<s>
In	O
mathematics	O
,	O
geometric	B-Algorithm
topology	I-Algorithm
is	O
the	O
study	O
of	O
manifolds	B-Architecture
and	O
maps	O
between	O
them	O
,	O
particularly	O
embeddings	O
of	O
one	O
manifold	B-Architecture
into	O
another	O
.	O
</s>
<s>
Geometric	B-Algorithm
topology	I-Algorithm
as	O
an	O
area	O
distinct	O
from	O
algebraic	O
topology	O
may	O
be	O
said	O
to	O
have	O
originated	O
in	O
the	O
1935	O
classification	O
of	O
lens	O
spaces	O
by	O
Reidemeister	O
torsion	O
,	O
which	O
required	O
distinguishing	O
spaces	O
that	O
are	O
homotopy	O
equivalent	O
but	O
not	O
homeomorphic	O
.	O
</s>
<s>
The	O
use	O
of	O
the	O
term	O
geometric	B-Algorithm
topology	I-Algorithm
to	O
describe	O
these	O
seems	O
to	O
have	O
originated	O
rather	O
recently	O
.	O
</s>
<s>
Manifolds	B-Architecture
differ	O
radically	O
in	O
behavior	O
in	O
high	O
and	O
low	O
dimension	O
.	O
</s>
<s>
High-dimensional	O
topology	O
refers	O
to	O
manifolds	B-Architecture
of	O
dimension	O
5	O
and	O
above	O
,	O
or	O
in	O
relative	O
terms	O
,	O
embeddings	O
in	O
codimension	O
3	O
and	O
above	O
.	O
</s>
<s>
Thus	O
the	O
topological	O
classification	O
of	O
4-manifolds	O
is	O
in	O
principle	O
tractable	O
,	O
and	O
the	O
key	O
questions	O
are	O
:	O
does	O
a	O
topological	O
manifold	B-Architecture
admit	O
a	O
differentiable	O
structure	O
,	O
and	O
if	O
so	O
,	O
how	O
many	O
?	O
</s>
<s>
The	O
distinction	O
is	O
because	O
surgery	O
theory	O
works	O
in	O
dimension	O
5	O
and	O
above	O
(	O
in	O
fact	O
,	O
in	O
many	O
cases	O
,	O
it	O
works	O
topologically	O
in	O
dimension	O
4	O
,	O
though	O
this	O
is	O
very	O
involved	O
to	O
prove	O
)	O
,	O
and	O
thus	O
the	O
behavior	O
of	O
manifolds	B-Architecture
in	O
dimension	O
5	O
and	O
above	O
may	O
be	O
studied	O
using	O
the	O
surgery	O
theory	O
program	O
.	O
</s>
<s>
Indeed	O
,	O
one	O
approach	O
to	O
discussing	O
low-dimensional	O
manifolds	B-Architecture
is	O
to	O
ask	O
"	O
what	O
would	O
surgery	O
theory	O
predict	O
to	O
be	O
true	O
,	O
were	O
it	O
to	O
work	O
?	O
"	O
</s>
<s>
In	O
all	O
dimensions	O
,	O
the	O
fundamental	O
group	O
of	O
a	O
manifold	B-Architecture
is	O
a	O
very	O
important	O
invariant	O
,	O
and	O
determines	O
much	O
of	O
the	O
structure	O
;	O
in	O
dimensions	O
1	O
,	O
2	O
and	O
3	O
,	O
the	O
possible	O
fundamental	O
groups	O
are	O
restricted	O
,	O
while	O
in	O
dimension	O
4	O
and	O
above	O
every	O
finitely	O
presented	O
group	O
is	O
the	O
fundamental	O
group	O
of	O
a	O
manifold	B-Architecture
(	O
note	O
that	O
it	O
is	O
sufficient	O
to	O
show	O
this	O
for	O
4	O
-	O
and	O
5-dimensional	O
manifolds	B-Architecture
,	O
and	O
then	O
to	O
take	O
products	O
with	O
spheres	O
to	O
get	O
higher	O
ones	O
)	O
.	O
</s>
<s>
A	O
manifold	B-Architecture
is	O
orientable	O
if	O
it	O
has	O
a	O
consistent	O
choice	O
of	O
orientation	O
,	O
and	O
a	O
connected	O
orientable	O
manifold	B-Architecture
has	O
exactly	O
two	O
different	O
possible	O
orientations	O
.	O
</s>
<s>
Formulations	O
applicable	O
to	O
general	O
topological	O
manifolds	B-Architecture
often	O
employ	O
methods	O
of	O
homology	O
theory	O
,	O
whereas	O
for	O
differentiable	O
manifolds	B-Architecture
more	O
structure	O
is	O
present	O
,	O
allowing	O
a	O
formulation	O
in	O
terms	O
of	O
differential	O
forms	O
.	O
</s>
<s>
A	O
handle	O
decomposition	O
is	O
to	O
a	O
manifold	B-Architecture
what	O
a	O
CW-decomposition	O
is	O
to	O
a	O
topological	O
space	O
—	O
in	O
many	O
regards	O
the	O
purpose	O
of	O
a	O
handle	O
decomposition	O
is	O
to	O
have	O
a	O
language	O
analogous	O
to	O
CW-complexes	O
,	O
but	O
adapted	O
to	O
the	O
world	O
of	O
smooth	O
manifolds	B-Architecture
.	O
</s>
<s>
Handle	O
decompositions	O
of	O
manifolds	B-Architecture
arise	O
naturally	O
via	O
Morse	O
theory	O
.	O
</s>
<s>
Local	O
flatness	O
is	O
a	O
property	O
of	O
a	O
submanifold	O
in	O
a	O
topological	O
manifold	B-Architecture
of	O
larger	O
dimension	O
.	O
</s>
<s>
In	O
the	O
category	O
of	O
topological	O
manifolds	B-Architecture
,	O
locally	O
flat	O
submanifolds	O
play	O
a	O
role	O
similar	O
to	O
that	O
of	O
embedded	O
submanifolds	O
in	O
the	O
category	O
of	O
smooth	O
manifolds	B-Architecture
.	O
</s>
<s>
Suppose	O
a	O
d	O
dimensional	O
manifold	B-Architecture
N	O
is	O
embedded	O
into	O
an	O
n	O
dimensional	O
manifold	B-Architecture
M	O
(	O
where	O
d	O
n	O
)	O
.	O
</s>
<s>
Low-dimensional	O
topology	O
is	O
strongly	O
geometric	O
,	O
as	O
reflected	O
in	O
the	O
uniformization	O
theorem	O
in	O
2	O
dimensions	O
–	O
every	O
surface	O
admits	O
a	O
constant	O
curvature	O
metric	O
;	O
geometrically	O
,	O
it	O
has	O
one	O
of	O
3	O
possible	O
geometries	O
:	O
positive	O
curvature/spherical	O
,	O
zero	O
curvature/flat	O
,	O
negative	O
curvature/hyperbolic	O
–	O
and	O
the	O
geometrization	O
conjecture	O
(	O
now	O
theorem	O
)	O
in	O
3	O
dimensions	O
–	O
every	O
3-manifold	O
can	O
be	O
cut	O
into	O
pieces	O
,	O
each	O
of	O
which	O
has	O
one	O
of	O
8	O
possible	O
geometries	O
.	O
</s>
<s>
2-dimensional	O
topology	O
can	O
be	O
studied	O
as	O
complex	O
geometry	O
in	O
one	O
variable	O
(	O
Riemann	O
surfaces	O
are	O
complex	O
curves	O
)	O
–	O
by	O
the	O
uniformization	O
theorem	O
every	O
conformal	O
class	O
of	O
metrics	O
is	O
equivalent	O
to	O
a	O
unique	O
complex	O
one	O
,	O
and	O
4-dimensional	O
topology	O
can	O
be	O
studied	O
from	O
the	O
point	O
of	O
view	O
of	O
complex	O
geometry	O
in	O
two	O
variables	O
(	O
complex	O
surfaces	O
)	O
,	O
though	O
not	O
every	O
4-manifold	O
admits	O
a	O
complex	O
structure	O
.	O
</s>
<s>
They	O
are	O
one	O
of	O
the	O
unifying	O
geometric	O
concepts	O
in	O
algebraic	O
topology	O
,	O
differential	B-Language
geometry	I-Language
and	O
algebraic	O
geometry	O
.	O
</s>
<s>
Surgery	O
theory	O
is	O
a	O
collection	O
of	O
techniques	O
used	O
to	O
produce	O
one	O
manifold	B-Architecture
from	O
another	O
in	O
a	O
'	O
controlled	O
 '	O
way	O
,	O
introduced	O
by	O
.	O
</s>
<s>
Surgery	O
refers	O
to	O
cutting	O
out	O
parts	O
of	O
the	O
manifold	B-Architecture
and	O
replacing	O
it	O
with	O
a	O
part	O
of	O
another	O
manifold	B-Architecture
,	O
matching	O
up	O
along	O
the	O
cut	O
or	O
boundary	O
.	O
</s>
<s>
It	O
is	O
a	O
major	O
tool	O
in	O
the	O
study	O
and	O
classification	O
of	O
manifolds	B-Architecture
of	O
dimension	O
greater	O
than	O
3	O
.	O
</s>
<s>
More	O
technically	O
,	O
the	O
idea	O
is	O
to	O
start	O
with	O
a	O
well-understood	O
manifold	B-Architecture
M	O
and	O
perform	O
surgery	O
on	O
it	O
to	O
produce	O
a	O
manifold	B-Architecture
M	O
′	O
having	O
some	O
desired	O
property	O
,	O
in	O
such	O
a	O
way	O
that	O
the	O
effects	O
on	O
the	O
homology	O
,	O
homotopy	O
groups	O
,	O
or	O
other	O
interesting	O
invariants	O
of	O
the	O
manifold	B-Architecture
are	O
known	O
.	O
</s>
