<s>
In	O
mathematics	O
,	O
a	O
manifold	B-Architecture
is	O
a	O
topological	O
space	O
that	O
locally	O
resembles	O
Euclidean	O
space	O
near	O
each	O
point	O
.	O
</s>
<s>
More	O
precisely	O
,	O
an	O
-dimensional	O
manifold	B-Architecture
,	O
or	O
-manifold	O
for	O
short	O
,	O
is	O
a	O
topological	O
space	O
with	O
the	O
property	O
that	O
each	O
point	O
has	O
a	O
neighborhood	O
that	O
is	O
homeomorphic	O
to	O
an	O
open	O
subset	O
of	O
-dimensional	O
Euclidean	O
space	O
.	O
</s>
<s>
One-dimensional	O
manifolds	B-Architecture
include	O
lines	O
and	O
circles	O
,	O
but	O
not	O
lemniscates	O
.	O
</s>
<s>
Two-dimensional	B-Architecture
manifolds	I-Architecture
are	O
also	O
called	O
surfaces	O
.	O
</s>
<s>
The	O
concept	O
of	O
a	O
manifold	B-Architecture
is	O
central	O
to	O
many	O
parts	O
of	O
geometry	O
and	O
modern	O
mathematical	O
physics	O
because	O
it	O
allows	O
complicated	O
structures	O
to	O
be	O
described	O
in	O
terms	O
of	O
well-understood	O
topological	O
properties	O
of	O
simpler	O
spaces	O
.	O
</s>
<s>
Manifolds	B-Architecture
naturally	O
arise	O
as	O
solution	O
sets	O
of	O
systems	O
of	O
equations	O
and	O
as	O
graphs	B-Application
of	I-Application
functions	I-Application
.	O
</s>
<s>
Manifolds	B-Architecture
can	O
be	O
equipped	O
with	O
additional	O
structure	O
.	O
</s>
<s>
One	O
important	O
class	O
of	O
manifolds	B-Architecture
are	O
differentiable	O
manifolds	B-Architecture
;	O
their	O
differentiable	O
structure	O
allows	O
calculus	O
to	O
be	O
done	O
.	O
</s>
<s>
A	O
Riemannian	O
metric	O
on	O
a	O
manifold	B-Architecture
allows	O
distances	O
and	O
angles	O
to	O
be	O
measured	O
.	O
</s>
<s>
Symplectic	O
manifolds	B-Architecture
serve	O
as	O
the	O
phase	O
spaces	O
in	O
the	O
Hamiltonian	O
formalism	O
of	O
classical	O
mechanics	O
,	O
while	O
four-dimensional	O
Lorentzian	O
manifolds	B-Architecture
model	O
spacetime	B-Protocol
in	O
general	O
relativity	O
.	O
</s>
<s>
The	O
study	O
of	O
manifolds	B-Architecture
requires	O
working	O
knowledge	O
of	O
calculus	O
and	O
topology	B-Architecture
.	O
</s>
<s>
After	O
a	O
line	O
,	O
a	O
circle	O
is	O
the	O
simplest	O
example	O
of	O
a	O
topological	O
manifold	B-Architecture
.	O
</s>
<s>
Topology	B-Architecture
ignores	O
bending	O
,	O
so	O
a	O
small	O
piece	O
of	O
a	O
circle	O
is	O
treated	O
the	O
same	O
as	O
a	O
small	O
piece	O
of	O
a	O
line	O
.	O
</s>
<s>
So	O
,	O
projection	O
onto	O
the	O
first	O
coordinate	O
is	O
a	O
continuous	O
and	O
invertible	O
mapping	B-Algorithm
from	O
the	O
upper	O
arc	O
to	O
the	O
open	O
interval	O
( −1	O
,	O
1	O
)	O
:	O
</s>
<s>
Together	O
,	O
these	O
parts	O
cover	O
the	O
whole	O
circle	O
,	O
and	O
the	O
four	O
charts	O
form	O
an	O
atlas	B-Device
for	O
the	O
circle	O
.	O
</s>
<s>
The	O
top	O
,	O
bottom	O
,	O
left	O
,	O
and	O
right	O
charts	O
do	O
not	O
form	O
the	O
only	O
possible	O
atlas	B-Device
.	O
</s>
<s>
may	O
be	O
covered	O
by	O
an	O
atlas	B-Device
of	O
six	O
charts	O
:	O
the	O
plane	O
divides	O
the	O
sphere	O
into	O
two	O
half	O
spheres	O
(	O
and	O
)	O
,	O
which	O
may	O
both	O
be	O
mapped	O
on	O
the	O
disc	O
by	O
the	O
projection	O
on	O
the	O
plane	O
of	O
coordinates	O
.	O
</s>
<s>
This	O
example	O
is	O
historically	O
significant	O
,	O
as	O
it	O
has	O
motivated	O
the	O
terminology	O
;	O
it	O
became	O
apparent	O
that	O
the	O
whole	O
surface	O
of	O
the	O
Earth	O
cannot	O
have	O
a	O
plane	O
representation	O
consisting	O
of	O
a	O
single	O
map	O
(	O
also	O
called	O
"	O
chart	O
"	O
,	O
see	O
nautical	B-Application
chart	I-Application
)	O
,	O
and	O
therefore	O
one	O
needs	O
atlases	O
for	O
covering	O
the	O
whole	O
Earth	O
surface	O
.	O
</s>
<s>
Manifolds	B-Architecture
need	O
not	O
be	O
connected	O
(	O
all	O
in	O
"	O
one	O
piece	O
"	O
)	O
;	O
an	O
example	O
is	O
a	O
pair	O
of	O
separate	O
circles	O
.	O
</s>
<s>
Manifolds	B-Architecture
need	O
not	O
be	O
closed	O
;	O
thus	O
a	O
line	O
segment	O
without	O
its	O
end	O
points	O
is	O
a	O
manifold	B-Architecture
.	O
</s>
<s>
They	O
are	O
never	O
countable	O
,	O
unless	O
the	O
dimension	O
of	O
the	O
manifold	B-Architecture
is	O
0	O
.	O
</s>
<s>
Putting	O
these	O
freedoms	O
together	O
,	O
other	O
examples	O
of	O
manifolds	B-Architecture
are	O
a	O
parabola	O
,	O
a	O
hyperbola	O
,	O
and	O
the	O
locus	O
of	O
points	O
on	O
a	O
cubic	O
curve	O
(	O
a	O
closed	O
loop	O
piece	O
and	O
an	O
open	O
,	O
infinite	O
piece	O
)	O
.	O
</s>
<s>
Even	O
with	O
the	O
bending	O
allowed	O
by	O
topology	B-Architecture
,	O
the	O
vicinity	O
of	O
the	O
shared	O
point	O
looks	O
like	O
a	O
"	O
+	O
"	O
,	O
not	O
a	O
line	O
.	O
</s>
<s>
Informally	O
,	O
a	O
manifold	B-Architecture
is	O
a	O
space	O
that	O
is	O
"	O
modeled	O
on	O
"	O
Euclidean	O
space	O
.	O
</s>
<s>
There	O
are	O
many	O
different	O
kinds	O
of	O
manifolds	B-Architecture
.	O
</s>
<s>
In	O
geometry	O
and	O
topology	B-Architecture
,	O
all	O
manifolds	B-Architecture
are	O
topological	O
manifolds	B-Architecture
,	O
possibly	O
with	O
additional	O
structure	O
.	O
</s>
<s>
A	O
manifold	B-Architecture
can	O
be	O
constructed	O
by	O
giving	O
a	O
collection	O
of	O
coordinate	O
charts	O
,	O
that	O
is	O
,	O
a	O
covering	O
by	O
open	O
sets	O
with	O
homeomorphisms	O
to	O
a	O
Euclidean	O
space	O
,	O
and	O
patching	O
functions	O
:	O
homeomorphisms	O
from	O
one	O
region	O
of	O
Euclidean	O
space	O
to	O
another	O
region	O
if	O
they	O
correspond	O
to	O
the	O
same	O
part	O
of	O
the	O
manifold	B-Architecture
in	O
two	O
different	O
coordinate	O
charts	O
.	O
</s>
<s>
A	O
manifold	B-Architecture
can	O
be	O
given	O
additional	O
structure	O
if	O
the	O
patching	O
functions	O
satisfy	O
axioms	O
beyond	O
continuity	O
.	O
</s>
<s>
For	O
instance	O
,	O
differentiable	O
manifolds	B-Architecture
have	O
homeomorphisms	O
on	O
overlapping	O
neighborhoods	O
diffeomorphic	O
with	O
each	O
other	O
,	O
so	O
that	O
the	O
manifold	B-Architecture
has	O
a	O
well-defined	O
set	O
of	O
functions	O
which	O
are	O
differentiable	O
in	O
each	O
neighborhood	O
,	O
thus	O
differentiable	O
on	O
the	O
manifold	B-Architecture
as	O
a	O
whole	O
.	O
</s>
<s>
Formally	O
,	O
a	O
(	O
topological	O
)	O
manifold	B-Architecture
is	O
a	O
second	O
countable	O
Hausdorff	O
space	O
that	O
is	O
locally	O
homeomorphic	O
to	O
a	O
Euclidean	O
space	O
.	O
</s>
<s>
Second	O
countable	O
and	O
Hausdorff	O
are	O
point-set	O
conditions	O
;	O
second	O
countable	O
excludes	O
spaces	O
which	O
are	O
in	O
some	O
sense	O
'	O
too	O
large	O
 '	O
such	O
as	O
the	O
long	O
line	O
,	O
while	O
Hausdorff	O
excludes	O
spaces	O
such	O
as	O
"	O
the	O
line	O
with	O
two	O
origins	O
"	O
(	O
these	O
generalizations	O
of	O
manifolds	B-Architecture
are	O
discussed	O
in	O
non-Hausdorff	O
manifolds	B-Architecture
)	O
.	O
</s>
<s>
The	O
that	O
appears	O
in	O
the	O
preceding	O
definition	O
is	O
called	O
the	O
local	O
dimension	O
of	O
the	O
manifold	B-Architecture
.	O
</s>
<s>
Generally	O
manifolds	B-Architecture
are	O
taken	O
to	O
have	O
a	O
constant	O
local	O
dimension	O
,	O
and	O
the	O
local	O
dimension	O
is	O
then	O
called	O
the	O
dimension	O
of	O
the	O
manifold	B-Architecture
.	O
</s>
<s>
This	O
is	O
,	O
in	O
particular	O
,	O
the	O
case	O
when	O
manifolds	B-Architecture
are	O
connected	O
.	O
</s>
<s>
However	O
,	O
some	O
authors	O
admit	O
manifolds	B-Architecture
that	O
are	O
not	O
connected	O
,	O
and	O
where	O
different	O
points	O
can	O
have	O
different	O
dimensions	O
.	O
</s>
<s>
If	O
a	O
manifold	B-Architecture
has	O
a	O
fixed	O
dimension	O
,	O
this	O
can	O
be	O
emphasized	O
by	O
calling	O
it	O
a	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
(	O
surface	O
of	O
a	O
)	O
sphere	O
has	O
a	O
constant	O
dimension	O
of	O
2	O
and	O
is	O
therefore	O
a	O
pure	O
manifold	B-Architecture
whereas	O
the	O
disjoint	O
union	O
of	O
a	O
sphere	O
and	O
a	O
line	O
in	O
three-dimensional	O
space	O
is	O
not	O
a	O
pure	O
manifold	B-Architecture
.	O
</s>
<s>
Sheaf-theoretically	O
,	O
a	O
manifold	B-Architecture
is	O
a	O
locally	O
ringed	O
space	O
,	O
whose	O
structure	O
sheaf	O
is	O
locally	O
isomorphic	O
to	O
the	O
sheaf	O
of	O
continuous	O
(	O
or	O
differentiable	O
,	O
or	O
complex-analytic	O
,	O
etc	O
.	O
)	O
</s>
<s>
This	O
definition	O
is	O
mostly	O
used	O
when	O
discussing	O
analytic	B-Language
manifolds	B-Architecture
in	O
algebraic	O
geometry	O
.	O
</s>
<s>
The	O
spherical	O
Earth	O
is	O
navigated	O
using	O
flat	O
maps	O
or	O
charts	O
,	O
collected	O
in	O
an	O
atlas	B-Device
.	O
</s>
<s>
Similarly	O
,	O
a	O
differentiable	O
manifold	B-Architecture
can	O
be	O
described	O
using	O
mathematical	B-Algorithm
maps	I-Algorithm
,	O
called	O
coordinate	O
charts	O
,	O
collected	O
in	O
a	O
mathematical	O
atlas	B-Device
.	O
</s>
<s>
It	O
is	O
not	O
generally	O
possible	O
to	O
describe	O
a	O
manifold	B-Architecture
with	O
just	O
one	O
chart	O
,	O
because	O
the	O
global	O
structure	O
of	O
the	O
manifold	B-Architecture
is	O
different	O
from	O
the	O
simple	O
structure	O
of	O
the	O
charts	O
.	O
</s>
<s>
When	O
a	O
manifold	B-Architecture
is	O
constructed	O
from	O
multiple	O
overlapping	O
charts	O
,	O
the	O
regions	O
where	O
they	O
overlap	O
carry	O
information	O
essential	O
to	O
understanding	O
the	O
global	O
structure	O
.	O
</s>
<s>
A	O
coordinate	O
map	O
,	O
a	O
coordinate	O
chart	O
,	O
or	O
simply	O
a	O
chart	O
,	O
of	O
a	O
manifold	B-Architecture
is	O
an	O
invertible	O
map	O
between	O
a	O
subset	O
of	O
the	O
manifold	B-Architecture
and	O
a	O
simple	O
space	O
such	O
that	O
both	O
the	O
map	O
and	O
its	O
inverse	O
preserve	O
the	O
desired	O
structure	O
.	O
</s>
<s>
For	O
a	O
topological	O
manifold	B-Architecture
,	O
the	O
simple	O
space	O
is	O
a	O
subset	O
of	O
some	O
Euclidean	O
space	O
and	O
interest	O
focuses	O
on	O
the	O
topological	O
structure	O
.	O
</s>
<s>
In	O
the	O
case	O
of	O
a	O
differentiable	O
manifold	B-Architecture
,	O
a	O
set	O
of	O
charts	O
called	O
an	O
atlas	B-Device
allows	O
us	O
to	O
do	O
calculus	O
on	O
manifolds	B-Architecture
.	O
</s>
<s>
The	O
description	O
of	O
most	O
manifolds	B-Architecture
requires	O
more	O
than	O
one	O
chart	O
.	O
</s>
<s>
A	O
specific	O
collection	O
of	O
charts	O
which	O
covers	O
a	O
manifold	B-Architecture
is	O
called	O
an	O
atlas	B-Device
.	O
</s>
<s>
An	O
atlas	B-Device
is	O
not	O
unique	O
as	O
all	O
manifolds	B-Architecture
can	O
be	O
covered	O
in	O
multiple	O
ways	O
using	O
different	O
combinations	O
of	O
charts	O
.	O
</s>
<s>
Two	O
atlases	O
are	O
said	O
to	O
be	O
equivalent	O
if	O
their	O
union	O
is	O
also	O
an	O
atlas	B-Device
.	O
</s>
<s>
The	O
atlas	B-Device
containing	O
all	O
possible	O
charts	O
consistent	O
with	O
a	O
given	O
atlas	B-Device
is	O
called	O
the	O
maximal	O
atlas	B-Device
(	O
i.e.	O
</s>
<s>
an	O
equivalence	O
class	O
containing	O
that	O
given	O
atlas	B-Device
)	O
.	O
</s>
<s>
Unlike	O
an	O
ordinary	O
atlas	B-Device
,	O
the	O
maximal	O
atlas	B-Device
of	O
a	O
given	O
manifold	B-Architecture
is	O
unique	O
.	O
</s>
<s>
Charts	O
in	O
an	O
atlas	B-Device
may	O
overlap	O
and	O
a	O
single	O
point	O
of	O
a	O
manifold	B-Architecture
may	O
be	O
represented	O
in	O
several	O
charts	O
.	O
</s>
<s>
If	O
two	O
charts	O
overlap	O
,	O
parts	O
of	O
them	O
represent	O
the	O
same	O
region	O
of	O
the	O
manifold	B-Architecture
,	O
just	O
as	O
a	O
map	O
of	O
Europe	O
and	O
a	O
map	O
of	O
Russia	O
may	O
both	O
contain	O
Moscow	O
.	O
</s>
<s>
Given	O
two	O
overlapping	O
charts	O
,	O
a	O
transition	O
function	O
can	O
be	O
defined	O
which	O
goes	O
from	O
an	O
open	O
ball	O
in	O
to	O
the	O
manifold	B-Architecture
and	O
then	O
back	O
to	O
another	O
(	O
or	O
perhaps	O
the	O
same	O
)	O
open	O
ball	O
in	O
.	O
</s>
<s>
An	O
atlas	B-Device
can	O
also	O
be	O
used	O
to	O
define	O
additional	O
structure	O
on	O
the	O
manifold	B-Architecture
.	O
</s>
<s>
If	O
all	O
transition	O
maps	O
are	O
compatible	O
with	O
this	O
structure	O
,	O
the	O
structure	O
transfers	O
to	O
the	O
manifold	B-Architecture
.	O
</s>
<s>
This	O
is	O
the	O
standard	O
way	O
differentiable	O
manifolds	B-Architecture
are	O
defined	O
.	O
</s>
<s>
If	O
the	O
transition	O
functions	O
of	O
an	O
atlas	B-Device
for	O
a	O
topological	O
manifold	B-Architecture
preserve	O
the	O
natural	O
differential	O
structure	O
of	O
(	O
that	O
is	O
,	O
if	O
they	O
are	O
diffeomorphisms	O
)	O
,	O
the	O
differential	O
structure	O
transfers	O
to	O
the	O
manifold	B-Architecture
and	O
turns	O
it	O
into	O
a	O
differentiable	O
manifold	B-Architecture
.	O
</s>
<s>
Complex	O
manifolds	B-Architecture
are	O
introduced	O
in	O
an	O
analogous	O
way	O
by	O
requiring	O
that	O
the	O
transition	O
functions	O
of	O
an	O
atlas	B-Device
are	O
holomorphic	O
functions	O
.	O
</s>
<s>
For	O
symplectic	O
manifolds	B-Architecture
,	O
the	O
transition	O
functions	O
must	O
be	O
symplectomorphisms	O
.	O
</s>
<s>
The	O
structure	O
on	O
the	O
manifold	B-Architecture
depends	O
on	O
the	O
atlas	B-Device
,	O
but	O
sometimes	O
different	O
atlases	O
can	O
be	O
said	O
to	O
give	O
rise	O
to	O
the	O
same	O
structure	O
.	O
</s>
<s>
A	O
manifold	B-Architecture
with	O
boundary	O
is	O
a	O
manifold	B-Architecture
with	O
an	O
edge	O
.	O
</s>
<s>
For	O
example	O
,	O
a	O
sheet	O
of	O
paper	O
is	O
a	O
2-manifold	O
with	O
a	O
1-dimensional	O
boundary	O
.	O
</s>
<s>
The	O
boundary	O
of	O
an	O
-manifold	O
with	O
boundary	O
is	O
an	O
-manifold	O
.	O
</s>
<s>
A	O
disk	O
(	O
circle	O
plus	O
interior	O
)	O
is	O
a	O
2-manifold	O
with	O
boundary	O
.	O
</s>
<s>
Its	O
boundary	O
is	O
a	O
circle	O
,	O
a	O
1-manifold	O
.	O
</s>
<s>
A	O
square	O
with	O
interior	O
is	O
also	O
a	O
2-manifold	O
with	O
boundary	O
.	O
</s>
<s>
A	O
ball	O
(	O
sphere	O
plus	O
interior	O
)	O
is	O
a	O
3-manifold	O
with	O
boundary	O
.	O
</s>
<s>
Its	O
boundary	O
is	O
a	O
sphere	O
,	O
a	O
2-manifold	O
.	O
</s>
<s>
(	O
Do	O
not	O
confuse	O
with	O
Boundary	O
(	O
topology	B-Architecture
)	O
)	O
.	O
</s>
<s>
In	O
technical	O
language	O
,	O
a	O
manifold	B-Architecture
with	O
boundary	O
is	O
a	O
space	O
containing	O
both	O
interior	O
points	O
and	O
boundary	O
points	O
.	O
</s>
<s>
Let	O
be	O
a	O
manifold	B-Architecture
with	O
boundary	O
.	O
</s>
<s>
If	O
is	O
a	O
manifold	B-Architecture
with	O
boundary	O
of	O
dimension	O
,	O
then	O
is	O
a	O
manifold	B-Architecture
(	O
without	O
boundary	O
)	O
of	O
dimension	O
and	O
is	O
a	O
manifold	B-Architecture
(	O
without	O
boundary	O
)	O
of	O
dimension	O
.	O
</s>
<s>
A	O
single	O
manifold	B-Architecture
can	O
be	O
constructed	O
in	O
different	O
ways	O
,	O
each	O
stressing	O
a	O
different	O
aspect	O
of	O
the	O
manifold	B-Architecture
,	O
thereby	O
leading	O
to	O
a	O
slightly	O
different	O
viewpoint	O
.	O
</s>
<s>
Perhaps	O
the	O
simplest	O
way	O
to	O
construct	O
a	O
manifold	B-Architecture
is	O
the	O
one	O
used	O
in	O
the	O
example	O
above	O
of	O
the	O
circle	O
.	O
</s>
<s>
First	O
,	O
a	O
subset	O
of	O
is	O
identified	O
,	O
and	O
then	O
an	O
atlas	B-Device
covering	O
this	O
subset	O
is	O
constructed	O
.	O
</s>
<s>
The	O
concept	O
of	O
manifold	B-Architecture
grew	O
historically	O
from	O
constructions	O
like	O
this	O
.	O
</s>
<s>
Together	O
with	O
two	O
charts	O
projecting	O
on	O
the	O
(	O
x	O
,	O
z	O
)	O
plane	O
and	O
two	O
charts	O
projecting	O
on	O
the	O
(	O
y	O
,	O
z	O
)	O
plane	O
,	O
an	O
atlas	B-Device
of	O
six	O
charts	O
is	O
obtained	O
which	O
covers	O
the	O
entire	O
sphere	O
.	O
</s>
<s>
A	O
manifold	B-Architecture
can	O
be	O
constructed	O
by	O
gluing	O
together	O
pieces	O
in	O
a	O
consistent	O
manner	O
,	O
making	O
them	O
into	O
overlapping	O
charts	O
.	O
</s>
<s>
This	O
construction	O
is	O
possible	O
for	O
any	O
manifold	B-Architecture
and	O
hence	O
it	O
is	O
often	O
used	O
as	O
a	O
characterisation	O
,	O
especially	O
for	O
differentiable	O
and	O
Riemannian	O
manifolds	B-Architecture
.	O
</s>
<s>
It	O
focuses	O
on	O
an	O
atlas	B-Device
,	O
as	O
the	O
patches	O
naturally	O
provide	O
charts	O
,	O
and	O
since	O
there	O
is	O
no	O
exterior	O
space	O
involved	O
it	O
leads	O
to	O
an	O
intrinsic	O
view	O
of	O
the	O
manifold	B-Architecture
.	O
</s>
<s>
The	O
manifold	B-Architecture
is	O
constructed	O
by	O
specifying	O
an	O
atlas	B-Device
,	O
which	O
is	O
itself	O
defined	O
by	O
transition	O
maps	O
.	O
</s>
<s>
A	O
point	O
of	O
the	O
manifold	B-Architecture
is	O
therefore	O
an	O
equivalence	O
class	O
of	O
points	O
which	O
are	O
mapped	O
to	O
each	O
other	O
by	O
transition	O
maps	O
.	O
</s>
<s>
For	O
topological	O
manifolds	B-Architecture
they	O
are	O
required	O
to	O
be	O
homeomorphisms	O
;	O
if	O
they	O
are	O
also	O
diffeomorphisms	O
,	O
the	O
resulting	O
manifold	B-Architecture
is	O
a	O
differentiable	O
manifold	B-Architecture
.	O
</s>
<s>
In	O
the	O
first	O
construction	O
,	O
the	O
manifold	B-Architecture
is	O
seen	O
as	O
embedded	O
in	O
some	O
Euclidean	O
space	O
.	O
</s>
<s>
When	O
a	O
manifold	B-Architecture
is	O
viewed	O
in	O
this	O
way	O
,	O
it	O
is	O
easy	O
to	O
use	O
intuition	O
from	O
Euclidean	O
spaces	O
to	O
define	O
additional	O
structure	O
.	O
</s>
<s>
The	O
patchwork	O
construction	O
does	O
not	O
use	O
any	O
embedding	O
,	O
but	O
simply	O
views	O
the	O
manifold	B-Architecture
as	O
a	O
topological	O
space	O
by	O
itself	O
.	O
</s>
<s>
As	O
the	O
transition	O
map	O
is	O
a	O
smooth	O
function	O
,	O
this	O
atlas	B-Device
defines	O
a	O
smooth	O
manifold	B-Architecture
.	O
</s>
<s>
It	O
is	O
possible	O
to	O
define	O
different	O
points	O
of	O
a	O
manifold	B-Architecture
to	O
be	O
same	O
.	O
</s>
<s>
There	O
is	O
,	O
however	O
,	O
no	O
reason	O
to	O
expect	O
such	O
quotient	O
spaces	O
to	O
be	O
manifolds	B-Architecture
.	O
</s>
<s>
Among	O
the	O
possible	O
quotient	O
spaces	O
that	O
are	O
not	O
necessarily	O
manifolds	B-Architecture
,	O
orbifolds	O
and	O
CW	O
complexes	O
are	O
considered	O
to	O
be	O
relatively	O
well-behaved	O
.	O
</s>
<s>
An	O
example	O
of	O
a	O
quotient	O
space	O
of	O
a	O
manifold	B-Architecture
that	O
is	O
also	O
a	O
manifold	B-Architecture
is	O
the	O
real	O
projective	O
space	O
,	O
identified	O
as	O
a	O
quotient	O
space	O
of	O
the	O
corresponding	O
sphere	O
.	O
</s>
<s>
One	O
method	O
of	O
identifying	O
points	O
(	O
gluing	O
them	O
together	O
)	O
is	O
through	O
a	O
right	O
(	O
or	O
left	O
)	O
action	O
of	O
a	O
group	O
,	O
which	O
acts	O
on	O
the	O
manifold	B-Architecture
.	O
</s>
<s>
If	O
M	O
is	O
the	O
manifold	B-Architecture
and	O
G	O
is	O
the	O
group	O
,	O
the	O
resulting	O
quotient	O
space	O
is	O
denoted	O
by	O
M	O
/	O
G	O
(	O
or	O
G	O
\	O
M	O
)	O
.	O
</s>
<s>
Manifolds	B-Architecture
which	O
can	O
be	O
constructed	O
by	O
identifying	O
points	O
include	O
tori	O
and	O
real	O
projective	O
spaces	O
(	O
starting	O
with	O
a	O
plane	O
and	O
a	O
sphere	O
,	O
respectively	O
)	O
.	O
</s>
<s>
Two	O
manifolds	B-Architecture
with	O
boundaries	O
can	O
be	O
glued	O
together	O
along	O
a	O
boundary	O
.	O
</s>
<s>
If	O
this	O
is	O
done	O
the	O
right	O
way	O
,	O
the	O
result	O
is	O
also	O
a	O
manifold	B-Architecture
.	O
</s>
<s>
Similarly	O
,	O
two	O
boundaries	O
of	O
a	O
single	O
manifold	B-Architecture
can	O
be	O
glued	O
together	O
.	O
</s>
<s>
Formally	O
,	O
the	O
gluing	O
is	O
defined	O
by	O
a	O
bijection	B-Algorithm
between	O
the	O
two	O
boundaries	O
.	O
</s>
<s>
For	O
a	O
topological	O
manifold	B-Architecture
,	O
this	O
bijection	B-Algorithm
should	O
be	O
a	O
homeomorphism	O
,	O
otherwise	O
the	O
result	O
will	O
not	O
be	O
a	O
topological	O
manifold	B-Architecture
.	O
</s>
<s>
Similarly	O
,	O
for	O
a	O
differentiable	O
manifold	B-Architecture
,	O
it	O
has	O
to	O
be	O
a	O
diffeomorphism	O
.	O
</s>
<s>
For	O
other	O
manifolds	B-Architecture
,	O
other	O
structures	O
should	O
be	O
preserved	O
.	O
</s>
<s>
A	O
finite	O
cylinder	O
may	O
be	O
constructed	O
as	O
a	O
manifold	B-Architecture
by	O
starting	O
with	O
a	O
strip	O
 [ 0 , 1 ] × [ 0 , 1 ] 	O
and	O
gluing	O
a	O
pair	O
of	O
opposite	O
edges	O
on	O
the	O
boundary	O
by	O
a	O
suitable	O
diffeomorphism	O
.	O
</s>
<s>
The	O
Cartesian	O
product	O
of	O
manifolds	B-Architecture
is	O
also	O
a	O
manifold	B-Architecture
.	O
</s>
<s>
The	O
dimension	O
of	O
the	O
product	O
manifold	B-Architecture
is	O
the	O
sum	O
of	O
the	O
dimensions	O
of	O
its	O
factors	O
.	O
</s>
<s>
Its	O
topology	B-Architecture
is	O
the	O
product	O
topology	B-Architecture
,	O
and	O
a	O
Cartesian	O
product	O
of	O
charts	O
is	O
a	O
chart	O
for	O
the	O
product	O
manifold	B-Architecture
.	O
</s>
<s>
Thus	O
,	O
an	O
atlas	B-Device
for	O
the	O
product	O
manifold	B-Architecture
can	O
be	O
constructed	O
using	O
atlases	O
for	O
its	O
factors	O
.	O
</s>
<s>
If	O
these	O
atlases	O
define	O
a	O
differential	O
structure	O
on	O
the	O
factors	O
,	O
the	O
corresponding	O
atlas	B-Device
defines	O
a	O
differential	O
structure	O
on	O
the	O
product	O
manifold	B-Architecture
.	O
</s>
<s>
If	O
one	O
of	O
the	O
factors	O
has	O
a	O
boundary	O
,	O
the	O
product	O
manifold	B-Architecture
also	O
has	O
a	O
boundary	O
.	O
</s>
<s>
The	O
study	O
of	O
manifolds	B-Architecture
combines	O
many	O
important	O
areas	O
of	O
mathematics	O
:	O
it	O
generalizes	O
concepts	O
such	O
as	O
curves	O
and	O
surfaces	O
as	O
well	O
as	O
ideas	O
from	O
linear	B-Language
algebra	I-Language
and	O
topology	B-Architecture
.	O
</s>
<s>
Before	O
the	O
modern	O
concept	O
of	O
a	O
manifold	B-Architecture
there	O
were	O
several	O
important	O
results	O
.	O
</s>
<s>
Non-Euclidean	O
geometry	O
considers	O
spaces	O
where	O
Euclid	B-Language
's	O
parallel	O
postulate	O
fails	O
.	O
</s>
<s>
In	O
the	O
modern	O
theory	O
of	O
manifolds	B-Architecture
,	O
these	O
notions	O
correspond	O
to	O
Riemannian	O
manifolds	B-Architecture
with	O
constant	O
negative	O
and	O
positive	O
curvature	O
,	O
respectively	O
.	O
</s>
<s>
Such	O
a	O
surface	O
would	O
,	O
in	O
modern	O
terminology	O
,	O
be	O
called	O
a	O
manifold	B-Architecture
;	O
and	O
in	O
modern	O
terms	O
,	O
the	O
theorem	O
proved	O
that	O
the	O
curvature	O
of	O
the	O
surface	O
is	O
an	O
intrinsic	O
property	O
.	O
</s>
<s>
Manifold	B-Architecture
theory	I-Architecture
has	O
come	O
to	O
focus	O
exclusively	O
on	O
these	O
intrinsic	O
properties	O
(	O
or	O
invariants	O
)	O
,	O
while	O
largely	O
ignoring	O
the	O
extrinsic	O
properties	O
of	O
the	O
ambient	O
space	O
.	O
</s>
<s>
Another	O
,	O
more	O
topological	O
example	O
of	O
an	O
intrinsic	O
property	O
of	O
a	O
manifold	B-Architecture
is	O
its	O
Euler	O
characteristic	O
.	O
</s>
<s>
Investigations	O
of	O
Niels	O
Henrik	O
Abel	O
and	O
Carl	O
Gustav	O
Jacobi	O
on	O
inversion	O
of	O
elliptic	O
integrals	O
in	O
the	O
first	O
half	O
of	O
19th	O
century	O
led	O
them	O
to	O
consider	O
special	O
types	O
of	O
complex	O
manifolds	B-Architecture
,	O
now	O
known	O
as	O
Jacobians	O
.	O
</s>
<s>
Bernhard	O
Riemann	O
further	O
contributed	O
to	O
their	O
theory	O
,	O
clarifying	O
the	O
geometric	O
meaning	O
of	O
the	O
process	O
of	O
analytic	B-Algorithm
continuation	I-Algorithm
of	O
functions	O
of	O
complex	O
variables	O
.	O
</s>
<s>
Another	O
important	O
source	O
of	O
manifolds	B-Architecture
in	O
19th	O
century	O
mathematics	O
was	O
analytical	O
mechanics	O
,	O
as	O
developed	O
by	O
Siméon	O
Poisson	O
,	O
Jacobi	O
,	O
and	O
William	O
Rowan	O
Hamilton	O
.	O
</s>
<s>
This	O
space	O
is	O
,	O
in	O
fact	O
,	O
a	O
high-dimensional	O
manifold	B-Architecture
,	O
whose	O
dimension	O
corresponds	O
to	O
the	O
degrees	O
of	O
freedom	O
of	O
the	O
system	O
and	O
where	O
the	O
points	O
are	O
specified	O
by	O
their	O
generalized	O
coordinates	O
.	O
</s>
<s>
For	O
an	O
unconstrained	O
movement	O
of	O
free	O
particles	O
the	O
manifold	B-Architecture
is	O
equivalent	O
to	O
the	O
Euclidean	O
space	O
,	O
but	O
various	O
conservation	O
laws	O
constrain	O
it	O
to	O
more	O
complicated	O
formations	O
,	O
e.g.	O
</s>
<s>
The	O
theory	O
of	O
a	O
rotating	O
solid	O
body	O
,	O
developed	O
in	O
the	O
18th	O
century	O
by	O
Leonhard	O
Euler	O
and	O
Joseph-Louis	O
Lagrange	O
,	O
gives	O
another	O
example	O
where	O
the	O
manifold	B-Architecture
is	O
nontrivial	O
.	O
</s>
<s>
Geometrical	O
and	O
topological	O
aspects	O
of	O
classical	O
mechanics	O
were	O
emphasized	O
by	O
Henri	O
Poincaré	O
,	O
one	O
of	O
the	O
founders	O
of	O
topology	B-Architecture
.	O
</s>
<s>
The	O
name	O
manifold	B-Architecture
comes	O
from	O
Riemann	O
's	O
original	O
German	O
term	O
,	O
Mannigfaltigkeit	O
,	O
which	O
William	O
Kingdon	O
Clifford	O
translated	O
as	O
"	O
manifoldness	O
"	O
.	O
</s>
<s>
Using	O
induction	B-Algorithm
,	O
Riemann	O
constructs	O
an	O
n-fach	O
ausgedehnte	O
Mannigfaltigkeit	O
(	O
n	O
times	O
extended	O
manifoldness	O
or	O
n-dimensional	O
manifoldness	O
)	O
as	O
a	O
continuous	O
stack	O
of	O
(	O
n−1	O
)	O
dimensional	O
manifoldnesses	O
.	O
</s>
<s>
Riemann	O
's	O
intuitive	O
notion	O
of	O
a	O
Mannigfaltigkeit	O
evolved	O
into	O
what	O
is	O
today	O
formalized	O
as	O
a	O
manifold	B-Architecture
.	O
</s>
<s>
Riemannian	O
manifolds	B-Architecture
and	O
Riemann	O
surfaces	O
are	O
named	O
after	O
Riemann	O
.	O
</s>
<s>
In	O
his	O
very	O
influential	O
paper	O
,	O
Analysis	O
Situs	O
,	O
Henri	O
Poincaré	O
gave	O
a	O
definition	O
of	O
a	O
differentiable	O
manifold	B-Architecture
(	O
variété	O
)	O
which	O
served	O
as	O
a	O
precursor	O
to	O
the	O
modern	O
concept	O
of	O
a	O
manifold	B-Architecture
.	O
</s>
<s>
In	O
the	O
first	O
section	O
of	O
Analysis	O
Situs	O
,	O
Poincaré	O
defines	O
a	O
manifold	B-Architecture
as	O
the	O
level	O
set	O
of	O
a	O
continuously	O
differentiable	O
function	O
between	O
Euclidean	O
spaces	O
that	O
satisfies	O
the	O
nondegeneracy	O
hypothesis	O
of	O
the	O
implicit	O
function	O
theorem	O
.	O
</s>
<s>
In	O
the	O
third	O
section	O
,	O
he	O
begins	O
by	O
remarking	O
that	O
the	O
graph	B-Application
of	O
a	O
continuously	O
differentiable	O
function	O
is	O
a	O
manifold	B-Architecture
in	O
the	O
latter	O
sense	O
.	O
</s>
<s>
He	O
then	O
proposes	O
a	O
new	O
,	O
more	O
general	O
,	O
definition	O
of	O
manifold	B-Architecture
based	O
on	O
a	O
'	O
chain	O
of	O
manifolds	B-Architecture
 '	O
(	O
une	O
chaîne	O
des	O
variétés	O
)	O
.	O
</s>
<s>
Poincaré	O
's	O
notion	O
of	O
a	O
chain	O
of	O
manifolds	B-Architecture
is	O
a	O
precursor	O
to	O
the	O
modern	O
notion	O
of	O
atlas	B-Device
.	O
</s>
<s>
In	O
particular	O
,	O
he	O
considers	O
two	O
manifolds	B-Architecture
defined	O
respectively	O
as	O
graphs	B-Application
of	I-Application
functions	I-Application
and	O
.	O
</s>
<s>
If	O
these	O
manifolds	B-Architecture
overlap	O
(	O
a	O
une	O
partie	O
commune	O
)	O
,	O
then	O
he	O
requires	O
that	O
the	O
coordinates	O
depend	O
continuously	O
differentiably	O
on	O
the	O
coordinates	O
and	O
vice	O
versa	O
( 	O
 '	O
les	O
...	O
sont	O
fonctions	O
analytiques	O
des	O
et	O
inversement	O
 '	O
)	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
unit	O
circle	O
in	O
the	O
plane	O
can	O
be	O
thought	O
of	O
as	O
the	O
graph	B-Application
of	O
the	O
function	O
or	O
else	O
the	O
function	O
in	O
a	O
neighborhood	O
of	O
every	O
point	O
except	O
the	O
points	O
(	O
1	O
,	O
0	O
)	O
and	O
( −1	O
,	O
0	O
)	O
;	O
and	O
in	O
a	O
neighborhood	O
of	O
those	O
points	O
,	O
it	O
can	O
be	O
thought	O
of	O
as	O
the	O
graph	B-Application
of	O
,	O
respectively	O
,	O
and	O
.	O
</s>
<s>
The	O
circle	O
can	O
be	O
represented	O
by	O
a	O
graph	B-Application
in	O
the	O
neighborhood	O
of	O
every	O
point	O
because	O
the	O
left	O
hand	O
side	O
of	O
its	O
defining	O
equation	O
has	O
nonzero	O
gradient	O
at	O
every	O
point	O
of	O
the	O
circle	O
.	O
</s>
<s>
By	O
the	O
implicit	O
function	O
theorem	O
,	O
every	O
submanifold	O
of	O
Euclidean	O
space	O
is	O
locally	O
the	O
graph	B-Application
of	I-Application
a	I-Application
function	I-Application
.	O
</s>
<s>
Hermann	O
Weyl	O
gave	O
an	O
intrinsic	O
definition	O
for	O
differentiable	O
manifolds	B-Architecture
in	O
his	O
lecture	O
course	O
on	O
Riemann	O
surfaces	O
in	O
1911	O
–	O
1912	O
,	O
opening	O
the	O
road	O
to	O
the	O
general	O
concept	O
of	O
a	O
topological	O
space	O
that	O
followed	O
shortly	O
.	O
</s>
<s>
During	O
the	O
1930s	O
Hassler	O
Whitney	O
and	O
others	O
clarified	O
the	O
foundational	O
aspects	O
of	O
the	O
subject	O
,	O
and	O
thus	O
intuitions	O
dating	O
back	O
to	O
the	O
latter	O
half	O
of	O
the	O
19th	O
century	O
became	O
precise	O
,	O
and	O
developed	O
through	O
differential	B-Language
geometry	I-Language
and	O
Lie	O
group	O
theory	O
.	O
</s>
<s>
Two-dimensional	B-Architecture
manifolds	I-Architecture
,	O
also	O
known	O
as	O
a	O
2D	O
surfaces	O
embedded	O
in	O
our	O
common	O
3D	O
space	O
,	O
were	O
considered	O
by	O
Riemann	O
under	O
the	O
guise	O
of	O
Riemann	O
surfaces	O
,	O
and	O
rigorously	O
classified	O
in	O
the	O
beginning	O
of	O
the	O
20th	O
century	O
by	O
Poul	O
Heegaard	O
and	O
Max	O
Dehn	O
.	O
</s>
<s>
Poincaré	O
pioneered	O
the	O
study	O
of	O
three-dimensional	O
manifolds	B-Architecture
and	O
raised	O
a	O
fundamental	O
question	O
about	O
them	O
,	O
today	O
known	O
as	O
the	O
Poincaré	O
conjecture	O
.	O
</s>
<s>
William	O
Thurston	O
's	O
geometrization	O
program	O
,	O
formulated	O
in	O
the	O
1970s	O
,	O
provided	O
a	O
far-reaching	O
extension	O
of	O
the	O
Poincaré	O
conjecture	O
to	O
the	O
general	O
three-dimensional	O
manifolds	B-Architecture
.	O
</s>
<s>
Four-dimensional	O
manifolds	B-Architecture
were	O
brought	O
to	O
the	O
forefront	O
of	O
mathematical	O
research	O
in	O
the	O
1980s	O
by	O
Michael	O
Freedman	O
and	O
in	O
a	O
different	O
setting	O
,	O
by	O
Simon	O
Donaldson	O
,	O
who	O
was	O
motivated	O
by	O
the	O
then	O
recent	O
progress	O
in	O
theoretical	O
physics	O
(	O
Yang	O
–	O
Mills	O
theory	O
)	O
,	O
where	O
they	O
serve	O
as	O
a	O
substitute	O
for	O
ordinary	O
'	O
flat	O
 '	O
spacetime	B-Protocol
.	O
</s>
<s>
Andrey	O
Markov	O
Jr.	O
showed	O
in	O
1960	O
that	O
no	O
algorithm	O
exists	O
for	O
classifying	O
four-dimensional	O
manifolds	B-Architecture
.	O
</s>
<s>
Important	O
work	O
on	O
higher-dimensional	O
manifolds	B-Architecture
,	O
including	O
analogues	O
of	O
the	O
Poincaré	O
conjecture	O
,	O
had	O
been	O
done	O
earlier	O
by	O
René	O
Thom	O
,	O
John	O
Milnor	O
,	O
Stephen	O
Smale	O
and	O
Sergei	O
Novikov	O
.	O
</s>
<s>
A	O
very	O
pervasive	O
and	O
flexible	O
technique	O
underlying	O
much	O
work	O
on	O
the	O
topology	B-Architecture
of	O
manifolds	B-Architecture
is	O
Morse	O
theory	O
.	O
</s>
<s>
The	O
simplest	O
kind	O
of	O
manifold	B-Architecture
to	O
define	O
is	O
the	O
topological	O
manifold	B-Architecture
,	O
which	O
looks	O
locally	O
like	O
some	O
"	O
ordinary	O
"	O
Euclidean	O
space	O
.	O
</s>
<s>
By	O
definition	O
,	O
all	O
manifolds	B-Architecture
are	O
topological	O
manifolds	B-Architecture
,	O
so	O
the	O
phrase	O
"	O
topological	O
manifold	B-Architecture
"	O
is	O
usually	O
used	O
to	O
emphasize	O
that	O
a	O
manifold	B-Architecture
lacks	O
additional	O
structure	O
,	O
or	O
that	O
only	O
its	O
topological	O
properties	O
are	O
being	O
considered	O
.	O
</s>
<s>
Formally	O
,	O
a	O
topological	O
manifold	B-Architecture
is	O
a	O
topological	O
space	O
locally	O
homeomorphic	O
to	O
a	O
Euclidean	O
space	O
.	O
</s>
<s>
This	O
means	O
that	O
every	O
point	O
has	O
a	O
neighbourhood	O
for	O
which	O
there	O
exists	O
a	O
homeomorphism	O
(	O
a	O
bijective	B-Algorithm
continuous	O
function	O
whose	O
inverse	O
is	O
also	O
continuous	O
)	O
mapping	B-Algorithm
that	O
neighbourhood	O
to	O
.	O
</s>
<s>
These	O
homeomorphisms	O
are	O
the	O
charts	O
of	O
the	O
manifold	B-Architecture
.	O
</s>
<s>
A	O
topological	O
manifold	B-Architecture
looks	O
locally	O
like	O
a	O
Euclidean	O
space	O
in	O
a	O
rather	O
weak	O
manner	O
:	O
while	O
for	O
each	O
individual	O
chart	O
it	O
is	O
possible	O
to	O
distinguish	O
differentiable	O
functions	O
or	O
measure	O
distances	O
and	O
angles	O
,	O
merely	O
by	O
virtue	O
of	O
being	O
a	O
topological	O
manifold	B-Architecture
a	O
space	O
does	O
not	O
have	O
any	O
particular	O
and	O
consistent	O
choice	O
of	O
such	O
concepts	O
.	O
</s>
<s>
In	O
order	O
to	O
discuss	O
such	O
properties	O
for	O
a	O
manifold	B-Architecture
,	O
one	O
needs	O
to	O
specify	O
further	O
structure	O
and	O
consider	O
differentiable	O
manifolds	B-Architecture
and	O
Riemannian	O
manifolds	B-Architecture
discussed	O
below	O
.	O
</s>
<s>
In	O
particular	O
,	O
the	O
same	O
underlying	O
topological	O
manifold	B-Architecture
can	O
have	O
several	O
mutually	O
incompatible	O
classes	O
of	O
differentiable	O
functions	O
and	O
an	O
infinite	O
number	O
of	O
ways	O
to	O
specify	O
distances	O
and	O
angles	O
.	O
</s>
<s>
The	O
dimension	O
of	O
the	O
manifold	B-Architecture
at	O
a	O
certain	O
point	O
is	O
the	O
dimension	O
of	O
the	O
Euclidean	O
space	O
that	O
the	O
charts	O
at	O
that	O
point	O
map	O
to	O
(	O
number	O
n	O
in	O
the	O
definition	O
)	O
.	O
</s>
<s>
All	O
points	O
in	O
a	O
connected	O
manifold	B-Architecture
have	O
the	O
same	O
dimension	O
.	O
</s>
<s>
Some	O
authors	O
require	O
that	O
all	O
charts	O
of	O
a	O
topological	O
manifold	B-Architecture
map	O
to	O
Euclidean	O
spaces	O
of	O
same	O
dimension	O
.	O
</s>
<s>
In	O
that	O
case	O
every	O
topological	O
manifold	B-Architecture
has	O
a	O
topological	O
invariant	O
,	O
its	O
dimension	O
.	O
</s>
<s>
For	O
most	O
applications	O
,	O
a	O
special	O
kind	O
of	O
topological	O
manifold	B-Architecture
,	O
namely	O
,	O
a	O
differentiable	O
manifold	B-Architecture
,	O
is	O
used	O
.	O
</s>
<s>
If	O
the	O
local	O
charts	O
on	O
a	O
manifold	B-Architecture
are	O
compatible	O
in	O
a	O
certain	O
sense	O
,	O
one	O
can	O
define	O
directions	O
,	O
tangent	O
spaces	O
,	O
and	O
differentiable	O
functions	O
on	O
that	O
manifold	B-Architecture
.	O
</s>
<s>
In	O
particular	O
it	O
is	O
possible	O
to	O
use	O
calculus	O
on	O
a	O
differentiable	O
manifold	B-Architecture
.	O
</s>
<s>
Each	O
point	O
of	O
an	O
n-dimensional	O
differentiable	O
manifold	B-Architecture
has	O
a	O
tangent	O
space	O
.	O
</s>
<s>
Two	O
important	O
classes	O
of	O
differentiable	O
manifolds	B-Architecture
are	O
smooth	O
and	O
analytic	B-Language
manifolds	B-Architecture
.	O
</s>
<s>
For	O
smooth	O
manifolds	B-Architecture
the	O
transition	O
maps	O
are	O
smooth	O
,	O
that	O
is	O
,	O
infinitely	O
differentiable	O
.	O
</s>
<s>
Analytic	B-Language
manifolds	B-Architecture
are	O
smooth	O
manifolds	B-Architecture
with	O
the	O
additional	O
condition	O
that	O
the	O
transition	O
maps	O
are	O
analytic	B-Language
(	O
they	O
can	O
be	O
expressed	O
as	O
power	O
series	O
)	O
.	O
</s>
<s>
The	O
sphere	O
can	O
be	O
given	O
analytic	B-Language
structure	O
,	O
as	O
can	O
most	O
familiar	O
curves	O
and	O
surfaces	O
.	O
</s>
<s>
A	O
rectifiable	O
set	O
generalizes	O
the	O
idea	O
of	O
a	O
piecewise	O
smooth	O
or	O
rectifiable	O
curve	O
to	O
higher	O
dimensions	O
;	O
however	O
,	O
rectifiable	O
sets	O
are	O
not	O
in	O
general	O
manifolds	B-Architecture
.	O
</s>
<s>
To	O
measure	O
distances	O
and	O
angles	O
on	O
manifolds	B-Architecture
,	O
the	O
manifold	B-Architecture
must	O
be	O
Riemannian	O
.	O
</s>
<s>
A	O
Riemannian	O
manifold	B-Architecture
is	O
a	O
differentiable	O
manifold	B-Architecture
in	O
which	O
each	O
tangent	O
space	O
is	O
equipped	O
with	O
an	O
inner	O
product	O
in	O
a	O
manner	O
which	O
varies	O
smoothly	O
from	O
point	O
to	O
point	O
.	O
</s>
<s>
This	O
allows	O
one	O
to	O
define	O
various	O
notions	O
such	O
as	O
length	O
,	O
angles	O
,	O
areas	O
(	O
or	O
volumes	O
)	O
,	O
curvature	O
and	O
divergence	B-Application
of	O
vector	O
fields	O
.	O
</s>
<s>
All	O
differentiable	O
manifolds	B-Architecture
(	O
of	O
constant	O
dimension	O
)	O
can	O
be	O
given	O
the	O
structure	O
of	O
a	O
Riemannian	O
manifold	B-Architecture
.	O
</s>
<s>
The	O
Euclidean	O
space	O
itself	O
carries	O
a	O
natural	O
structure	O
of	O
Riemannian	O
manifold	B-Architecture
(	O
the	O
tangent	O
spaces	O
are	O
naturally	O
identified	O
with	O
the	O
Euclidean	O
space	O
itself	O
and	O
carry	O
the	O
standard	O
scalar	O
product	O
of	O
the	O
space	O
)	O
.	O
</s>
<s>
A	O
Finsler	O
manifold	B-Architecture
allows	O
the	O
definition	O
of	O
distance	O
but	O
does	O
not	O
require	O
the	O
concept	O
of	O
angle	O
;	O
it	O
is	O
an	O
analytic	B-Language
manifold	B-Architecture
in	O
which	O
each	O
tangent	O
space	O
is	O
equipped	O
with	O
a	O
norm	O
,	O
,	O
in	O
a	O
manner	O
which	O
varies	O
smoothly	O
from	O
point	O
to	O
point	O
.	O
</s>
<s>
Any	O
Riemannian	O
manifold	B-Architecture
is	O
a	O
Finsler	O
manifold	B-Architecture
.	O
</s>
<s>
Lie	O
groups	O
,	O
named	O
after	O
Sophus	O
Lie	O
,	O
are	O
differentiable	O
manifolds	B-Architecture
that	O
carry	O
also	O
the	O
structure	O
of	O
a	O
group	O
which	O
is	O
such	O
that	O
the	O
group	O
operations	O
are	O
defined	O
by	O
smooth	O
maps	O
.	O
</s>
<s>
Other	O
examples	O
of	O
Lie	O
groups	O
include	O
special	O
groups	O
of	O
matrices	B-Architecture
,	O
which	O
are	O
all	O
subgroups	O
of	O
the	O
general	O
linear	O
group	O
,	O
the	O
group	O
of	O
matrices	B-Architecture
with	O
non-zero	O
determinant	O
.	O
</s>
<s>
If	O
the	O
matrix	O
entries	O
are	O
real	O
numbers	O
,	O
this	O
will	O
be	O
an	O
-dimensional	O
disconnected	O
manifold	B-Architecture
.	O
</s>
<s>
The	O
orthogonal	O
groups	O
,	O
the	O
symmetry	O
groups	O
of	O
the	O
sphere	O
and	O
hyperspheres	O
,	O
are	O
dimensional	O
manifolds	B-Architecture
,	O
where	O
is	O
the	O
dimension	O
of	O
the	O
sphere	O
.	O
</s>
<s>
A	O
complex	O
manifold	B-Architecture
is	O
a	O
manifold	B-Architecture
whose	O
charts	O
take	O
values	O
in	O
and	O
whose	O
transition	O
functions	O
are	O
holomorphic	O
on	O
the	O
overlaps	O
.	O
</s>
<s>
These	O
manifolds	B-Architecture
are	O
the	O
basic	O
objects	O
of	O
study	O
in	O
complex	O
geometry	O
.	O
</s>
<s>
A	O
one-complex-dimensional	O
manifold	B-Architecture
is	O
called	O
a	O
Riemann	O
surface	O
.	O
</s>
<s>
An	O
-dimensional	O
complex	O
manifold	B-Architecture
has	O
dimension	O
as	O
a	O
real	O
differentiable	O
manifold	B-Architecture
.	O
</s>
<s>
A	O
CR	O
manifold	B-Architecture
is	O
a	O
manifold	B-Architecture
modeled	O
on	O
boundaries	O
of	O
domains	O
in	O
.	O
</s>
<s>
'	O
Infinite	O
dimensional	O
manifolds	B-Architecture
 '	O
:	O
to	O
allow	O
for	O
infinite	O
dimensions	O
,	O
one	O
may	O
consider	O
Banach	O
manifolds	B-Architecture
which	O
are	O
locally	O
homeomorphic	O
to	O
Banach	O
spaces	O
.	O
</s>
<s>
Similarly	O
,	O
Fréchet	O
manifolds	B-Architecture
are	O
locally	O
homeomorphic	O
to	O
Fréchet	B-Algorithm
spaces	I-Algorithm
.	O
</s>
<s>
A	O
symplectic	O
manifold	B-Architecture
is	O
a	O
kind	O
of	O
manifold	B-Architecture
which	O
is	O
used	O
to	O
represent	O
the	O
phase	O
spaces	O
in	O
classical	O
mechanics	O
.	O
</s>
<s>
A	O
closely	O
related	O
type	O
of	O
manifold	B-Architecture
is	O
a	O
contact	O
manifold	B-Architecture
.	O
</s>
<s>
A	O
combinatorial	B-Algorithm
manifold	I-Algorithm
is	O
a	O
kind	O
of	O
manifold	B-Architecture
which	O
is	O
discretization	O
of	O
a	O
manifold	B-Architecture
.	O
</s>
<s>
It	O
usually	O
means	O
a	O
piecewise	O
linear	O
manifold	B-Architecture
made	O
by	O
simplicial	O
complexes	O
.	O
</s>
<s>
A	O
digital	B-Algorithm
manifold	I-Algorithm
is	O
a	O
special	O
kind	O
of	O
combinatorial	B-Algorithm
manifold	I-Algorithm
which	O
is	O
defined	O
in	O
digital	O
space	O
.	O
</s>
<s>
See	O
digital	B-Algorithm
topology	I-Algorithm
.	O
</s>
<s>
Different	O
notions	O
of	O
manifolds	B-Architecture
have	O
different	O
notions	O
of	O
classification	O
and	O
invariant	O
;	O
in	O
this	O
section	O
we	O
focus	O
on	O
smooth	O
closed	O
manifolds	B-Architecture
.	O
</s>
<s>
The	O
classification	O
of	O
smooth	O
closed	O
manifolds	B-Architecture
is	O
well	O
understood	O
in	O
principle	O
,	O
except	O
in	O
dimension	O
4	O
:	O
in	O
low	O
dimensions	O
(	O
2	O
and	O
3	O
)	O
it	O
is	O
geometric	O
,	O
via	O
the	O
uniformization	O
theorem	O
and	O
the	O
solution	O
of	O
the	O
Poincaré	O
conjecture	O
,	O
and	O
in	O
high	O
dimension	O
(	O
5	O
and	O
above	O
)	O
it	O
is	O
algebraic	O
,	O
via	O
surgery	O
theory	O
.	O
</s>
<s>
This	O
is	O
a	O
classification	O
in	O
principle	O
:	O
the	O
general	O
question	O
of	O
whether	O
two	O
smooth	O
manifolds	B-Architecture
are	O
diffeomorphic	O
is	O
not	O
computable	O
in	O
general	O
.	O
</s>
<s>
This	O
is	O
much	O
harder	O
in	O
higher	O
dimensions	O
:	O
higher-dimensional	O
manifolds	B-Architecture
cannot	O
be	O
directly	O
visualized	O
(	O
though	O
visual	O
intuition	O
is	O
useful	O
in	O
understanding	O
them	O
)	O
,	O
nor	O
can	O
their	O
diffeomorphism	O
classes	O
be	O
enumerated	O
,	O
nor	O
can	O
one	O
in	O
general	O
determine	O
if	O
two	O
different	O
descriptions	O
of	O
a	O
higher-dimensional	O
manifold	B-Architecture
refer	O
to	O
the	O
same	O
object	O
.	O
</s>
<s>
However	O
,	O
one	O
can	O
determine	O
if	O
two	O
manifolds	B-Architecture
are	O
different	O
if	O
there	O
is	O
some	O
intrinsic	O
characteristic	O
that	O
differentiates	O
them	O
.	O
</s>
<s>
Such	O
criteria	O
are	O
commonly	O
referred	O
to	O
as	O
invariants	O
,	O
because	O
,	O
while	O
they	O
may	O
be	O
defined	O
in	O
terms	O
of	O
some	O
presentation	O
(	O
such	O
as	O
the	O
genus	O
in	O
terms	O
of	O
a	O
triangulation	O
)	O
,	O
they	O
are	O
the	O
same	O
relative	O
to	O
all	O
possible	O
descriptions	O
of	O
a	O
particular	O
manifold	B-Architecture
:	O
they	O
are	O
invariant	O
under	O
different	O
descriptions	O
.	O
</s>
<s>
Naively	O
,	O
one	O
could	O
hope	O
to	O
develop	O
an	O
arsenal	O
of	O
invariant	O
criteria	O
that	O
would	O
definitively	O
classify	O
all	O
manifolds	B-Architecture
up	O
to	O
isomorphism	O
.	O
</s>
<s>
Unfortunately	O
,	O
it	O
is	O
known	O
that	O
for	O
manifolds	B-Architecture
of	O
dimension	O
4	O
and	O
higher	O
,	O
no	O
program	O
exists	O
that	O
can	O
decide	O
whether	O
two	O
manifolds	B-Architecture
are	O
diffeomorphic	O
.	O
</s>
<s>
Smooth	O
manifolds	B-Architecture
have	O
a	O
rich	O
set	O
of	O
invariants	O
,	O
coming	O
from	O
point-set	O
topology	B-Architecture
,	O
classic	O
algebraic	O
topology	B-Architecture
,	O
and	O
geometric	B-Algorithm
topology	I-Algorithm
.	O
</s>
<s>
Smooth	O
closed	O
manifolds	B-Architecture
have	O
no	O
local	O
invariants	O
(	O
other	O
than	O
dimension	O
)	O
,	O
though	O
geometric	O
manifolds	B-Architecture
have	O
local	O
invariants	O
,	O
notably	O
the	O
curvature	O
of	O
a	O
Riemannian	O
manifold	B-Architecture
and	O
the	O
torsion	O
of	O
a	O
manifold	B-Architecture
equipped	O
with	O
an	O
affine	O
connection	O
.	O
</s>
<s>
This	O
distinction	O
between	O
local	O
invariants	O
and	O
no	O
local	O
invariants	O
is	O
a	O
common	O
way	O
to	O
distinguish	O
between	O
geometry	O
and	O
topology	B-Architecture
.	O
</s>
<s>
All	O
invariants	O
of	O
a	O
smooth	O
closed	O
manifold	B-Architecture
are	O
thus	O
global	O
.	O
</s>
<s>
Algebraic	O
topology	B-Architecture
is	O
a	O
source	O
of	O
a	O
number	O
of	O
important	O
global	O
invariant	O
properties	O
.	O
</s>
<s>
Indeed	O
,	O
several	O
branches	O
of	O
mathematics	O
,	O
such	O
as	O
homology	O
and	O
homotopy	O
theory	O
,	O
and	O
the	O
theory	O
of	O
characteristic	O
classes	O
were	O
founded	O
in	O
order	O
to	O
study	O
invariant	O
properties	O
of	O
manifolds	B-Architecture
.	O
</s>
<s>
In	O
dimensions	O
two	O
and	O
higher	O
,	O
a	O
simple	O
but	O
important	O
invariant	O
criterion	O
is	O
the	O
question	O
of	O
whether	O
a	O
manifold	B-Architecture
admits	O
a	O
meaningful	O
orientation	O
.	O
</s>
<s>
Consider	O
a	O
topological	O
manifold	B-Architecture
with	O
charts	O
mapping	B-Algorithm
to	O
.	O
</s>
<s>
Given	O
an	O
ordered	O
basis	O
for	O
,	O
a	O
chart	O
causes	O
its	O
piece	O
of	O
the	O
manifold	B-Architecture
to	O
itself	O
acquire	O
a	O
sense	O
of	O
ordering	O
,	O
which	O
in	O
3-dimensions	O
can	O
be	O
viewed	O
as	O
either	O
right-handed	O
or	O
left-handed	O
.	O
</s>
<s>
Overlapping	O
charts	O
are	O
not	O
required	O
to	O
agree	O
in	O
their	O
sense	O
of	O
ordering	O
,	O
which	O
gives	O
manifolds	B-Architecture
an	O
important	O
freedom	O
.	O
</s>
<s>
For	O
some	O
manifolds	B-Architecture
,	O
like	O
the	O
sphere	O
,	O
charts	O
can	O
be	O
chosen	O
so	O
that	O
overlapping	O
regions	O
agree	O
on	O
their	O
"	O
handedness	O
"	O
;	O
these	O
are	O
orientable	O
manifolds	B-Architecture
.	O
</s>
<s>
Some	O
illustrative	O
examples	O
of	O
non-orientable	O
manifolds	B-Architecture
include	O
:	O
(	O
1	O
)	O
the	O
Möbius	O
strip	O
,	O
which	O
is	O
a	O
manifold	B-Architecture
with	O
boundary	O
,	O
(	O
2	O
)	O
the	O
Klein	O
bottle	O
,	O
which	O
must	O
intersect	O
itself	O
in	O
its	O
3-space	O
representation	O
,	O
and	O
(	O
3	O
)	O
the	O
real	O
projective	O
plane	O
,	O
which	O
arises	O
naturally	O
in	O
geometry	O
.	O
</s>
<s>
Begin	O
with	O
an	O
infinite	O
circular	O
cylinder	O
standing	O
vertically	O
,	O
a	O
manifold	B-Architecture
without	O
boundary	O
.	O
</s>
<s>
This	O
is	O
an	O
orientable	O
manifold	B-Architecture
with	O
boundary	O
,	O
upon	O
which	O
"	O
surgery	O
"	O
will	O
be	O
performed	O
.	O
</s>
<s>
Gluing	O
the	O
circles	O
together	O
will	O
produce	O
a	O
new	O
,	O
closed	O
manifold	B-Architecture
without	O
boundary	O
,	O
the	O
Klein	O
bottle	O
.	O
</s>
<s>
For	O
two	O
dimensional	O
manifolds	B-Architecture
a	O
key	O
invariant	O
property	O
is	O
the	O
genus	O
,	O
or	O
"	O
number	O
of	O
handles	O
"	O
present	O
in	O
a	O
surface	O
.	O
</s>
<s>
Indeed	O
,	O
it	O
is	O
possible	O
to	O
fully	O
characterize	O
compact	O
,	O
two-dimensional	B-Architecture
manifolds	I-Architecture
on	O
the	O
basis	O
of	O
genus	O
and	O
orientability	O
.	O
</s>
<s>
In	O
higher-dimensional	O
manifolds	B-Architecture
genus	O
is	O
replaced	O
by	O
the	O
notion	O
of	O
Euler	O
characteristic	O
,	O
and	O
more	O
generally	O
Betti	O
numbers	O
and	O
homology	O
and	O
cohomology	O
.	O
</s>
<s>
Just	O
as	O
there	O
are	O
various	O
types	O
of	O
manifolds	B-Architecture
,	O
there	O
are	O
various	O
types	O
of	O
maps	O
of	O
manifolds	B-Architecture
.	O
</s>
<s>
In	O
geometric	B-Algorithm
topology	I-Algorithm
a	O
basic	O
type	O
are	O
embeddings	O
,	O
of	O
which	O
knot	O
theory	O
is	O
a	O
central	O
example	O
,	O
and	O
generalizations	O
such	O
as	O
immersions	O
,	O
submersions	O
,	O
covering	O
spaces	O
,	O
and	O
ramified	O
covering	O
spaces	O
.	O
</s>
<s>
A	O
basic	O
example	O
of	O
maps	O
between	O
manifolds	B-Architecture
are	O
scalar-valued	O
functions	O
on	O
a	O
manifold	B-Architecture
,	O
</s>
<s>
sometimes	O
called	O
regular	O
functions	O
or	O
functionals	O
,	O
by	O
analogy	O
with	O
algebraic	O
geometry	O
or	O
linear	B-Language
algebra	I-Language
.	O
</s>
<s>
These	O
are	O
of	O
interest	O
both	O
in	O
their	O
own	O
right	O
,	O
and	O
to	O
study	O
the	O
underlying	O
manifold	B-Architecture
.	O
</s>
<s>
In	O
geometric	B-Algorithm
topology	I-Algorithm
,	O
most	O
commonly	O
studied	O
are	O
Morse	O
functions	O
,	O
which	O
yield	O
handlebody	O
decompositions	O
,	O
while	O
in	O
mathematical	O
analysis	O
,	O
one	O
often	O
studies	O
solution	O
to	O
partial	O
differential	O
equations	O
,	O
an	O
important	O
example	O
of	O
which	O
is	O
harmonic	O
analysis	O
,	O
where	O
one	O
studies	O
harmonic	O
functions	O
:	O
the	O
kernel	O
of	O
the	O
Laplace	O
operator	O
.	O
</s>
<s>
This	O
leads	O
to	O
such	O
functions	O
as	O
the	O
spherical	O
harmonics	O
,	O
and	O
to	O
heat	O
kernel	O
methods	O
of	O
studying	O
manifolds	B-Architecture
,	O
such	O
as	O
hearing	O
the	O
shape	O
of	O
a	O
drum	O
and	O
some	O
proofs	O
of	O
the	O
Atiyah	O
–	O
Singer	O
index	O
theorem	O
.	O
</s>
<s>
Infinite	O
dimensional	O
manifolds	B-Architecture
The	O
definition	O
of	O
a	O
manifold	B-Architecture
can	O
be	O
generalized	O
by	O
dropping	O
the	O
requirement	O
of	O
finite	O
dimensionality	O
.	O
</s>
<s>
Thus	O
an	O
infinite	O
dimensional	O
manifold	B-Architecture
is	O
a	O
topological	O
space	O
locally	O
homeomorphic	O
to	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
over	O
the	O
reals	O
.	O
</s>
<s>
This	O
omits	O
the	O
point-set	O
axioms	O
,	O
allowing	O
higher	O
cardinalities	O
and	O
non-Hausdorff	O
manifolds	B-Architecture
;	O
and	O
it	O
omits	O
finite	O
dimension	O
,	O
allowing	O
structures	O
such	O
as	O
Hilbert	B-Algorithm
manifolds	I-Algorithm
to	O
be	O
modeled	O
on	O
Hilbert	O
spaces	O
,	O
Banach	O
manifolds	B-Architecture
to	O
be	O
modeled	O
on	O
Banach	O
spaces	O
,	O
and	O
Fréchet	O
manifolds	B-Architecture
to	O
be	O
modeled	O
on	O
Fréchet	B-Algorithm
spaces	I-Algorithm
.	O
</s>
<s>
Usually	O
one	O
relaxes	O
one	O
or	O
the	O
other	O
condition	O
:	O
manifolds	B-Architecture
with	O
the	O
point-set	O
axioms	O
are	O
studied	O
in	O
general	O
topology	B-Architecture
,	O
while	O
infinite-dimensional	O
manifolds	B-Architecture
are	O
studied	O
in	O
functional	B-Application
analysis	I-Application
.	O
</s>
<s>
Orbifolds	O
An	O
orbifold	O
is	O
a	O
generalization	O
of	O
manifold	B-Architecture
allowing	O
for	O
certain	O
kinds	O
of	O
"	O
singularities	O
"	O
in	O
the	O
topology	B-Architecture
.	O
</s>
<s>
Algebraic	O
varieties	O
and	O
schemes	O
Non-singular	O
algebraic	O
varieties	O
over	O
the	O
real	O
or	O
complex	O
numbers	O
are	O
manifolds	B-Architecture
.	O
</s>
<s>
One	O
generalizes	O
this	O
first	O
by	O
allowing	O
singularities	O
,	O
secondly	O
by	O
allowing	O
different	O
fields	O
,	O
and	O
thirdly	O
by	O
emulating	O
the	O
patching	O
construction	O
of	O
manifolds	B-Architecture
:	O
just	O
as	O
a	O
manifold	B-Architecture
is	O
glued	O
together	O
from	O
open	O
subsets	O
of	O
Euclidean	O
space	O
,	O
an	O
algebraic	O
variety	O
is	O
glued	O
together	O
from	O
affine	O
algebraic	O
varieties	O
,	O
which	O
are	O
zero	O
sets	O
of	O
polynomials	O
over	O
algebraically	O
closed	O
fields	O
.	O
</s>
<s>
Both	O
are	O
related	O
to	O
manifolds	B-Architecture
,	O
but	O
are	O
constructed	O
algebraically	O
using	O
sheaves	O
instead	O
of	O
atlases	O
.	O
</s>
<s>
Because	O
of	O
singular	O
points	O
,	O
a	O
variety	O
is	O
in	O
general	O
not	O
a	O
manifold	B-Architecture
,	O
though	O
linguistically	O
the	O
French	O
variété	O
,	O
German	O
Mannigfaltigkeit	O
and	O
English	O
manifold	B-Architecture
are	O
largely	O
synonymous	O
.	O
</s>
<s>
In	O
French	O
an	O
algebraic	O
variety	O
is	O
called	O
une	O
variété	O
algébrique	O
(	O
an	O
algebraic	O
variety	O
)	O
,	O
while	O
a	O
smooth	O
manifold	B-Architecture
is	O
called	O
une	O
variété	O
différentielle	O
(	O
a	O
differential	O
variety	O
)	O
.	O
</s>
<s>
Stratified	O
space	O
A	O
"	O
stratified	O
space	O
"	O
is	O
a	O
space	O
that	O
can	O
be	O
divided	O
into	O
pieces	O
(	O
"	O
strata	O
"	O
)	O
,	O
with	O
each	O
stratum	O
a	O
manifold	B-Architecture
,	O
with	O
the	O
strata	O
fitting	O
together	O
in	O
prescribed	O
ways	O
(	O
formally	O
,	O
a	O
filtration	O
by	O
closed	O
subsets	O
)	O
.	O
</s>
<s>
There	O
are	O
various	O
technical	O
definitions	O
,	O
notably	O
a	O
Whitney	O
stratified	O
space	O
(	O
see	O
Whitney	O
conditions	O
)	O
for	O
smooth	O
manifolds	B-Architecture
and	O
a	O
topologically	O
stratified	O
space	O
for	O
topological	O
manifolds	B-Architecture
.	O
</s>
<s>
Basic	O
examples	O
include	O
manifold	B-Architecture
with	O
boundary	O
(	O
top	O
dimensional	O
manifold	B-Architecture
and	O
codimension	O
1	O
boundary	O
)	O
and	O
manifolds	B-Architecture
with	O
corners	O
(	O
top	O
dimensional	O
manifold	B-Architecture
,	O
codimension	O
1	O
boundary	O
,	O
codimension	O
2	O
corners	O
)	O
.	O
</s>
<s>
Whitney	O
stratified	O
spaces	O
are	O
a	O
broad	O
class	O
of	O
spaces	O
,	O
including	O
algebraic	O
varieties	O
,	O
analytic	B-Language
varieties	O
,	O
semialgebraic	O
sets	O
,	O
and	O
subanalytic	O
sets	O
.	O
</s>
<s>
In	O
general	O
the	O
resulting	O
space	O
is	O
singular	O
,	O
hence	O
not	O
a	O
manifold	B-Architecture
.	O
</s>
<s>
However	O
,	O
they	O
are	O
of	O
central	O
interest	O
in	O
algebraic	O
topology	B-Architecture
,	O
especially	O
in	O
homotopy	O
theory	O
.	O
</s>
<s>
Homology	O
manifolds	B-Architecture
A	O
homology	O
manifold	B-Architecture
is	O
a	O
space	O
that	O
behaves	O
like	O
a	O
manifold	B-Architecture
from	O
the	O
point	O
of	O
view	O
of	O
homology	O
theory	O
.	O
</s>
<s>
These	O
are	O
not	O
all	O
manifolds	B-Architecture
,	O
but	O
(	O
in	O
high	O
dimension	O
)	O
can	O
be	O
analyzed	O
by	O
surgery	O
theory	O
similarly	O
to	O
manifolds	B-Architecture
,	O
and	O
failure	O
to	O
be	O
a	O
manifold	B-Architecture
is	O
a	O
local	O
obstruction	O
,	O
as	O
in	O
surgery	O
theory	O
.	O
</s>
<s>
Let	O
be	O
equipped	O
with	O
the	O
topology	B-Architecture
induced	O
by	O
.	O
</s>
