Top Banner
Universal Magnification Invariants and Lefschetz Fixed Point Theory Marcus Werner Department of Mathematics Duke University Gravitational Lensing Workshop Department of Mathematics and Statistics University of South Florida April 2, 2010
43

Universal Magnification Invariants and Lefschetz Fixed ...

Nov 29, 2021

Download

Documents

dariahiddleston
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Page 1: Universal Magnification Invariants and Lefschetz Fixed ...

Universal Magnification Invariants and Lefschetz Fixed Point Theory

Marcus WernerDepartment of Mathematics

Duke University

Gravitational Lensing WorkshopDepartment of Mathematics and Statistics

University of South Florida

April 2, 2010

Page 2: Universal Magnification Invariants and Lefschetz Fixed ...

Other related talks today

Amir Aazami, at 12.15pm:Orbifolds, the A, D, E Classification, and Gravitational Lensing

Arlie Petters, at 3.00pm:Magnification Relations and Their Violation

Page 3: Universal Magnification Invariants and Lefschetz Fixed ...

● Astronomical motivation● Gravitational lensing theory● Topology in gravitational lensing● Properties of image magnification● Caustic singularities● Lefschetz fixed point theory● Application to universal magnification invariants

Outline

Page 4: Universal Magnification Invariants and Lefschetz Fixed ...

Astronomical motivation:the flux ratio anomaly

Gravitational lens system CLASS B2045+265:

Quasar at z=1.28 lensed by galaxy at z=0.87, in H line.

Indicative of substructure?

Cf. Fassnacht et al. (1999), Koopmans et al. (2003), Keeton et al. (2003), McKean et al. (2007). Image: CASTLES lensing database.

ABC

∣A∣∣B∣∣C∣=0.51

Page 5: Universal Magnification Invariants and Lefschetz Fixed ...

Gravitational lensing theory

Quasi-Newtonian impulse approximation: a useful framework for lensing problems in astrophysics, in the limit of

● Geometrical optics● Small deflection angles● Asymptotically Euclidean space● Thin lenses, compared to the length of the light ray

Consider the parallel lens plane and source plane containing deflecting masses and light sources,

respectively, at .

L=ℝ2

S=ℝ2

x∈L ,y∈S

Page 6: Universal Magnification Invariants and Lefschetz Fixed ...

Gravitational lensing theory

Page 7: Universal Magnification Invariants and Lefschetz Fixed ...

Gravitational lensing theory:the lensing map

Geometrical and gravitational time delay combined yields the Fermat potential given by

Then the Fermat's principle implies the lens equation

mapping images at surjectively to the source.

The lensing map is .

y x : L×S ℝ

y x =12

∣x−y∣2−x

∇ y x =0

y=x−∇ x

x

: L S ,x y

Page 8: Universal Magnification Invariants and Lefschetz Fixed ...

Gravitational lensing theory:the time delay surface

Page 9: Universal Magnification Invariants and Lefschetz Fixed ...

Gravitational lensing theory:image properties

Page 10: Universal Magnification Invariants and Lefschetz Fixed ...

Topology in gravitational lensing

Images are non-degenerate critical points of the Morse function , with Morse index .

Theorem [Morse, 1925]. Let be a smooth closed real manifold with dimension and Euler characteristic , and non-degenerate critical points with Morse index . Then

y x

MM d

n

∑=0

d

−1 n=M

Page 11: Universal Magnification Invariants and Lefschetz Fixed ...

Topology in gravitational lensing:closing the time delay surface

Page 12: Universal Magnification Invariants and Lefschetz Fixed ...

Topology in gravitational lensing: odd number theorem

Hence from Morse theory:

Total number of images:

Therefore, the odd number theorem follows:

Cf. Burke (1981), Petters (1995). Spacetime version: McKenzie (1985)

nmin−nsadnmax1==2

nminnsadnmax=ntot

ntot=2n sad1

Page 13: Universal Magnification Invariants and Lefschetz Fixed ...

Topology in gravitational lensing

SDSS J1004+4112. Cluster: z=0.68; quasar: z=1.73; galaxy: z=3.33.

Image: ESA, NASA, K. Sharon, E. Ofek.

Page 14: Universal Magnification Invariants and Lefschetz Fixed ...

Properties of image magnification

Due to Liouville's Theorem in curved spacetime, the intensity obeys

Achromaticity of gravity:

Hence flux is proportional to solid angle, and the signed image magnification is

where

The sign defines image parity.

I /3=const.

=const.

=1

det Jac Jac =

∂ y1

∂ x1

∂ y1

∂ x2

∂ y2

∂ x1

∂ y2

∂ x2

Page 15: Universal Magnification Invariants and Lefschetz Fixed ...

Properties of image magnification

The critical set of the lensing map is defined by in , corresponding to infinite magnification .

The critical set is mapped to caustics in by the lensing map, .

Caustics delimit domains of constant image number in .

According to singularity theory, only certain types of caustics occur generically.

L

S

Crit

Caustic =Crit

det Jac =0

S

Page 16: Universal Magnification Invariants and Lefschetz Fixed ...

Magnification invariants

Constant sums of signed image magnifications, independent of lens parameters and image positions.

Global magnification invariants:

● Apply to all images

● Dependent on the lens model

● Source in maximal domain

Example: for Schwarzschild and Plummer model.

Cf. Witt & Mao (1995), Dalal & Rabin (2001), Hunter & Evans (2001), Evans & Hunter (2002), Werner & Evans (2006), Werner (2007).

∑i=1

n

i=1

n

Page 17: Universal Magnification Invariants and Lefschetz Fixed ...

Universal magnification invariants

● Apply to a subset of images: image multiplets

● Source near a caustic singularity

● Independent of lens model: genericity of caustics

Well-known for folds and cusps, recently extended to higher singularities by Aazami and Petters:

Blandford & Narayan (1986), Schneider & Weiss (1992), Zakharov (1999), Aazami & Petters (2009, 2010). Application of Lefschetz: Werner (2009).

∑i=1

n

i=0

n

Page 18: Universal Magnification Invariants and Lefschetz Fixed ...

Caustic singularities: generic case

Polynomial generating potential: , with

state variables and control parameters ,

inducing a generic map and caustics as discussed above.

Stable caustics in a space of control parameters :

● Fold:

● Cusp:

where is the number of images in maximal caustic domain.

u x :ℝ2×ℝ pℝ

x= x1 , x2 u∈ℝ p , p≥2

u=y , p=2

n

n=2

n=3

x =x1 , x22

x =x1 , x1 x2x23

Page 19: Universal Magnification Invariants and Lefschetz Fixed ...

Caustic singularities: the cusp

Page 20: Universal Magnification Invariants and Lefschetz Fixed ...

Higher caustic singularities

Stable caustics in a space of control parameters

, e. g. a 1-parameter family of lens models:

● Swallowtail:

and umbilics for which ,

● Elliptic umbilic:

● Hyperbolic umbilic:

all of which have . Caustics trace out big caustics in parameter space .

u= y1 , y2 , c , p=3

x =3 x22−3 x1

2−2 c x1 ,6 x1 x2−2 c x2

x =−3 x12−c x2 ,−3 x2

2−c x1

x =x1 x2c x12x1

4 , x2

rank Jac =0

n=4ℝ3

Page 21: Universal Magnification Invariants and Lefschetz Fixed ...

Singularities: big caustic of the swallowtail

Page 22: Universal Magnification Invariants and Lefschetz Fixed ...

Singularities: big caustic of the elliptic umbilic

Page 23: Universal Magnification Invariants and Lefschetz Fixed ...

Singularities: big caustic of the hyperbolic umbilic

Page 24: Universal Magnification Invariants and Lefschetz Fixed ...

Caustic singularities: image multiplets

For fixed parameters , images are real solutions of the equation

Calculate the intersection numbers for the maximal caustic domain to find the number of images there.

Example: elliptic umbilic, e. g. for

Hence, there are simple real solutions at

y1 , y2 , c

x =1x1 , x2 ,2 x1 , x2= y1 , y2

nI 1x− y1 ,2x − y2

y=0 ,

I x1 ,2 x2 I −3 x1−2c ,2 x2 I 3 x2−c 3 x2c ,3 x1−c

n=4

0 ,0 , −2c /3 ,0 , c /3 , c /3 , c /3 ,−c /3

Page 25: Universal Magnification Invariants and Lefschetz Fixed ...

Solomon Lefschetz

Born 1884 in Moscow, died 1972 in Princeton, NJ.

Engineering in Paris, then emigration to USA at 21.

Turned to mathematics after accident. PhD at Clark, MA, in 1911.

A founding father of algebraic topology, in Topology (AMS, 1930).

Image: from the St. Andrews, UK, History of Mathematics Site

Page 26: Universal Magnification Invariants and Lefschetz Fixed ...

Lefschetz fixed point theory

Let be a closed manifold with dimension , and be differentiable. Then the fixed points of are

Intersection of graph of , , with diagonal in , :

Connection of local, geometrical properties at fixed points, the fixed point indices , with global, topological properties, the Lefschetz number , gives a Lefschetz fixed point formula:

M dim M =dff :M M

fM×M

Graph f ={x , f x ∈M×M }

Fix f ={x∈M : f x =x }

Diag M ={x , x ∈M×M }

Fix f =Graph f ∩Diag M

L

L= ∑Fix f

Page 27: Universal Magnification Invariants and Lefschetz Fixed ...

Lefschetz fixed point theory: real manifolds

The pull-back of obeys , and therefore induces a map on the de Rham cohomology classes of , , where

Then the global form of the Lefschetz number is defined by

and is a homotopy invariant.

Example: if is homotopic to the identity , then

f * f d ° f *= f *°dM

H k M ={[]} []={ '∈k M :d '=0,− '=d ' ' }

L f =∑k=0

d

−1k trace f *H k M

f I

L f =∑k=0

d

−1k bk=M

Page 28: Universal Magnification Invariants and Lefschetz Fixed ...

Lefschetz fixed point theory: real manifolds

The fixed point indices are well-defined if the intersection of with is transversal, that is, stable:

Positively oriented tangent space bases:

Left-hand side:

Rearranging,

{ e1

0 , , ed

0 , 0e1

, , 0ed }

{ e1

D f e1 , , ed

D f e1 , e1

e1 , , ed

ed}

{ e1

0 , , ed

0 , 0 I−D f e1

, , 0 I−D f ed }

T x , x M×M :

T x , x Graph f T x , x Diag M =T x , x M×M

Graph f Diag M

Page 29: Universal Magnification Invariants and Lefschetz Fixed ...

Lefschetz fixed point theory: real manifolds

Hence, the local form of the Lefschetz fixed point formula:

so the fixed point indices are .

Example: map on the closed unit ball homotopic to

identity, so .

L f = ∑Fix f

sgndet I−D f

∈{−1,1}

f : B1 B1

L f =B1=1

Page 30: Universal Magnification Invariants and Lefschetz Fixed ...

Lefschetz fixed point theory: a simple example

Page 31: Universal Magnification Invariants and Lefschetz Fixed ...

Lefschetz fixed point theory: complex manifolds

Let be complex and holomorphic, that is .

Then the pull-back of obeys , and therefore induces a map on the Dolbeault cohomology classes of , , where

The holomorphic Lefschetz number is defined by

and the fixed point formula becomes

f *

f

∂° f *= f *°∂

M

H r , s M ={[]}

[]={ '∈r , sM :∂ '=0,− '=∂ ' ' , ' '∈r , s−1M }

Lhol f =∑s=0

d

−1s trace f * H 0, sM

f

∂ f =0

M

Lhol f = ∑Fix f

1det I−D f

Page 32: Universal Magnification Invariants and Lefschetz Fixed ...

Lefschetz fixed point theory: complex manifolds

Example: the rational fixed point theorem in complex

dynamics. For and ,

Proof: Choose local holomorphic coordinates such that

and . Then

and the result follows

using .

f ≠ IM=ℂ=ℂ∪{∞}=ℂ P1

∑Fix f

1

1−dfdz

=1=Lhol f

f 0=0 f z =dfdz

0 zO z2

12 i∮C 0

dzz− f z

=1−dfdz

0−1

12 i∮C∞ 1

z− f z −

1z dz=0

Page 33: Universal Magnification Invariants and Lefschetz Fixed ...

Application to gravitational lensing

Lefschetz fixed point theory does not directly apply to the gravitational lensing formalism since

● for real manifolds, the fixed point indices are integers , which cannot account for magnifications;

● the standard complexification of the lens plane does not, in general, yield a holomorphic lensing map;

● the lens plane is not a closed manifold.

∈{−1,1}

L=ℝ2

z=x1i x2 ⇒= z ,z

Page 34: Universal Magnification Invariants and Lefschetz Fixed ...

Application to gravitational lensing

Allow the lens plane coordinates to take complex values so that and

is holomorphic. Then the lens equation becomes

Real solutions for fixed source position correspond to real images as before.

x1 , x2 x1 , x2≡ z1 , z2

ℂ z1 , z2=1

ℂ z1 , z2 ,2

ℂ z1 , z2 :ℂ2

ℂ2

1ℂ z1 , z2 ,2

ℂ z1 , z2= y1 , y2

y1 , y2

Page 35: Universal Magnification Invariants and Lefschetz Fixed ...

Application to generic maps of singularities

Bézout's Theorem implies

Fold Cusp Swallowtail Elliptic umbilic Hyperbolic umbilic

where are the degrees of the polynomials , respectively; the maximum number of solutions of , possibly complex; the number of real solutions in the maximal caustic domain.

deg 1ℂ deg 2

ℂ nmax n1 2 2 21 3 3 34 1 4 42 2 4 42 2 4 4

deg 1ℂ , deg 2

ℂ1

ℂ ,2ℂ nmax

1ℂ ,2

ℂ= y1, y2 n

Page 36: Universal Magnification Invariants and Lefschetz Fixed ...

Fixed point formalism

Construct a map such that images are fixed points,

The matrix of first partial derivatives with respect to the holomorphic coordinates is

f :ℂ2 ℂ2

f z1 , z2= z1−1ℂ z1 , z2 y1 , z1−2

ℂ z1 , z 2 y2

D f =1−∂1

∂ z1

−∂1

∂ z 2

−∂2

∂ z1

1−∂2

∂ z2

Page 37: Universal Magnification Invariants and Lefschetz Fixed ...

Fixed point formalism

Hence we obtain

so that the fixed point indices are

if evaluated in the maximal caustic domain, where all fixed points are real and correspond to images.

det I−D f =det ∂1

∂ z1

∂1ℂ

∂ z2

∂2ℂ

∂ z1

∂2ℂ

∂ z2

=det Jac =1

=1

det I−D f =

Page 38: Universal Magnification Invariants and Lefschetz Fixed ...

Fixed point formalism

Rewriting the lens equation in homogeneous coordinates

so that for ,

Fixed points of at infinity correspond to solutions as . However, in the maximal caustic domain, all fixed points of are at finite positions (i. e., image multiplets) by construction. So has no fixed points at infinity there.

z1=Z 1

Z 0

, z 2=Z 2

Z 0

Z 0 , Z 1 , Z 2 Z 0≠0

0=−1ℂZ 0 , Z 1 , Z 2 y1

0=−2ℂZ 0 , Z 1 , Z 2 y2

f

ff

Z 1 , Z 2≠0Z 0 0

Page 39: Universal Magnification Invariants and Lefschetz Fixed ...

Fixed point formalism

Consider the map

such that

where . Hence is holomorphic and well defined, that is, there is no such that , since has no fixed points at infinity. So the holomorphic Lefschetz formula applies.

Cf. Atiyah & Bott (1968)

F :ℂ P2 ℂ P2 ,Z 0 :Z 1:Z 2F 0 :F 1: F 2

F 0=Z 0m

F 1=Z 1Z 0m−1

Z 0m −1

ℂZ 0 , Z 1 , Z 2 y1

F 2=Z 2Z 0m−1

Z 0m −2

ℂZ 0 , Z 1 , Z 2 y2

m=max deg 1ℂ , deg 2

ℂ≥2 F

Z 0 :Z 1 :Z 2F Z 0:Z 1 :Z 2=0 : 0: 0∉ℂ P 2 f

Page 40: Universal Magnification Invariants and Lefschetz Fixed ...

Fixed point formalism

Consider , where and .

Then on , we recover , whence

Now the non-vanishing Dolbeault cohomology classes are

if , so

since only contributes.

ℂ P2=ℂ2∪ℂP1 ℂ2 :Z 0=1 ℂ P1:Z 0=0

ℂ2 F 1= f 1 , F 2= f 2

Fix Fℂ

2=Fix f

H r , sℂP n=ℂ r=s

Lhol F =∑s=0

2

−1s trace F *H 0, s ℂ P 2=1

H 0,0 ℂ P2=ℂ

Page 41: Universal Magnification Invariants and Lefschetz Fixed ...

Application to universal magnification invariants

Hence,

and the result follows (Werner 2009).

1=Lhol F = ∑Fix F

1det I−D F

= ∑Fix F

ℂ2

1det I−DF

∑Fix F

ℂP1

1det I−D F

= ∑Fix f

1det I−D f

1

=∑i=1

n

i1

Page 42: Universal Magnification Invariants and Lefschetz Fixed ...

Conclusions

● A fixed point approach to lensing theory seems natural because of the split of terms corresponding to the geometrical and gravitational time delay:

● The transversality condition for fixed points, that is, well-defined fixed point indices, becomes the usual regularity condition for images:

x=y∇ x = f x

det I−D f ≠0⇒∣∣∞

Page 43: Universal Magnification Invariants and Lefschetz Fixed ...

Conclusions

● Lefschetz fixed point formulae are special cases of the Atiyah-Bott Theorem, which ties universal magnification invariants in with a deep result in fixed point theory and topology.

● The same formalism can be applied not only to the generic cases discussed here, but also to elliptic and hyperbolic umbilics in lensing (where there is a preferred coordinate system). Open problem: is there an extension to other magnification invariants?

● Does a spacetime version of global or universal magnification invariants exist?