December21,2016 HironobuNAOE Homology4-ballswithcomplexityzeroandcollapsingofshadows

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Preliminaries Main results Martelli’s graph and rational homology 4-balls

Homology 4-balls with complexity zero and collapsing of shadows

Hironobu NAOE

Tohoku University

December 21, 2016

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Introduction (1/4)

※In this talk we assume that manifolds arecompact,orientedand smoothotherwise mentioned.

Definition (exotic)

Two manifoldsX andY are said to be exoticif they are homeomorphic but NOT diﬀeomorphic.

Milnor found manifolds exotic to S⁷in 1956.

There are 28 diﬀerent smooth structures onS⁷. Any 3-manifold has no exotic pairs.

Euclidean spaceRⁿ has an exotic pair iﬀn= 4.

There are uncountably many smooth structures onR⁴.

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Introduction (1/4)

※In this talk we assume that manifolds arecompact,orientedand smoothotherwise mentioned.

Definition (exotic)

Two manifoldsX andY are said to be exoticif they are homeomorphic but NOT diﬀeomorphic.

Milnor found manifolds exotic to S⁷in 1956.

There are 28 diﬀerent smooth structures onS⁷. Any 3-manifold has no exotic pairs.

Euclidean spaceRⁿ has an exotic pair iﬀn= 4.

There are uncountably many smooth structures onR⁴.

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Introduction (2/4)

Theorem (Matveyev, Curtis-Freedman-Hsiang-Stong ’96)

Let(M, M^′)be an exotic pair of simply-connected closed 4-manifolds.

Then there exists a contractible submanifoldC ofM s.t. M^′ is obtained fromM by a surgery alongC.

Such a contractible submanifoldC is called acork.

M cork

cork

M^′ cork

Motivation

Classification and characterization of corks.

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Introduction (2/4)

Theorem (Matveyev, Curtis-Freedman-Hsiang-Stong ’96)

Let(M, M^′)be an exotic pair of simply-connected closed 4-manifolds.

Then there exists a contractible submanifoldC ofM s.t. M^′ is obtained fromM by a surgery alongC.

Such a contractible submanifoldC is called acork.

M cork

cork

M^′ cork

Motivation

Classification and characterization of corks.

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Introduction (3/4)

Description of 4-manifolds:

Kirby diagram · · · link + framings

shadow· · · 2-dimensional (simple) polyhedron +gleams Example(simple polyhedra)

compact surface Bing’s house

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Introduction (3/4)

Description of 4-manifolds:

Kirby diagram · · · link + framings

shadow· · · 2-dimensional (simple) polyhedron +gleams Definition (shadow-complexitysc)

Theshadow-complexitysc(M)of a 4-manifold M is defined as the minimum number of true vertices of a shadow ofM.

∼=

true vertex

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Introduction (4/4)

Main theorem (N.)

M is an integral homology 4-ball w/sc(M) = 0⇐⇒M ∼=D⁴. Corollary (N.)

There are no corks w/sc= 0.

Remark.

There are (infinitely) many rational homology 4-balls w/sc= 0.

There are (infinitely) many corks w/sc= 1(N. ’15).

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The plan of this talk

1 Preliminaries

2 Main results

3 Martelli’s graph and rational homology 4-balls

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Simple polyhedron Shadows of 4-manifolds

§ 1 Preliminaries - Simple polyhedron (1/2)

simple polyhedron: a compact topological spaceX s.t. each point ofX has a neighborhood homeo. to one of:

(i) (ii) (iii)

(iv) (v)

. singular set: set of points of type (ii), (iii) and (v).

true vertex: a point of type (iii).

region: a conn. component of X\Sing(X).

boundary: set of points of type (iv) and (v).

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§ 1 Preliminaries - Simple polyhedron (2/2)

Example(simple polyhedron) D²∪(two annuli):

(iii)

(ii)

true vertex

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§ 1 Preliminaries - Shadow (1/3)

Definition (shadow)

LetM be a 4-manifold w/ boundary andX⊂M be a simple polyhedron.

IfM ↘X, and ifX is proper and locally flat inM, thenX is called ashadowofM.

M

Examples: X

D²-bundle overΣhas a shadowΣ.

Let M →D²be a Lefschetz fibration whose regular fiber isF w/

vanishing cyclesγ1, . . . , γn. Then a simple polyhedron obtained fromF by attaching n2-disks alongγ1, . . . , γn is a shadow ofM.

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§ 1 Preliminaries - Shadow (2/3)

M

X Studies of shadows:

■Spin^c structure

■(almost) complex structure

■Stein structure



 · · · 4-manifolds

■hyperbolic structure (volume)

■stable maps intoR²

}

· · · 3-manifolds

The boundary∂X is a link (or a trivalent graph) in the 3-manifold∂M.

■ (quantum) invariants

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§ 1 Preliminaries - Shadow (2/3)

M

■Spin^c structure

■Stein structure



}

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§ 1 Preliminaries - Shadow (2/3)

M

■Spin^c structure

■Stein structure



}

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§ 1 Preliminaries - Shadow (3/3)

Definition (shadow-complexitysc)

Theorem (Costantino ’06)

M is a closed 4-manifold w/sc^sp(M) = 0

⇐⇒M ∼=S⁴, S²×S²,CP²,CP²,CP²#CP²,CP²#CP²orCP²#CP².

Theorem (Martelli ’11)

M is a closed 4-manifold w/sc(M) = 0

⇐⇒M ∼=M^′#_kCP², whereM^′ is a “4-dimensional graph manifold”.

Thus theclosed4-manifolds w/sc= 0 have been completely classified.

Our main result is the case of the 4-manifolds with boundary.

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§ 1 Preliminaries - Shadow (3/3)

⇐⇒M ∼=S⁴, S²×S²,CP²,CP²,CP²#CP²,CP²#CP²orCP²#CP². Theorem (Martelli ’11)

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§ 1 Preliminaries - Shadow (3/3)

⇐⇒M ∼=S⁴, S²×S²,CP²,CP²,CP²#CP²,CP²#CP²orCP²#CP². Theorem (Martelli ’11)

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§ 2 Main results

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§ 2 Main results (1/4)

Theorem A (N.)

LetX be a shadow ofM⁴, and letY ⊂X be a simple polyhedron.

IfX ↘Y, thenY is a shadow of a 4-manifold diﬀeo. toM. Theorem B (N.)

Every acyclic simple polyhedron w/ no true vertices collapses ontoD².

Remark.

A PL-manifold has a unique smoothing in dim≤6.

X is said to be acyclic ifH_∗(X;Z)∼=H_∗({pt.};Z).

There are 3 types of collapsing in Theorem B.

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§ 2 Main results (1/4)

Theorem A (N.)

Every acyclic simple polyhedron w/ no true vertices collapses ontoD². Remark.

A PL-manifold has a unique smoothing in dim≤6.

X is said to be acyclic ifH_∗(X;Z)∼=H_∗({pt.};Z).

There are 3 types of collapsing in Theorem B.

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§ 2 Main results (2/4)

collapse (a)

collapse (b)

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§ 2 Main results (3/4)

collapse (c)

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§ 2 Main results (4/4)

Theorem A (N.) (recall)

IfX ↘Y, thenY is a shadow of a 4-manifold diﬀeo. toM. Theorem B (N.) (recall)

Every acyclic simple polyhedron w/ no true vertices collapses ontoD². The main theorem is a consequence of Theorem A and B.

Main theorem (N.) (recall)

M is an integral homology 4-ball w/sc(M) = 0iﬀM ∼=D⁴. Fact

D⁴ is a unique 4-manifold having a shadowD².

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§ 2 Main results (4/4)

Theorem A (N.) (recall)

IfX ↘Y, thenY is a shadow of a 4-manifold diﬀeo. toM. Theorem B (N.) (recall)

Every acyclic simple polyhedron w/ no true vertices collapses ontoD². The main theorem is a consequence of Theorem A and B.

Main theorem (N.) (recall)

M is an integral homology 4-ball w/sc(M) = 0iﬀM ∼=D⁴. Fact

D⁴ is a unique 4-manifold having a shadowD².

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§ 3 Martelli’s graph and

rational homology 4-balls

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§ 3 Martelli’s graph and rational homology 4-balls (1/2)

Proposition (Martelli ’11)

Every simple polyhedronX whose singular set consists of circles can be represented by a graph w/ vertices of the following:

∂X

D² P Y2

Y111 Y12 Y3

Theorem B (N.) (recall)

Every acyclic simple polyhedron w/ no true vertices collapses ontoD². We used Martelli’s graph to prove Theorem B.

IfX is acyclic, then its graph is a tree.

Introduce some moves of graphs which corresponds to collapsing.

etc...

A graph ofX is a tree≠⇒X is acyclic.

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§ 3 Martelli’s graph and rational homology 4-balls (1/2)

∂X

D² P Y2

Y111 Y12 Y3

IfX is acyclic, then its graph is a tree.

etc...

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§ 3 Martelli’s graph and rational homology 4-balls (1/2)

∂X

D² P Y2

Y111 Y12 Y3

IfX is acyclic, then its graph is atree.

etc...

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§ 3 Martelli’s graph and rational homology 4-balls (1/2)

∂X

D² P Y2

Y111 Y12 Y3

IfX is acyclic, then its graph is atree.

etc...

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§ 3 Martelli’s graph and rational homology 4-balls (2/2)

Proposition (N.)

Suppose that a graphGof a simple polyhedronX is a tree. Then the following moves ofGpreserve the rational homology ofX.

, and .

∂X

Y2 Y3 ∂X Y12 (edge)

This immediately yields the following:

Proposition (N.)

X is a shadow of QHB⁴w/o true vertices.

⇐⇒A graph ofX is a tree s.t. the resulting graph after applying the above moves encodes an acyclic polyhedron.

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Summary

Theorem A (N.)

Every acyclic simple polyhedron w/ no true vertices collapses ontoD². Main theorem (N.)

M is an integral homology 4-ball w/sc(M) = 0iﬀM ∼=D⁴. Corollary (N.)

There are no corks w/sc= 0.