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F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
ω
:
π
→
{
±
1
}
{\displaystyle \omega \colon \pi \to \left\{\pm 1\right\}}
This notion is of particular significance in surgery theory .
Motivation
[ edit ]
Given a manifold M , one takes
π
=
π
1
M
{\displaystyle \pi =\pi _{1}M}
(the fundamental group ), and then
ω
{\displaystyle \omega }
sends an element of
π
{\displaystyle \pi }
to
−
1
{\displaystyle -1}
if and only if the class it represents is orientation-reversing.
This map
ω
{\displaystyle \omega }
is trivial if and only if M is orientable .
The orientation character is an algebraic structure on the fundamental group of a manifold, which captures which loops are orientation reversing and which are orientation preserving.
Twisted group algebra
[ edit ]
The orientation character defines a twisted involution (*-ring structure) on the group ring
Z
[
π
]
{\displaystyle \mathbf {Z} [\pi ]}
, by
g
↦
ω
(
g
)
g
−
1
{\displaystyle g\mapsto \omega (g )g^{-1}}
(i.e.,
±
g
−
1
{\displaystyle \pm g^{-1}}
, accordingly as
g
{\displaystyle g}
is orientation preserving or reversing). This is denoted
Z
[
π
]
ω
{\displaystyle \mathbf {Z} [\pi ]^{\omega }}
.
Examples
[ edit ]
In real projective spaces , the orientation character evaluates trivially on loops if the dimension is odd, and assigns -1 to noncontractible loops in even dimension.
Properties
[ edit ]
The orientation character is either trivial or has kernel an index 2 subgroup, which determines the map completely.
See also
[ edit ]
References
[ edit ]
External links
[ edit ]
t
e
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Orientation_character&oldid=1170151790 "
C a t e g o r i e s :
● G e o m e t r i c t o p o l o g y
● G r o u p t h e o r y
● M o r p h i s m s
● S u r g e r y t h e o r y
● G r o u p t h e o r y s t u b s
H i d d e n c a t e g o r i e s :
● W i k i p e d i a a r t i c l e s t h a t a r e t o o t e c h n i c a l f r o m J u n e 2 0 2 3
● A l l a r t i c l e s t h a t a r e t o o t e c h n i c a l
● A r t i c l e s l a c k i n g s o u r c e s f r o m J u n e 2 0 2 3
● A l l a r t i c l e s l a c k i n g s o u r c e s
● A l l s t u b a r t i c l e s
● T h i s p a g e w a s l a s t e d i t e d o n 1 3 A u g u s t 2 0 2 3 , a t 1 2 : 1 1 ( U T C ) .
● T e x t i s a v a i l a b l e u n d e r t h e C r e a t i v e C o m m o n s A t t r i b u t i o n - S h a r e A l i k e L i c e n s e 4 . 0 ;
a d d i t i o n a l t e r m s m a y a p p l y . B y u s i n g t h i s s i t e , y o u a g r e e t o t h e T e r m s o f U s e a n d P r i v a c y P o l i c y . W i k i p e d i a ® i s a r e g i s t e r e d t r a d e m a r k o f t h e W i k i m e d i a F o u n d a t i o n , I n c . , a n o n - p r o f i t o r g a n i z a t i o n .
● P r i v a c y p o l i c y
● A b o u t W i k i p e d i a
● D i s c l a i m e r s
● C o n t a c t W i k i p e d i a
● C o d e o f C o n d u c t
● D e v e l o p e r s
● S t a t i s t i c s
● C o o k i e s t a t e m e n t
● M o b i l e v i e w