Characterization of Branched Covers with Simplicial Branch...

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Characterization of Branched Covers with Simplicial Branch Sets Eden Prywes Princeton University Differential Geometry & Geometric Analysis Seminar May 27th, 2020 Joint work with Rami Luisto Eden Prywes Branched Covers

Transcript of Characterization of Branched Covers with Simplicial Branch...

Page 1: Characterization of Branched Covers with Simplicial Branch ...eprywes/Branched_Cover_Slides.pdfcover. Theorem (Sto low) Let f : S2!bCbe a branched cover. Then there exists a homeomorphism

Characterization of Branched Covers withSimplicial Branch Sets

Eden Prywes

Princeton University

Differential Geometry & Geometric Analysis Seminar

May 27th, 2020

Joint work with Rami Luisto

Eden Prywes Branched Covers

Page 2: Characterization of Branched Covers with Simplicial Branch ...eprywes/Branched_Cover_Slides.pdfcover. Theorem (Sto low) Let f : S2!bCbe a branched cover. Then there exists a homeomorphism

Branched Covers

Definition

A branched cover is a continuous map f : Ω→ Rn, where Ω is adomain in Rn, that is discrete and open.

At most points f is a local homeomorphism. The branch setof f , denoted Bf , is the set of points where f fails to be alocal homeomorphism.

Branched covers are topological generalization of quasiregularmaps.

Eden Prywes Branched Covers

Page 3: Characterization of Branched Covers with Simplicial Branch ...eprywes/Branched_Cover_Slides.pdfcover. Theorem (Sto low) Let f : S2!bCbe a branched cover. Then there exists a homeomorphism

Branched Covers

Definition

A branched cover is a continuous map f : Ω→ Rn, where Ω is adomain in Rn, that is discrete and open.

At most points f is a local homeomorphism. The branch setof f , denoted Bf , is the set of points where f fails to be alocal homeomorphism.

Branched covers are topological generalization of quasiregularmaps.

Eden Prywes Branched Covers

Page 4: Characterization of Branched Covers with Simplicial Branch ...eprywes/Branched_Cover_Slides.pdfcover. Theorem (Sto low) Let f : S2!bCbe a branched cover. Then there exists a homeomorphism

Branched Covers

Definition

A branched cover is a continuous map f : Ω→ Rn, where Ω is adomain in Rn, that is discrete and open.

At most points f is a local homeomorphism. The branch setof f , denoted Bf , is the set of points where f fails to be alocal homeomorphism.

Branched covers are topological generalization of quasiregularmaps.

Eden Prywes Branched Covers

Page 5: Characterization of Branched Covers with Simplicial Branch ...eprywes/Branched_Cover_Slides.pdfcover. Theorem (Sto low) Let f : S2!bCbe a branched cover. Then there exists a homeomorphism

Branched Covers in Dimension Two

In two dimensions the typical example of a branched cover is arational map f : C→ C.

The branch set is the finite set of critical points of f .Near the branch points, f behaves like the map zd , where d isthe degree of the map.

Topologically, this map is equivalent to a winding map:(r , θ) 7→ (r , dθ).

Eden Prywes Branched Covers

Page 6: Characterization of Branched Covers with Simplicial Branch ...eprywes/Branched_Cover_Slides.pdfcover. Theorem (Sto low) Let f : S2!bCbe a branched cover. Then there exists a homeomorphism

Branched Covers in Dimension Two

In two dimensions the typical example of a branched cover is arational map f : C→ C.

The branch set is the finite set of critical points of f .Near the branch points, f behaves like the map zd , where d isthe degree of the map.Topologically, this map is equivalent to a winding map:(r , θ) 7→ (r , dθ).

Eden Prywes Branched Covers

Page 7: Characterization of Branched Covers with Simplicial Branch ...eprywes/Branched_Cover_Slides.pdfcover. Theorem (Sto low) Let f : S2!bCbe a branched cover. Then there exists a homeomorphism

Example

Define f : C/Z2 → C as:

This is topologically the same as the Weierstrass p-function.

Locally near the branch point the maps behaves like a windingmap.

Can extend this to a PL-map F : R2n → S2 × · · · × S2.

Eden Prywes Branched Covers

Page 8: Characterization of Branched Covers with Simplicial Branch ...eprywes/Branched_Cover_Slides.pdfcover. Theorem (Sto low) Let f : S2!bCbe a branched cover. Then there exists a homeomorphism

Example

Define f : C/Z2 → C as:

This is topologically the same as the Weierstrass p-function.

Locally near the branch point the maps behaves like a windingmap.

Can extend this to a PL-map F : R2n → S2 × · · · × S2.

Eden Prywes Branched Covers

Page 9: Characterization of Branched Covers with Simplicial Branch ...eprywes/Branched_Cover_Slides.pdfcover. Theorem (Sto low) Let f : S2!bCbe a branched cover. Then there exists a homeomorphism

Branched Covers in Dimension Two

Up to homeomorphism, rational maps characterize every branchedcover.

Theorem (Stoılow)

Let f : S2 → C be a branched cover. Then there exists ahomeomorphism h : C→ S2 so that f h is a rational map.

Corollary

Every branched cover from S2 → S2 is equivalent up to ahomeomorphism to a piecewise linear (PL) map.

Eden Prywes Branched Covers

Page 10: Characterization of Branched Covers with Simplicial Branch ...eprywes/Branched_Cover_Slides.pdfcover. Theorem (Sto low) Let f : S2!bCbe a branched cover. Then there exists a homeomorphism

Branched Covers in Dimension Two

Up to homeomorphism, rational maps characterize every branchedcover.

Theorem (Stoılow)

Let f : S2 → C be a branched cover. Then there exists ahomeomorphism h : C→ S2 so that f h is a rational map.

Corollary

Every branched cover from S2 → S2 is equivalent up to ahomeomorphism to a piecewise linear (PL) map.

Eden Prywes Branched Covers

Page 11: Characterization of Branched Covers with Simplicial Branch ...eprywes/Branched_Cover_Slides.pdfcover. Theorem (Sto low) Let f : S2!bCbe a branched cover. Then there exists a homeomorphism

Motivation

Definition

A map f : Ω→ Rn is K -quasiregular if f ∈W 1,nloc (Ω) and for

almost every x ∈ Ω,

‖Df ‖n ≤ KJf ,

where Df is the derivative of f and Jf = det(Df ).

By a theorem due to Reshetnyak, quasiregular maps arebranched covers.

The converse is false in general and it is difficult to constructquasiregular maps.

PL maps are typically quasiregular.

Eden Prywes Branched Covers

Page 12: Characterization of Branched Covers with Simplicial Branch ...eprywes/Branched_Cover_Slides.pdfcover. Theorem (Sto low) Let f : S2!bCbe a branched cover. Then there exists a homeomorphism

Motivation

Definition

A map f : Ω→ Rn is K -quasiregular if f ∈W 1,nloc (Ω) and for

almost every x ∈ Ω,

‖Df ‖n ≤ KJf ,

where Df is the derivative of f and Jf = det(Df ).

By a theorem due to Reshetnyak, quasiregular maps arebranched covers.

The converse is false in general and it is difficult to constructquasiregular maps.

PL maps are typically quasiregular.

Eden Prywes Branched Covers

Page 13: Characterization of Branched Covers with Simplicial Branch ...eprywes/Branched_Cover_Slides.pdfcover. Theorem (Sto low) Let f : S2!bCbe a branched cover. Then there exists a homeomorphism

Motivation

Definition

A map f : Ω→ Rn is K -quasiregular if f ∈W 1,nloc (Ω) and for

almost every x ∈ Ω,

‖Df ‖n ≤ KJf ,

where Df is the derivative of f and Jf = det(Df ).

By a theorem due to Reshetnyak, quasiregular maps arebranched covers.

The converse is false in general and it is difficult to constructquasiregular maps.

PL maps are typically quasiregular.

Eden Prywes Branched Covers

Page 14: Characterization of Branched Covers with Simplicial Branch ...eprywes/Branched_Cover_Slides.pdfcover. Theorem (Sto low) Let f : S2!bCbe a branched cover. Then there exists a homeomorphism

Quasiregular Ellipticity

An n-dimensional manifold M is Quasiregularly Elliptic if thereexists a quasiregular map f : Rn → M.In dimension 2, M is homeomorphic to C, C,S1 × R or S1 × S1.In dimension 3, closed quasiregularly elliptic manifolds arequotients of either S3, S1 × S1 × S1 or S2 × S1.

Theorem (P., ’19)

If M is a closed, orientable Riemannian manifold of dimension dthat admits a quasiregular map from Rd , then dimH`(M) ≤

(d`

).

If ` = d/2, then b+d/2(M), b−d/2(M) ≤ 1

2

( dd/2

).

Eden Prywes Branched Covers

Page 15: Characterization of Branched Covers with Simplicial Branch ...eprywes/Branched_Cover_Slides.pdfcover. Theorem (Sto low) Let f : S2!bCbe a branched cover. Then there exists a homeomorphism

Quasiregular Ellipticity

An n-dimensional manifold M is Quasiregularly Elliptic if thereexists a quasiregular map f : Rn → M.In dimension 2, M is homeomorphic to C, C,S1 × R or S1 × S1.In dimension 3, closed quasiregularly elliptic manifolds arequotients of either S3, S1 × S1 × S1 or S2 × S1.

Theorem (P., ’19)

If M is a closed, orientable Riemannian manifold of dimension dthat admits a quasiregular map from Rd , then dimH`(M) ≤

(d`

).

If ` = d/2, then b+d/2(M), b−d/2(M) ≤ 1

2

( dd/2

).

Eden Prywes Branched Covers

Page 16: Characterization of Branched Covers with Simplicial Branch ...eprywes/Branched_Cover_Slides.pdfcover. Theorem (Sto low) Let f : S2!bCbe a branched cover. Then there exists a homeomorphism

Flat Branch Sets

The geometry of the branch set can give information on thebehavior of the map.

Theorem (Church and Hemmingsen, ’60)

Let f : Ω→ Rn be a branched cover, where Ω is a domain in Rn. Iff (Bf ) can be embedded into a codimension 2 subspace, then f istopologically equivalent to a winding map.

The k-winding map for k ∈ N iswk(r , θ, x2, . . . , xn) := (r , kθ, x2, . . . , xn).

By a theorem due to Cernavskii and Vaisala, Bf and f (Bf )have topological dimension less than or equal to n − 2.

In dimension 2 this hypothesis is always satisfied, but it is notalways satisfied in higher dimensions.

Eden Prywes Branched Covers

Page 17: Characterization of Branched Covers with Simplicial Branch ...eprywes/Branched_Cover_Slides.pdfcover. Theorem (Sto low) Let f : S2!bCbe a branched cover. Then there exists a homeomorphism

Flat Branch Sets

The geometry of the branch set can give information on thebehavior of the map.

Theorem (Church and Hemmingsen, ’60)

Let f : Ω→ Rn be a branched cover, where Ω is a domain in Rn. Iff (Bf ) can be embedded into a codimension 2 subspace, then f istopologically equivalent to a winding map.

The k-winding map for k ∈ N iswk(r , θ, x2, . . . , xn) := (r , kθ, x2, . . . , xn).

By a theorem due to Cernavskii and Vaisala, Bf and f (Bf )have topological dimension less than or equal to n − 2.

In dimension 2 this hypothesis is always satisfied, but it is notalways satisfied in higher dimensions.

Eden Prywes Branched Covers

Page 18: Characterization of Branched Covers with Simplicial Branch ...eprywes/Branched_Cover_Slides.pdfcover. Theorem (Sto low) Let f : S2!bCbe a branched cover. Then there exists a homeomorphism

Counterexample to Church and Hemmingsen

Let P be the Poincare homology sphere.

S3 is the universal covering space of P.

If π : S3 → P is the covering map, then we can take thesuspension of both sides to get a map

Σπ : S4 → ΣP.

ΣP is not a topological manifold, but ΣΣP ' S5. So

ΣΣπ : S5 → S5

is a branched cover with branch set BΣΣπ,ΣΣπ(BΣΣπ) ' S1.

Note that π1(S5 \ ΣΣπ(BΣΣπ)) has order 120.

The natural choice for an open neighborhood of a point inΣΣπ(BΣΣπ) has boundary that is homeomorphic to ΣP not S4.

Eden Prywes Branched Covers

Page 19: Characterization of Branched Covers with Simplicial Branch ...eprywes/Branched_Cover_Slides.pdfcover. Theorem (Sto low) Let f : S2!bCbe a branched cover. Then there exists a homeomorphism

Counterexample to Church and Hemmingsen

Let P be the Poincare homology sphere.

S3 is the universal covering space of P.

If π : S3 → P is the covering map, then we can take thesuspension of both sides to get a map

Σπ : S4 → ΣP.

ΣP is not a topological manifold, but ΣΣP ' S5. So

ΣΣπ : S5 → S5

is a branched cover with branch set BΣΣπ,ΣΣπ(BΣΣπ) ' S1.

Note that π1(S5 \ ΣΣπ(BΣΣπ)) has order 120.

The natural choice for an open neighborhood of a point inΣΣπ(BΣΣπ) has boundary that is homeomorphic to ΣP not S4.

Eden Prywes Branched Covers

Page 20: Characterization of Branched Covers with Simplicial Branch ...eprywes/Branched_Cover_Slides.pdfcover. Theorem (Sto low) Let f : S2!bCbe a branched cover. Then there exists a homeomorphism

Counterexample to Church and Hemmingsen

Let P be the Poincare homology sphere.

S3 is the universal covering space of P.

If π : S3 → P is the covering map, then we can take thesuspension of both sides to get a map

Σπ : S4 → ΣP.

ΣP is not a topological manifold, but ΣΣP ' S5. So

ΣΣπ : S5 → S5

is a branched cover with branch set BΣΣπ,ΣΣπ(BΣΣπ) ' S1.

Note that π1(S5 \ ΣΣπ(BΣΣπ)) has order 120.

The natural choice for an open neighborhood of a point inΣΣπ(BΣΣπ) has boundary that is homeomorphic to ΣP not S4.

Eden Prywes Branched Covers

Page 21: Characterization of Branched Covers with Simplicial Branch ...eprywes/Branched_Cover_Slides.pdfcover. Theorem (Sto low) Let f : S2!bCbe a branched cover. Then there exists a homeomorphism

Counterexample to Church and Hemmingsen

Let P be the Poincare homology sphere.

S3 is the universal covering space of P.

If π : S3 → P is the covering map, then we can take thesuspension of both sides to get a map

Σπ : S4 → ΣP.

ΣP is not a topological manifold, but ΣΣP ' S5. So

ΣΣπ : S5 → S5

is a branched cover with branch set BΣΣπ,ΣΣπ(BΣΣπ) ' S1.

Note that π1(S5 \ ΣΣπ(BΣΣπ)) has order 120.

The natural choice for an open neighborhood of a point inΣΣπ(BΣΣπ) has boundary that is homeomorphic to ΣP not S4.

Eden Prywes Branched Covers

Page 22: Characterization of Branched Covers with Simplicial Branch ...eprywes/Branched_Cover_Slides.pdfcover. Theorem (Sto low) Let f : S2!bCbe a branched cover. Then there exists a homeomorphism

Counterexample to Church and Hemmingsen

Let P be the Poincare homology sphere.

S3 is the universal covering space of P.

If π : S3 → P is the covering map, then we can take thesuspension of both sides to get a map

Σπ : S4 → ΣP.

ΣP is not a topological manifold, but ΣΣP ' S5. So

ΣΣπ : S5 → S5

is a branched cover with branch set BΣΣπ,ΣΣπ(BΣΣπ) ' S1.

Note that π1(S5 \ ΣΣπ(BΣΣπ)) has order 120.

The natural choice for an open neighborhood of a point inΣΣπ(BΣΣπ) has boundary that is homeomorphic to ΣP not S4.

Eden Prywes Branched Covers

Page 23: Characterization of Branched Covers with Simplicial Branch ...eprywes/Branched_Cover_Slides.pdfcover. Theorem (Sto low) Let f : S2!bCbe a branched cover. Then there exists a homeomorphism

Generalizing Church and Hemmingsen

Theorem (Martio and Srebro, ’79)

Let f : Ω→ R3 be a branched cover and x0 ∈ Bf . If there existsand open neighborhood V of x0 so that the image of the branchset f (Bf ∩ V ) can be embedded into a union of finitely many linesegments originating from f (x0), then f is topologically equivalenton V to a cone of a rational map g : C→ C.

A cone of a map g is the map

g × id : cone(C)→ cone(C),

cone(C) =C× [0, 1]

(z , 0) ∼ (w , 0)

(C× [0, 1] with this identification is homeomorphic to B3).

This implies that f is topologically equivalent to a PL map.

Eden Prywes Branched Covers

Page 24: Characterization of Branched Covers with Simplicial Branch ...eprywes/Branched_Cover_Slides.pdfcover. Theorem (Sto low) Let f : S2!bCbe a branched cover. Then there exists a homeomorphism

Generalizing Church and Hemmingsen

Theorem (Martio and Srebro, ’79)

Let f : Ω→ R3 be a branched cover and x0 ∈ Bf . If there existsand open neighborhood V of x0 so that the image of the branchset f (Bf ∩ V ) can be embedded into a union of finitely many linesegments originating from f (x0), then f is topologically equivalenton V to a cone of a rational map g : C→ C.

A cone of a map g is the map

g × id : cone(C)→ cone(C),

cone(C) =C× [0, 1]

(z , 0) ∼ (w , 0)

(C× [0, 1] with this identification is homeomorphic to B3).

This implies that f is topologically equivalent to a PL map.

Eden Prywes Branched Covers

Page 25: Characterization of Branched Covers with Simplicial Branch ...eprywes/Branched_Cover_Slides.pdfcover. Theorem (Sto low) Let f : S2!bCbe a branched cover. Then there exists a homeomorphism

Main Result

Theorem (Luisto and P., ’19)

Let f : Ω→ Rn be a branched cover and x0 ∈ Bf . If there exists anopen neighborhood V of x0 so that the image of the branch setf (Bf ∩ V ) can be embedded into an (n − 2)-simplicial complex,then f is topologically equivalent on V to a cone of a PL mapg : Sn−1 → Sn−1.

This implies that f is topologically equivalent to a PL map.

This theorem also extends to a global result for f : Sn → Sn.

Eden Prywes Branched Covers

Page 26: Characterization of Branched Covers with Simplicial Branch ...eprywes/Branched_Cover_Slides.pdfcover. Theorem (Sto low) Let f : S2!bCbe a branched cover. Then there exists a homeomorphism

Main Result

Theorem (Luisto and P., ’19)

Let f : Ω→ Rn be a branched cover and x0 ∈ Bf . If there exists anopen neighborhood V of x0 so that the image of the branch setf (Bf ∩ V ) can be embedded into an (n − 2)-simplicial complex,then f is topologically equivalent on V to a cone of a PL mapg : Sn−1 → Sn−1.

This implies that f is topologically equivalent to a PL map.

This theorem also extends to a global result for f : Sn → Sn.

Eden Prywes Branched Covers

Page 27: Characterization of Branched Covers with Simplicial Branch ...eprywes/Branched_Cover_Slides.pdfcover. Theorem (Sto low) Let f : S2!bCbe a branched cover. Then there exists a homeomorphism

Main Result

Theorem (Luisto and P., ’19)

Let f : Ω→ Rn be a branched cover and x0 ∈ Bf . If there exists anopen neighborhood V of x0 so that the image of the branch setf (Bf ∩ V ) can be embedded into an (n − 2)-simplicial complex,then f is topologically equivalent on V to a cone of a PL mapg : Sn−1 → Sn−1.

This implies that f is topologically equivalent to a PL map.

This theorem also extends to a global result for f : Sn → Sn.

Eden Prywes Branched Covers

Page 28: Characterization of Branched Covers with Simplicial Branch ...eprywes/Branched_Cover_Slides.pdfcover. Theorem (Sto low) Let f : S2!bCbe a branched cover. Then there exists a homeomorphism

Construction of a QR map

We can use this result to construct quasiregular maps.

Corollary

For each n ∈ N there exists a quasiregular map f : R2n → CPn.

There exists a quasiregular map from R2n → (CP1)n.

The map

([z1 : w1], · · · , [zn : wn]) 7→

[z1 · · · zn :n∑

i=1

z1 · · · zi · · · znwi : · · · : w1 · · ·wn]

is a branched cover from CP1 × · · · × CP1 → CPn.

Eden Prywes Branched Covers

Page 29: Characterization of Branched Covers with Simplicial Branch ...eprywes/Branched_Cover_Slides.pdfcover. Theorem (Sto low) Let f : S2!bCbe a branched cover. Then there exists a homeomorphism

Construction of a QR map

We can use this result to construct quasiregular maps.

Corollary

For each n ∈ N there exists a quasiregular map f : R2n → CPn.

There exists a quasiregular map from R2n → (CP1)n.

The map

([z1 : w1], · · · , [zn : wn]) 7→

[z1 · · · zn :n∑

i=1

z1 · · · zi · · · znwi : · · · : w1 · · ·wn]

is a branched cover from CP1 × · · · × CP1 → CPn.

Eden Prywes Branched Covers

Page 30: Characterization of Branched Covers with Simplicial Branch ...eprywes/Branched_Cover_Slides.pdfcover. Theorem (Sto low) Let f : S2!bCbe a branched cover. Then there exists a homeomorphism

([z1 : w1], · · · , [zn : wn]) 7→

[z1 · · · zn :n∑

i=1

z1 · · · zi · · · znwi : · · · : w1 · · ·wn]

The map can be thought of as the coefficients of the polynomial

p(u, v) = (z1u + w1v) · · · (znu + wnv).

So the branch set is

([z1 : w1], · · · , [zn : wn]) ∈ (CP1)n : [zi : wi ] = [zj : wj ] for some i 6= j.

The image of this can be given a simplicial structure and so thereis a PL version of the map.

Eden Prywes Branched Covers

Page 31: Characterization of Branched Covers with Simplicial Branch ...eprywes/Branched_Cover_Slides.pdfcover. Theorem (Sto low) Let f : S2!bCbe a branched cover. Then there exists a homeomorphism

Branched Covers of CPn

For dimension 4:

Theorem (Piergallini and Zuddas, ’18)

If M is of the form #mCP2#nCP2 or #n(S2 × S2), then N admitsa PL (and quasiregular) map from R4 when b+

2 (M), b−2 (M) ≤ 3.

If M is quasiregularly elliptic, then b+2 (M), b−2 (M) ≤ 1

2

( nn/2

).

Is there a manifold that admits a quasiregular (PL) map fromRd , but does not admit a quasiregular (PL) map from T d?

All the above examples factor through the torus.

Eden Prywes Branched Covers

Page 32: Characterization of Branched Covers with Simplicial Branch ...eprywes/Branched_Cover_Slides.pdfcover. Theorem (Sto low) Let f : S2!bCbe a branched cover. Then there exists a homeomorphism

Branched Covers of CPn

For dimension 4:

Theorem (Piergallini and Zuddas, ’18)

If M is of the form #mCP2#nCP2 or #n(S2 × S2), then N admitsa PL (and quasiregular) map from R4 when b+

2 (M), b−2 (M) ≤ 3.

If M is quasiregularly elliptic, then b+2 (M), b−2 (M) ≤ 1

2

( nn/2

).

Is there a manifold that admits a quasiregular (PL) map fromRd , but does not admit a quasiregular (PL) map from T d?

All the above examples factor through the torus.

Eden Prywes Branched Covers

Page 33: Characterization of Branched Covers with Simplicial Branch ...eprywes/Branched_Cover_Slides.pdfcover. Theorem (Sto low) Let f : S2!bCbe a branched cover. Then there exists a homeomorphism

Normal Neighborhoods

Theorem (Luisto and P., ’19)

Let f : Ω→ Rn be a branched cover and x0 ∈ Bf . If there exists anopen neighborhood V of x0 so that the image of the branch setf (Bf ∩ V ) can be embedded into an (n − 2)-simplicial complex,then f is topologically equivalent on V to a cone of a PL mapg : Sn−1 → Sn−1.

Let f : Ω→ Rn be a branched cover and x0 ∈ Ω be a point. Thereexists a radius r0 > 0 and a family of neighborhoods, denotedU(x0, r), such that for 0 < r ≤ r0

x0 ∈ U(x0, r)

f (U(x0, r)) = B(f (x0), r)

f (∂U(x0, r) = ∂B(f (x0), r)

f −1f (x0) ∩ U(x0, r) = x0

Eden Prywes Branched Covers

Page 34: Characterization of Branched Covers with Simplicial Branch ...eprywes/Branched_Cover_Slides.pdfcover. Theorem (Sto low) Let f : S2!bCbe a branched cover. Then there exists a homeomorphism

Outline of Proof

Suppose that near x0, ∂U(x0, r) is homeomorphic to Sn−1.

It is a fact that restricted to ∂U(x0, r), f is still a branchedcover. So if we induct on the dimension, f : ∂U(x0, r)→ Sn−1

is equivalent to a PL map.

By a path lifting argument we show that f behaves the sameway topologically on the boundaries of U(x0, r) for allsufficiently small r .

So f is equivalent to a cone of a PL map.

It is not clear that ∂U(x0, r) ' Sn−1, in fact it may not evenbe a manifold.

Eden Prywes Branched Covers

Page 35: Characterization of Branched Covers with Simplicial Branch ...eprywes/Branched_Cover_Slides.pdfcover. Theorem (Sto low) Let f : S2!bCbe a branched cover. Then there exists a homeomorphism

Outline of Proof

Suppose that near x0, ∂U(x0, r) is homeomorphic to Sn−1.

It is a fact that restricted to ∂U(x0, r), f is still a branchedcover. So if we induct on the dimension, f : ∂U(x0, r)→ Sn−1

is equivalent to a PL map.

By a path lifting argument we show that f behaves the sameway topologically on the boundaries of U(x0, r) for allsufficiently small r .

So f is equivalent to a cone of a PL map.

It is not clear that ∂U(x0, r) ' Sn−1, in fact it may not evenbe a manifold.

Eden Prywes Branched Covers

Page 36: Characterization of Branched Covers with Simplicial Branch ...eprywes/Branched_Cover_Slides.pdfcover. Theorem (Sto low) Let f : S2!bCbe a branched cover. Then there exists a homeomorphism

Outline of Proof

Suppose that near x0, ∂U(x0, r) is homeomorphic to Sn−1.

It is a fact that restricted to ∂U(x0, r), f is still a branchedcover. So if we induct on the dimension, f : ∂U(x0, r)→ Sn−1

is equivalent to a PL map.

By a path lifting argument we show that f behaves the sameway topologically on the boundaries of U(x0, r) for allsufficiently small r .

So f is equivalent to a cone of a PL map.

It is not clear that ∂U(x0, r) ' Sn−1, in fact it may not evenbe a manifold.

Eden Prywes Branched Covers

Page 37: Characterization of Branched Covers with Simplicial Branch ...eprywes/Branched_Cover_Slides.pdfcover. Theorem (Sto low) Let f : S2!bCbe a branched cover. Then there exists a homeomorphism

Outline of Proof

Suppose that near x0, ∂U(x0, r) is homeomorphic to Sn−1.

It is a fact that restricted to ∂U(x0, r), f is still a branchedcover. So if we induct on the dimension, f : ∂U(x0, r)→ Sn−1

is equivalent to a PL map.

By a path lifting argument we show that f behaves the sameway topologically on the boundaries of U(x0, r) for allsufficiently small r .

So f is equivalent to a cone of a PL map.

It is not clear that ∂U(x0, r) ' Sn−1, in fact it may not evenbe a manifold.

Eden Prywes Branched Covers

Page 38: Characterization of Branched Covers with Simplicial Branch ...eprywes/Branched_Cover_Slides.pdfcover. Theorem (Sto low) Let f : S2!bCbe a branched cover. Then there exists a homeomorphism

Path Lifting

r

f(x0)

β

f(z0)R

x0

z0

U(x0, f, r)

α1 α2

U(z0, f, R)

f

r

f(x0)

β(s0)

f(z0)R

x0

z0

α1(s0) α2(s0)

U(z0, f, R)

f

f(z0)

R

S

ζ

f γ3z0

U(z0, f, R)

U1

U2

γ1 γ2γ3

f

Eden Prywes Branched Covers

Page 39: Characterization of Branched Covers with Simplicial Branch ...eprywes/Branched_Cover_Slides.pdfcover. Theorem (Sto low) Let f : S2!bCbe a branched cover. Then there exists a homeomorphism

Back to Dimensions Two and Three

In dimensions two and three the proof simplifies.

In dimension two, f is locally injective on ∂U(x0, r) and so∂U(x0, r) is a manifold and is homeomorphic to S1.

In dimension three Martio and Srebro first show that∂U(x0, r) is a manifold.

Like in dimension 2, ∂U(x0, r) is a manifold away from thebranch set of f .The image of the branch set is ”ray-like” by assumption and sointersects B(f (x0), r) at a discrete set of points.Topologically f behaves like the power map z 7→ zd on∂U(x0, r) near the intersection by Church and Hemmingsen’stheorem and so still can be used to define a chart for ∂U(x0, r),

∂U(x0, r) is homeomorphic to S2 if it is simply connected.This follows because U(x0, r) is contractible.

Eden Prywes Branched Covers

Page 40: Characterization of Branched Covers with Simplicial Branch ...eprywes/Branched_Cover_Slides.pdfcover. Theorem (Sto low) Let f : S2!bCbe a branched cover. Then there exists a homeomorphism

Back to Dimensions Two and Three

In dimensions two and three the proof simplifies.

In dimension two, f is locally injective on ∂U(x0, r) and so∂U(x0, r) is a manifold and is homeomorphic to S1.

In dimension three Martio and Srebro first show that∂U(x0, r) is a manifold.

Like in dimension 2, ∂U(x0, r) is a manifold away from thebranch set of f .The image of the branch set is ”ray-like” by assumption and sointersects B(f (x0), r) at a discrete set of points.Topologically f behaves like the power map z 7→ zd on∂U(x0, r) near the intersection by Church and Hemmingsen’stheorem and so still can be used to define a chart for ∂U(x0, r),

∂U(x0, r) is homeomorphic to S2 if it is simply connected.This follows because U(x0, r) is contractible.

Eden Prywes Branched Covers

Page 41: Characterization of Branched Covers with Simplicial Branch ...eprywes/Branched_Cover_Slides.pdfcover. Theorem (Sto low) Let f : S2!bCbe a branched cover. Then there exists a homeomorphism

Back to Dimensions Two and Three

In dimensions two and three the proof simplifies.

In dimension two, f is locally injective on ∂U(x0, r) and so∂U(x0, r) is a manifold and is homeomorphic to S1.

In dimension three Martio and Srebro first show that∂U(x0, r) is a manifold.

Like in dimension 2, ∂U(x0, r) is a manifold away from thebranch set of f .The image of the branch set is ”ray-like” by assumption and sointersects B(f (x0), r) at a discrete set of points.Topologically f behaves like the power map z 7→ zd on∂U(x0, r) near the intersection by Church and Hemmingsen’stheorem and so still can be used to define a chart for ∂U(x0, r),

∂U(x0, r) is homeomorphic to S2 if it is simply connected.This follows because U(x0, r) is contractible.

Eden Prywes Branched Covers

Page 42: Characterization of Branched Covers with Simplicial Branch ...eprywes/Branched_Cover_Slides.pdfcover. Theorem (Sto low) Let f : S2!bCbe a branched cover. Then there exists a homeomorphism

∂U(x0, r) is a Manifold

In the general case when f : Ω→ Rn,

f restricted to ∂U(x0, r) is a branched cover and away fromthe branch set is a covering map. So ∂U(x0, r) \ Bf is amanifold.

As in dimension 3, if x ∈ ∂U(x0, r) ∩ Bf , then we consider themap f restricted to a normal neighborhood of x in ∂U(x0, r).

We continue to go down in dimension considering more andmore nested normal neighborhoods.

Eden Prywes Branched Covers

Page 43: Characterization of Branched Covers with Simplicial Branch ...eprywes/Branched_Cover_Slides.pdfcover. Theorem (Sto low) Let f : S2!bCbe a branched cover. Then there exists a homeomorphism

∂U(x0, r) is homeomorphic to a sphere

We show a stronger fact that for any point x , if we considernormal neighborhoods of dimension k + 1, then their boundarieswill be homeomorphic to Sk when taken sufficiently close to x .

By the path lifting argument, normal neighborhoods will havea cone structure. So if U is a normal neighborhood of a pointx , then U ' cone(∂U).

Let γ : S` → ∂U ' U \ x0. There is a homotopy sending γto a point in U × (0, 1)n−k−1. It can be chosen to avoidx0 × (0, 1)n−k−1 when 1 ≤ ` < k . So π`(V ) = 0 .

Eden Prywes Branched Covers

Page 44: Characterization of Branched Covers with Simplicial Branch ...eprywes/Branched_Cover_Slides.pdfcover. Theorem (Sto low) Let f : S2!bCbe a branched cover. Then there exists a homeomorphism

∂U(x0, r) is homeomorphic to a sphere

We show a stronger fact that for any point x , if we considernormal neighborhoods of dimension k + 1, then their boundarieswill be homeomorphic to Sk when taken sufficiently close to x .

By the path lifting argument, normal neighborhoods will havea cone structure. So if U is a normal neighborhood of a pointx , then U ' cone(∂U).

Let γ : S` → ∂U ' U \ x0. There is a homotopy sending γto a point in U × (0, 1)n−k−1. It can be chosen to avoidx0 × (0, 1)n−k−1 when 1 ≤ ` < k . So π`(V ) = 0 .

Eden Prywes Branched Covers

Page 45: Characterization of Branched Covers with Simplicial Branch ...eprywes/Branched_Cover_Slides.pdfcover. Theorem (Sto low) Let f : S2!bCbe a branched cover. Then there exists a homeomorphism

Converse Result

There is a partial converse to the Martio-Srebro result.

Theorem (Martio and Srebro, ’79)

Let f : Ω→ R3 be a branched cover so that at x ∈ Ω there existsan r0 > 0 with the property that for all r ≤ r0, ∂U(x0, r) is amanifold. Then at x0, f is equivalent to a path of rational maps.

We show a corresponding result:

Theorem (Luisto and P, ’19)

Let f : Ω→ Rn be a branched cover so that at x ∈ Ω there existsan r0 > 0 with the property that for all r ≤ r0, U(x0, r) is amanifold with boundary. Then at x0, f is equivalent to a path ofbranched covers.

Eden Prywes Branched Covers

Page 46: Characterization of Branched Covers with Simplicial Branch ...eprywes/Branched_Cover_Slides.pdfcover. Theorem (Sto low) Let f : S2!bCbe a branched cover. Then there exists a homeomorphism

Converse Result

There is a partial converse to the Martio-Srebro result.

Theorem (Martio and Srebro, ’79)

Let f : Ω→ R3 be a branched cover so that at x ∈ Ω there existsan r0 > 0 with the property that for all r ≤ r0, ∂U(x0, r) is amanifold. Then at x0, f is equivalent to a path of rational maps.

We show a corresponding result:

Theorem (Luisto and P, ’19)

Let f : Ω→ Rn be a branched cover so that at x ∈ Ω there existsan r0 > 0 with the property that for all r ≤ r0, U(x0, r) is amanifold with boundary. Then at x0, f is equivalent to a path ofbranched covers.

Eden Prywes Branched Covers

Page 47: Characterization of Branched Covers with Simplicial Branch ...eprywes/Branched_Cover_Slides.pdfcover. Theorem (Sto low) Let f : S2!bCbe a branched cover. Then there exists a homeomorphism

Converse Result

We can iterate the previous result to get a lower bound on thetopological dimension of Bf .

Corollary (Luisto and P, ’19)

Let f : Ω→ Rn be a branched cover so that for some k ,2 ≤ k ≤ n− 2, all the normal domains of dimension less than k aremanifolds with boundary, then dimtop(Bf ) ≥ n − k.

It is not possible to show that if all the normal domains aremanifolds, then f is equivalent to a PL-map. Let w : R3 → R3 be awinding map and let h : R3 → R3 be a homeomorphism that takesthe set B = (0, t2 cos(1/t), t), t ∈ R to the z-axis near 0. Definef := w h w . The branch set is a union of the z-axis and B.

Eden Prywes Branched Covers

Page 48: Characterization of Branched Covers with Simplicial Branch ...eprywes/Branched_Cover_Slides.pdfcover. Theorem (Sto low) Let f : S2!bCbe a branched cover. Then there exists a homeomorphism

Converse Result

We can iterate the previous result to get a lower bound on thetopological dimension of Bf .

Corollary (Luisto and P, ’19)

Let f : Ω→ Rn be a branched cover so that for some k ,2 ≤ k ≤ n− 2, all the normal domains of dimension less than k aremanifolds with boundary, then dimtop(Bf ) ≥ n − k.

It is not possible to show that if all the normal domains aremanifolds, then f is equivalent to a PL-map. Let w : R3 → R3 be awinding map and let h : R3 → R3 be a homeomorphism that takesthe set B = (0, t2 cos(1/t), t), t ∈ R to the z-axis near 0. Definef := w h w . The branch set is a union of the z-axis and B.

Eden Prywes Branched Covers

Page 49: Characterization of Branched Covers with Simplicial Branch ...eprywes/Branched_Cover_Slides.pdfcover. Theorem (Sto low) Let f : S2!bCbe a branched cover. Then there exists a homeomorphism

Thank you!

Eden Prywes Branched Covers