Equivariant geometry of Alexandrov 3-spacesPlan: 1 Alexandrov spaces. 2 Group actions on them 3...

Equivariant geometry of Alexandrov3-spaces

Jesus Nunez-Zimbron

Jesus Nunez-Zimbron (UCSB) Equivariant geometry of Alexandrov 3-spaces October 14, 2016 1 / 31

1 Alexandrov spaces.

2 Group actions on them →.3 Circle actions on Alexandrov 3-spaces.

4 Applications.

Alexandrov spaces

The model space M2k with constant

curvature k ∈ R is:

(a) (R2, g0), if k = 0.

(b) (S2(1/√k), g0), if k > 0.

(c) (H2(1/√−k), g0) if k < 0.

diam(M2k ) =

π/√k si k > 0;

∞ si k ≤ 0.

Alexandrov spaces (X , d) are lengthspaces, i.e. metric spaces such that

dist(p, q) = infγ∈Ωpq

Length(γ)

Length(γ) = supn∑

dist(γ(ti ), γ(ti+1))

They are defined by comparing geodesictriangles to those of the model space:

4pqr : geodesic triangle in X

k ∈ R4pqr : comparison triangle =triangle in M2

k whose sides havethe same lengths as thecorresponding sides of 4pqr .

4pqr exists and is unique if k ≤ 0or k > 0 and

d(p, q)+d(p, r)+d(q, r) < 2π/√k.

Alexandrov spaces

(a) (R2, g0), if k = 0.

(b) (S2(1/√k), g0), if k > 0.

(c) (H2(1/√−k), g0) if k < 0.

diam(M2k ) =

π/√k si k > 0;

∞ si k ≤ 0.

Length(γ)

d(p, q)+d(p, r)+d(q, r) < 2π/√k.

Alexandrov spaces

(a) (R2, g0), if k = 0.

(b) (S2(1/√k), g0), if k > 0.

(c) (H2(1/√−k), g0) if k < 0.

diam(M2k ) =

π/√k si k > 0;

∞ si k ≤ 0.

Length(γ)

d(p, q)+d(p, r)+d(q, r) < 2π/√k.

Alexandrov spaces

(a) (R2, g0), if k = 0.

(b) (S2(1/√k), g0), if k > 0.

(c) (H2(1/√−k), g0) if k < 0.

diam(M2k ) =

π/√k si k > 0;

∞ si k ≤ 0.

Length(γ)

d(p, q)+d(p, r)+d(q, r) < 2π/√k.

Alexandrov spaces (cont’d)

Definition (T-property) Given 4pqr inX and a comparison triangle 4pqr , forany s ∈ [qr ] and the correspondingpoint s ∈ [qr ]

dist(p, s) ≥ dist(p, s).

A length space has curvX ≥ k if eachx ∈ X has a neighborhood U in whichthe T-property holds for every4pqr ⊂ U.

Definition An Alexandrov space is acomplete, locally compact length spacewith curv ≥ k for some k ∈ R.

Examples & constructions

(i) Riemannian manifolds with sec ≥ k .

(ii) Euclidean cones

K (Y ) := Y × [0,∞)/Y × 0

d((p, t), (q, s)) :=√t2 + s2 + 2ts cos dY (p, q)

curvY ≥ 1 =⇒ curvK (Y ) ≥ 0.

(iii) Suspensions

Susp(Y ) = Y × [0, π]/Y ×0,Y ×π

cos d((p, t), (q, s)) :=

cos t cos s + sin t sin s cos dY (p, q).

curvY ≥ 1 =⇒ curv Susp(Y ) ≥ 1

(iv) Quotients of isometric actions.

G compact Lie group y by isometrieson a Riemannian manifold M withsec ≥ k . Then curvM/G ≥ k.

K (Y ) := Y × [0,∞)/Y × 0

curvY ≥ 1 =⇒ curvK (Y ) ≥ 0.

(iii) Suspensions

Susp(Y ) = Y × [0, π]/Y ×0,Y ×π

cos d((p, t), (q, s)) :=

K (Y ) := Y × [0,∞)/Y × 0

curvY ≥ 1 =⇒ curvK (Y ) ≥ 0.

(iii) Suspensions

Susp(Y ) = Y × [0, π]/Y ×0,Y ×π

cos d((p, t), (q, s)) :=

K (Y ) := Y × [0,∞)/Y × 0

curvY ≥ 1 =⇒ curvK (Y ) ≥ 0.

(iii) Suspensions

Susp(Y ) = Y × [0, π]/Y ×0,Y ×π

cos d((p, t), (q, s)) :=

Examples & constructions (cont’d)

(v) Gromov-Hausdorff limits.curvXi ≥ k and Xi →GH Y thencurvY ≥ k.

(iii) Suspensions

Susp(Y ) = Y × [0, π]/Y ×0,Y ×π

cos d((p, t), (q, s)) :=

Examples & constructions (cont’d)

(v) Gromov-Hausdorff limits.curvXi ≥ k and Xi →GH Y thencurvY ≥ k.

(vi) Gluings:If X ,Y are Alexandrov spaces withcurv ≥ k and we have an isometry∂X → ∂Y then X ∪∂ Y is anAlexandrov space of curv ≥ k.

(iii) Suspensions

Susp(Y ) = Y × [0, π]/Y ×0,Y ×π

cos d((p, t), (q, s)) :=

Properties

(i) Geodesics do not branch.

Not an Alexandrov space:

(ii) Toponogov Theorem: T -property issatisfied globally.(iii) Hausdorff dimension of X (dimX )is a non-negative integer or ∞.(iv) Angles between geodesics at thesame point are well defined:

Properties

(ii) Toponogov Theorem: T -property issatisfied globally.

(iii) Hausdorff dimension of X (dimX )is a non-negative integer or ∞.(iv) Angles between geodesics at thesame point are well defined:

Properties

(ii) Toponogov Theorem: T -property issatisfied globally.(iii) Hausdorff dimension of X (dimX )is a non-negative integer or ∞.

(iv) Angles between geodesics at thesame point are well defined:

Properties

Properties (cont’d)

β(s)β(s1)

α(t1)

β(si )

α(ti )=⇒

pβ(si ) β(s1)

α(t1)

α(ti )

∠(α, β) = limti ,si→0

∠α(ti )pβ(si ).

tangent direction at p: geodesics/anglezero(Sp,∠) possibly noncomplete metricspace.(Σp,∠) metric completion is called thespace of directions of X at p.

∠(α, β) = limti ,si→0

∠α(ti )pβ(si ).

tangent direction at p: geodesics/anglezero(Sp,∠) possibly noncomplete metricspace.(Σp,∠) metric completion is called thespace of directions of X at p.

(v) Thm. (Burago, Gromov, Perelman)1. Σp is an Alexandrov space withcurv ≥ 12. dim Σp = dimX − 13. Σp is homeomorphic to Sn−1 on adense set. (topologically regular points)

(vi) Thm. (Conical neighborhood(Perelman))Every p ∈ X has a neighborhoodpointed-homeomorphic to the cone overΣp.

(vii) Boundary of an Alexandrov space:1. If dimX = 1, ∂X is the topologicalboundary.2. For dimX = n > 1, p ∈ ∂X if andonly if ∂Σp 6= ∅.∂X is a closed subset of codim. 1.

∂X ⊂ S(X ). If ∂X = ∅, thencodimS(X ) ≥ 3. And so:· dimX = 1, 2 =⇒ X is homeomorphicto a topological manifold.

Closed Alexandrov 3-spaces

If X is closed and dimX = 3, thendim Σp = 2.

Bonnet-Myers Thm =⇒ π1Σp is finite=⇒ Σp

∼= S2 or Σp∼= RP2.

Conical Nhgb Thm & codim. of S(X )=⇒ Finite number of points withΣp∼= RP2.

=⇒ X ∼= M3 ∪2si=1 K (RP2).

Other description Grove-Wilking,Harvey-Searle:non-manifold X 3, then there exist:

- Alexandrov manifold X 3

(orientable branched double cover)

- Orientation-reversing isometricinvolution ι : X → X with onlyfixed points

such that X is isometric to X/ι.Ramification locus → topologicallysingular points

∼= S2 or Σp∼= RP2.

=⇒ X ∼= M3 ∪2si=1 K (RP2).

∼= S2 or Σp∼= RP2.

=⇒ X ∼= M3 ∪2si=1 K (RP2).

Group actions on Alexandrov spaces

Thm. (Fukaya, Yamaguchi)Isom(X ) is a Lie group.X compact =⇒ Iso(X ) is compact.

Measures of the size of Iso(X ):

- Symmetry degree: dim Iso(X )- Symmetry rank: rankIso(X )- Cohomogeneity: dimX/G whereG ≤ Iso(X ).

Cohomogeneity 0: Thm. (Berestovskii)A homogeneous Alexandrov space isisometric to a Riemannian manifold.

Cohomogeneity 1: Here, X/G must beeither a circle or an interval.

Group actions on Alexandrov spaces (cont’d)

Thm. (Galaz-Garcıa, Searle)

- If X/G is a circle then M isequivariantly homeomorphic to afiber bundle over S1 with fiberG/H and structure groupN(H)/H. In particular X is amanifold.

- If X/G ∼= [−1, 1], then there is agroup diagram

``@@@@@@@@

``AAAAAAA i+

>>~~~~~~~

Cohomogeneity 1: Here, X/G must beeither a circle or an interval.

Group actions on Alexandrov spaces (cont’d)

Thm. (Galaz-Garcıa, Searle)

- If X/G is a circle then M isequivariantly homeomorphic to afiber bundle over S1 with fiberG/H and structure groupN(H)/H. In particular X is amanifold.

- If X/G ∼= [−1, 1], then there is agroup diagram

``@@@@@@@@

``AAAAAAA i+

>>~~~~~~~

- L± istropies at ±1, H principalisotropy, L±/H are isometric to ahomogeneous space with sec > 0.

- X is the union of two fiber bundleswith base G/L± and fiberK (L±/H).

Note: This allows them to classifytopologically cohomogeneity oneAlexandrov spaces up to dimension 4.The only non-manifold one in dim. 3 isSusp(RP2).

Cohomogeneity 2 Alexandrov spaces

We focus on the case X closed,dimX = 3 and effective action, i.e.principal isotropy is e. Because of thedimensions G = S1.

(i) The action leaves the topologicallyregular and topologically singular setsinvariant.

(ii) The possible isotropies are theclosed subgroups of S1: e,Zk and S1

itself.

(iii) The manifold case was done byOrlik, Raymond: M/G is a 2-manifoldcarrying a set of topological/equivariantinvariants.

itself.

Circle actions on 3-manifolds

Analysis of nghbs of the orbits andassociate invariants:

b ∈ H2(O∗,Z)

O is the principal stratum. Openand dense by the Principal orbitThm. (Alexandrov version by:Galaz-Garcıa, Guijarro).

g is the genus of M∗

ε ∈ o, n depending oforientabilities.

f is the number of boundary cpts.of fixed points.

t is the number of boundary cpts.of isotropy Z2

(αi , βi ) are the Seifert invariants oforbits with isotropy Zk .

b ∈ H2(O∗,Z)

Thm. (Orlik, Raymond)

a circle action on a closed M3 isuniquely determined up toequivariant homeomorphism by theset of invariants

b; (ε, g , f , t); (αi , βi )ni=1

If f > 0 then M3 is equivariantlyhomeomorphic to

M(ε,g ,f ,t)#L(α1, β1)# . . .#L(αm, βm),

where M(ε,g ,f ,t) is

i=1S2 × S1)

i=1RP2 × S1)

N ∼= S3 or S2xS1.

Thm. (–,2014)

A circle action on a closedAlexandrov space X 3 is uniquelydetermined by the set of invariants

b; (ε, g , f , t); (αi , βi ); (r1, ..., rs)

ri even non-negative integers.

X is equivariantly homeomorphicto

i=1 Susp(RP2))

where M =b; (ε, g , f + s, t); (αi , βi )ni=1

Thm. (Orlik, Raymond)

a circle action on a closed M3 isuniquely determined up toequivariant homeomorphism by theset of invariants

b; (ε, g , f , t); (αi , βi )ni=1

If f > 0 then M3 is equivariantlyhomeomorphic to

M(ε,g ,f ,t)#L(α1, β1)# . . .#L(αm, βm),

where M(ε,g ,f ,t) is

i=1S2 × S1)

i=1RP2 × S1)

N ∼= S3 or S2xS1.

Thm. (–,2014)

A circle action on a closedAlexandrov space X 3 is uniquelydetermined by the set of invariants

b; (ε, g , f , t); (αi , βi ); (r1, ..., rs)

ri even non-negative integers.

i=1 Susp(RP2))

where M =b; (ε, g , f + s, t); (αi , βi )ni=1

Sketch of proof.

The analysis of the action attopologically regular points is that ofOrlik-Raymond. We need to see whathappens at topologically singular points.

(i) top. singular points are fixed points.

Thm. Slice Theorem (Harvey-Searle)Let a compact Lie group G act byisometries on an Alexandrov space X ,then a small nghbd. of the orbit G/Gp

is equivariantly homeomorphic toG ×Gp K (S⊥p ). (S⊥p is the subset of Σp

orthogonal to the orbit).

Sketch of proof.

Thm. Slice Theorem (Harvey-Searle)Let a compact Lie group G act byisometries on an Alexandrov space X ,then a small nghbd. of the orbit G/Gp

is equivariantly homeomorphic toG ×Gp K (S⊥p ). (S⊥p is the subset of Σp

orthogonal to the orbit).

Sketch of proof.

(ii) Slice Thm =⇒ action commuteswith cone construction at p and=⇒ K (RP2)/S1 ∼= K (RP2/S1).

(iii) Only one S1 action on RP2; theone induced by rotations on S2.

Sketch of proof.

Sketch of proof (cont’d)

Orbit space in the Alexandrov case:

New invariants: (r1, . . . , rs):Number of top. sing. points oneach boundary component.

(iv) For each action we can “read off”the invariants.

(v) Now, given a set of invariants wewant an Alexandrov space with anS1-action with those invariants andprove that two Alexandrov spaces withthe same invariants are equivariantlyhomeomorphic.

(vi) First we think about the case X/Gis homeomorphic to a 2-disk and noisolated Zk orbits.

S1 Z2P∗

(x+)∗

(x−)∗

U∗RF U∗SE

B∗ε (x+)

(vii) We can construct a cross-sectionX/S1 → X to π : X → X/S1:

- P → P∗ is a trivial principalS1-bundle.

- Extend the cross-section to U∗RFand U∗SE “radially”.

- Extend to the conicalneighborhoods by “copying thebase”.

S1 Z2P∗

(x+)∗

(x−)∗

U∗RF U∗SE

B∗ε (x+)

(viii) With this, if there are twoAlexandrov spaces X , Y whose orbitspace is a 2-disk and with sameinvariants we can give an equivarianthomeomorphism via the sections:

Xϕ //____________

π1 ''

π2wwX/S1 ∼= Y /S1

99rrrrrrrrrrr

eeLLLLLLLLLLL

It follows by induction that this is truefor any number of top. singular points.

S1 Z2P∗

(x+)∗

(x−)∗

U∗RF U∗SE

B∗ε (x+)

(ix) This shows thatSusp(RP2)# . . .# Susp(RP2) is theonly Alexandrov 3-space with such anorbit space.

(x) The manifold

M = b; (ε, g , f + s, t); (αi , βi )ni=1has at least s circles of fixed points sowe can take equivariant connected sumswith Susp(RP2)# . . .# Susp(RP2). .

(ix) This shows thatSusp(RP2)# . . .# Susp(RP2) is theonly Alexandrov 3-space with such anorbit space.

(x) The manifold

M = b; (ε, g , f + s, t); (αi , βi )ni=1has at least s circles of fixed points sowe can take equivariant connected sumswith Susp(RP2)# . . .# Susp(RP2). .

Application

The decomposition in equivariantconnected sums makes it plausible tocompute topological invariants andobtain the following application:

Recall that X is aspherical ifπq(X ) = 0, q > 1.

Borel conjecture: If two closedaspherical manifolds are homotopyequivalent then they arehomeomorphic. (false in thesmooth category because of exoticmanifolds, but true for n ≤ 3 atleast).

π2(#si=1Susp(RP2)) ∼= ⊕s

i=1Z2.

Mayer-Vietoris ⇒H2(M#

i=1 Susp(RP2))) ∼=

H2(M)⊕ H2(#si=1Susp(RP2))

ThenH2(M#

i=1 Susp(RP2))) ∼= 0

can only happen if there are noSusp(RP2) summands.

Therefore, the only asphericalAlexandrov spaces with anisometric S1 action are3-manifolds, and the Borelconjecture is true there.

Application

i=1Z2.

i=1 Susp(RP2))) ∼=

ThenH2(M#

i=1 Susp(RP2))) ∼= 0

Application

i=1Z2.

i=1 Susp(RP2))) ∼=

ThenH2(M#

i=1 Susp(RP2))) ∼= 0

Application

i=1Z2.

i=1 Susp(RP2))) ∼=

ThenH2(M#

i=1 Susp(RP2))) ∼= 0

Application

i=1Z2.

i=1 Susp(RP2))) ∼=

ThenH2(M#

i=1 Susp(RP2))) ∼= 0

Application

i=1Z2.

i=1 Susp(RP2))) ∼=

ThenH2(M#

i=1 Susp(RP2))) ∼= 0

Local circle actions and Thurston geometries

An isometric local circle action is adecomposition of X into disjoint curves(possibly points) such that each curvehas a nhgb. with an isometric circleaction whose fibers are the curves ofthe decomposition.

Classified in the manifold case byFintushel, Orlik-Raymond. In theAlexandrov case we obtained

Thm (Galaz-Garcıa, –, 2016)An isometric local circle action onX is determined up to equivarianthomeomorphism by

(b; ε, g , (f , k1), (t, k2); (αi , βi ); (r1, r2, . . . , rs)) ,

M# Susp(RP2)# · · ·# Susp(RP2),

where M is the closed 3-manifolddetermined by the set of invariants

(b; ε, g , (f , k1), (t, k2); (αi , βi )ni=1)

in the manifold case.

Local circle actions and Thurston geometries (cont’d)

Mitsuishi-Yamaguchi consideredcollapsing sequences of closedAlexandrov 3-spaces, i.e.

Xi →GH Y

with diam(Xi ) ≤ D, curvXi ≥ −1 anddimY < 3.

When dimY = 2 and ∂Y = ∅, theyobtained that Xi is a kind of

generalized Seifert fiber space admittingsingular interval fibers at sometopologically singular points.

Thm (Galaz-Garcıa, –, 2016) If thesequence does not have singular intervalfibers, the collapse occurs along thefibers of an isometric local circle actionon the Xi for i big enough.

An Alexandrov 3-space is geometric ifit is a quotient of the correspondingThurston geometry by some cocompactlattice.

We will say a closed Alexandrov 3-spaceis irreducible if every embedded spherebounds a 3-ball and if the space hastopologically singular points we furtherrequire that every RP2 bounds aK (RP2).

Thm. (Galaz-Garcıa, Guijarro, –, 2016)For any D > 0 there exists ε(D) > 0such that if X is a closed irreducibleAlexandrov 3-space with diamD andvolX < ε then X is geometric.

sketch of part of the proof: Assumethat this is not the case. Then for asequence εi → 0, there is a sequence Xi

coverging in GH to Y .

Case dimY = 2, ∂Y = ∅. Then byMitsuishi-Yamaguchi’s work, Xi is ageneralized Seifert fiber space. If Xi

does not have singular interval fibers,the collapse occurs along the fibers of alocal action.=⇒ decomposition into equivariantconnected sums.=⇒ condition of irreducibility rules outeverything but one connectedsummand, which then is geometric.

If Xi does have singular fibers then wetake its branched double cover. Thenuse that:

Thm.(Galaz-Garcıa, –,2016) Anisometric local circle action on anon-manifold Alexandrov 3-space canbe lifted to an isometric local circle

action on its branched double cover.

The branched double cover doesn’thave singular fibers. Although X mightnot be irreducible, it is possible to stillsee that we can rule out everything butone summand.

Thank you!

Equivariant geometry of Alexandrov 3-spacesPlan: 1 Alexandrov spaces. 2 Group actions on them 3...

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