is	O
essential	O
to	O
applications	O
of	O
the	O
theory	O
,	O
including	O
the	O
Brouwer	O
fixed	O
point	O
theorem	O
and	O
the	O
topological	O
invariance	O
of	O
simplicial	B-Algorithm
homology	I-Algorithm
.	O
</s>
<s>
Singular	O
homology	O
is	O
defined	O
for	O
all	O
topological	O
spaces	O
and	O
depends	O
only	O
on	O
the	O
topology	O
,	O
not	O
any	O
triangulation	O
;	O
and	O
it	O
agrees	O
with	O
simplicial	B-Algorithm
homology	I-Algorithm
for	O
spaces	O
which	O
can	O
be	O
triangulated	O
.	O
</s>
<s>
Nonetheless	O
,	O
because	O
it	O
is	O
possible	O
to	O
compute	O
the	O
simplicial	B-Algorithm
homology	I-Algorithm
of	O
a	O
simplicial	O
complex	O
automatically	O
and	O
efficiently	O
,	O
simplicial	B-Algorithm
homology	I-Algorithm
has	O
become	O
important	O
for	O
application	O
to	O
real-life	O
situations	O
,	O
such	O
as	O
image	B-General_Concept
analysis	I-General_Concept
,	O
medical	B-Application
imaging	I-Application
,	O
and	O
data	B-General_Concept
analysis	I-General_Concept
in	O
general	O
.	O
</s>
<s>
Computation	O
of	O
persistent	B-Algorithm
homology	I-Algorithm
involves	O
analysis	O
of	O
homology	O
at	O
different	O
resolutions	O
,	O
registering	O
homology	O
classes	O
(	O
holes	O
)	O
that	O
persist	O
as	O
the	O
resolution	O
is	O
changed	O
.	O
</s>
<s>
More	O
generally	O
,	O
simplicial	B-Algorithm
homology	I-Algorithm
plays	O
a	O
central	O
role	O
in	O
topological	B-Algorithm
data	I-Algorithm
analysis	I-Algorithm
,	O
a	O
technique	O
in	O
the	O
field	O
of	O
data	B-Application
mining	I-Application
.	O
</s>
<s>
Stand-alone	O
implementations	O
in	O
C++	B-Language
are	O
available	O
as	O
part	O
of	O
the	O
,	O
and	O
software	O
projects	O
.	O
</s>
<s>
For	O
Python	B-Language
,	O
there	O
are	O
libraries	O
such	O
as	O
,	O
,	O
and	O
,	O
the	O
latter	O
aimed	O
at	O
generating	O
topological	O
features	O
for	O
machine	O
learning	O
.	O
</s>
<s>
These	O
can	O
be	O
found	O
at	O
the	O
PyPI	B-Language
repository	O
.	O
</s>
