homology of planar polygon spaces
DESCRIPTION
Homology of Planar Polygon Spaces. Based on the paper by M. Farber and D. Schütz Presented by Yair Carmon. Contents. Linkages: Introduction/Reminder Main result: Betti numbers of a planar linkage Remarks on main results (very informal) Outline of proof. Acknowledgement. - PowerPoint PPT PresentationTRANSCRIPT
M. Farber and D. Schütz | Homology of Planar Polygon Spaces | Presented by Yair Carmon
Homology of Planar
Polygon SpacesBased on the paper by M. Farber and D. Schütz
Presented by Yair Carmon
M. Farber and D. Schütz | Homology of Planar Polygon Spaces | Presented by Yair Carmon 2
Contents
• Linkages: Introduction/Reminder
• Main result: Betti numbers of a planar linkage
• Remarks on main results
• (very informal) Outline of proof
M. Farber and D. Schütz | Homology of Planar Polygon Spaces | Presented by Yair Carmon 3
Acknowledgement• I would like to thank my
brother Dan for helpfuldiscussions
M. Farber and D. Schütz | Homology of Planar Polygon Spaces | Presented by Yair Carmon 4
The mechanical linkage• “rigid links connected with joints to form a closed chain,
or a series of closed chains”
• Linkages are useful!• We limit ourselves to a single 2d chain w/ two fixed joints
Definition and animations courtesy of Wikipedia
Klann linkage Peaucellier–Lipkin linkage
Chebyshev linkage
M. Farber and D. Schütz | Homology of Planar Polygon Spaces | Presented by Yair Carmon 5
Configuration space of a planar linkage• Informally: the set of angles in an n–edges polygon• Formally:
• Modulo SO(2) is equivalent to two fixed joints
( ) ( )1 2, ,..., n
nl l l += Îl ¡
( ) 1 11
1,..., | 0
n
n i ii
M u u S S l u=
ì üï ïï ï= Î ´ ´ =í ýï ïï ïî þål L ( )/ 2SO
1l
2l
5l
4l
3l
1u2u
5u4u 3u
1 1u =% 2u%5u%
4u%3u%
( )*1
/ 2i i
SOu uuÛ ¬
M. Farber and D. Schütz | Homology of Planar Polygon Spaces | Presented by Yair Carmon 6
Configuration space – properties
• Ml is n–3 dimensional
• l is called “generic” if Ml contains no collinear cfgs
• For generic l, Ml is a closed smooth manifold
• For non–generic l , Ml is a compact manifold with finitely many singular points (the collinear configurations)
M. Farber and D. Schütz | Homology of Planar Polygon Spaces | Presented by Yair Carmon 7
Configuration space – examples
@ { } { }0 1È( )3,3,2=l
@4g =( )1,1,1,1,1=l
@ 1S( )5,2,2,2=l
M. Farber and D. Schütz | Homology of Planar Polygon Spaces | Presented by Yair Carmon 8
Betti numbers of a planar linkage• A subset is called short/median/long if
• Assume l1 is maximal, and define,sk (mk ) – the number of short (median) subsets of size
k+1 containing the index 1
• Then Hk(Ml ; Z) is free abelian of rank
{ }1,...,J nÍ
jj J
lÎå j
j Jl
Ïå<=>
( ) 3k k k n kb M s m s - -= + +l
M. Farber and D. Schütz | Homology of Planar Polygon Spaces | Presented by Yair Carmon 9
Back to the examples
( )3,3,2=l { }0 3 0
0 0
12 , 0
n
k
s s xb b
- -
>
= = =Þ = =
{ }0 1 0
0 0 1 1
1 , , 01
k ks l s mb s s b
f>= = = =Þ = + = =
( )5,2,2,2=l
( )1,1,1,1,1=l 0 1 2
0 0 0 2
1 1 1 1
2 2 1 0
1 , 4, , 0181
k ks s s mb s m sb s m sb s m s
>= = =Þ = + + =Þ = + + =Þ = + + =
M. Farber and D. Schütz | Homology of Planar Polygon Spaces | Presented by Yair Carmon 10
The non-generic cases (n=4)( )4,3,2,1=l
@1 2b =
M. Farber and D. Schütz | Homology of Planar Polygon Spaces | Presented by Yair Carmon 11
The non-generic cases (n=4)( )2,2,1,1=l
@1 3b =
M. Farber and D. Schütz | Homology of Planar Polygon Spaces | Presented by Yair Carmon 12
The non-generic cases (n=4)( )1,1,1,1=l
@1 4b =
M. Farber and D. Schütz | Homology of Planar Polygon Spaces | Presented by Yair Carmon 13
Remark (1) – order of the edges• The order of edges doesn’t matter – not surprising!• Ml is explicitly independent on edge order:
• Also, there is a clear isomorphism between permutations of l:
( ) 1 11
1,..., | 0
n
n i ii
M u u S S l u=
ì üï ïï ï= Î ´ ´ =í ýï ïï ïî þål L ( )/ 2SO
1l
2l
5l
4l
3l
1u2u
5u4u 3u
( )1,2,3,4,5
1l
2l
5l
4l3l
1u
2u
5u
4u3u
( )1,5,4,2,3
M. Farber and D. Schütz | Homology of Planar Polygon Spaces | Presented by Yair Carmon 14
Remark (2) – b0(Ml)• Assume w.l.o • For nontrivial Ml , s0 = 1 and m0 = 0• sn–3 = 1 iff {2,3} is long, and sn–3 = 0 otherwise
• b0(Ml) = 2 iff {2,3} is long, and = 1 otherwise• Geometric meaning: existence of path between a
configuration and its reflection
1 2 nl l l³ ³ ³L
M. Farber and D. Schütz | Homology of Planar Polygon Spaces | Presented by Yair Carmon 15
Remark (2) – b0(Ml), geometric proof• w.l.o assume the longest edge is connected to both
second longest edges, and their lengths are l1,l2,l3• In a path from a config to its reflection, each angle must
pass through 0 or • If {2,3} is long the angle between l1 and l2 can’t be • Let’s see if it can be zero:
1 2l l-2l
3l4l
L nlÛ
{ }
1 2 34
2,3 is not long
n
ii
l l l l=
- + ³åc
M. Farber and D. Schütz | Homology of Planar Polygon Spaces | Presented by Yair Carmon 16
Remark (2’) – topology when b0(Ml)• If b0(Ml) = 2 , every set that contains 1 and contains
neither 2 nor 3 is short, therefore:
• The Poincaré polynomial is 2(1+t)n–3
• In this case Ml was proven to have the topology of two disjoint tori of dimension n–3:
( )3 3 3 3, 0 23k k k
n n n ns m b Mk k n k k
æ ö æ ö æ ö æ ö- - - -÷ ÷ ÷ ÷ç ç ç ç÷ ÷ ÷ ÷ç ç ç ç= = Þ = + =÷ ÷ ÷ ÷ç ç ç ç÷ ÷ ÷ ÷- -÷ ÷ ÷ ÷ç ç ç çè ø è ø è ø è øl
3 3n nM T T- -@l ò
M. Farber and D. Schütz | Homology of Planar Polygon Spaces | Presented by Yair Carmon 17
Remark (3) – Equilateral case• Nongeneric for even n
• Betti numbers can be computed from simple combinatorics (boring)
• Sum of betti numbers is maximal in the equilateral case (proven in the paper)
M. Farber and D. Schütz | Homology of Planar Polygon Spaces | Presented by Yair Carmon 18
Remark (4) – Euler characteristic• In the generic case
• A striking difference between even and odd n!
• For the nongeneric case, add
( ) ( ) ( ) ( )33
00
2 1 , odd10 , even
n kn k k
kkk
s nM b Mn
c-
-
==
ìïï -ïï@ - = íïïïïî
åål l
( )3
01
n k
kk
m-
=-å
M. Farber and D. Schütz | Homology of Planar Polygon Spaces | Presented by Yair Carmon 19
(some of) The ideas behind the proof• Warning – hand waving ahead!
M. Farber and D. Schütz | Homology of Planar Polygon Spaces | Presented by Yair Carmon 20
The short version• Define W, the configuration space of an open chain• Define Wa, by putting a ball in the end of the chain
• The homology of W is simple to calculate (it’s a torus)• The homology of Wa is derived (for small enough a)
– using Morse theory and reflection symmetry
• The homology of Ml is derived from their “difference”
1l2l 4l
3l
aW aW
M. Farber and D. Schütz | Homology of Planar Polygon Spaces | Presented by Yair Carmon 21
W – “the robot arm”
• Not dependant on l• This is an n – 1 dimensional cfg space that contains Ml • WJ W is the subspace for which
i.e. locked robot arm• WJ Ml = iff J is long
( ){ } ( )( ){ }
1 11
1 1 11 1
,..., / 2,..., | 1
nn
n
W u u S S SOu u S S u T -
= Î ´ ´= Î ´ ´ = @
LL
1l2l
4l
3l
,n mu u n m J= " Î
M. Farber and D. Schütz | Homology of Planar Polygon Spaces | Presented by Yair Carmon 22
fl – the robot arm distance metric
•
• a < 0 chosen so that WJ Wa iff J is long
• fl is Morse on W, Wa
• The critical points of fl are collinear configurations pI (I long) such that,
• pI WJ iff J I , and has index n – | I | (for long J )• pJ is the global maximum of WJ (for long J )
( )2
11
: , ,...,n
n i ii
f W f u u l u=
® = - å¡
(1 ,aW f a- ù= - ¥ úûl
1 , , 1 , i iu i I u i I¢ ¢= ± " Î = " Ïm
M. Farber and D. Schütz | Homology of Planar Polygon Spaces | Presented by Yair Carmon 23
Homology ofW• Reminder: W is diffeomorphic to Tn–1
• Hi(W ; Z) is free abelian and generated by the homology classes of the sub–tori of dimension i
• These are exactly the manifolds WJ for which 1 J and i = n – |J|
• Therefore,
( ) s.t.
1
;i JJi n J
J
H W W= -
Î
é ù= Å ê úë û¢
M. Farber and D. Schütz | Homology of Planar Polygon Spaces | Presented by Yair Carmon 24
Homology of Wa
• Everything (W , Wa ,WJ , fl ) is invariant to reflection!
• Critical points of fl are fixed points for reflection!
• Every critical point of fl in Wa is contained in a WJ in which it’s a global maximum (all the long J’s)
• [Apply some fancy Morse theory…]
( ) s.t.
is long
;ai JJ
i n JJ
H W W= -
é ù= Å ê úë û¢
M. Farber and D. Schütz | Homology of Planar Polygon Spaces | Presented by Yair Carmon 25
Hi(Wa ) and Hi(W ) revisited• Rewrite the homology calsses
• Where, Si / Mi / Li are generated by the set of classes [WJ
] for which J which is short/median/long , |J| = n – i and 1 J
• is like Li but with 1 J
( )( )
;;
ai i i
i i i i
H W L LH W L S M
= Å= Å Å
%¢¢
iL%
M. Farber and D. Schütz | Homology of Planar Polygon Spaces | Presented by Yair Carmon 26
Homomorphism• Define a natural homomorphism,
By treating every homology class inWa as a class inW
• Clearly,
• It can be shown that
• Therefore,
( ) ( ): ; ;ai i iH W H Wf ®¢ ¢
( )i i iL Lf Í%( )i i iL Lf =
( ) ( )( )
11
1 1
ker rk kercoker rk coker
i i i in n i
i i i i n i n i
L s sS M s m
ffff
-- - -
- - - -
Þ = =Å Þ = +
%;;
M. Farber and D. Schütz | Homology of Planar Polygon Spaces | Presented by Yair Carmon 27
Homology of Ml• Ml is a deformation retract of • [somehow] Obtain the short exact sequence:
( )1\ ,0aW W f a- é ù= ê úë ûl
( )1 20 coker ker 0in i n iH Mff - - - -® ® ® ®l
Im Kerg h=
- 1:1g
- untoh
Every point in Hi(Ml) can expressed with two “coordinates”from the other groups, and it’s an isomorphism since we’re free abelian
( ) 1 2coker kerin i n iH M ff - - - -Þ Ål ;
( )iH M l 2ker n if - -1coker n if - -
M. Farber and D. Schütz | Homology of Planar Polygon Spaces | Presented by Yair Carmon 28
Homology of Ml – the end• Putting it all together,
( ) ( ) ( ) ( )( ) ( )
( )1 2
11
1 1
rk rk rk coker rk kerrk ker
rk coker
ii n i n i
i in n i
i n i n i
H M H Ms ss m
ffff
- - - -
-- - -
- - - -
= = += == +ß
l l
( ) ( ) 3rki i i i n ib M H M s m s - -= = + +l l
M. Farber and D. Schütz | Homology of Planar Polygon Spaces | Presented by Yair Carmon 29
?