Casson invariant and structure of the mapping class group

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Casson invariant and

structure of the mapping class group

Shigeyuki MORITA

January 25, 2017

Shigeyuki MORITA Casson invariant and structure of the mapping class group

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Contents

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1 Homology3-spheres and the Torelli group

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2 Casson invariant and the first MMM class

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3 Difference between two filtrations of the Torelli group

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4 Finite type invariants and the Torelli group

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5 Open problems

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6 Extending the above picture to a wider context

based on joint work with T. Sakasai and M. Suzuki

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7 Prospect

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Homology3-spheres and the Torelli group (1)

M(3) ={closed oriented3-manifold}/ori. pres. diffeo.

∪

H(3) ={closed orientedhomology3-sphere}/ori. pres. diffeo.

Heegaard decomposition:

M(3)3^∀[M], M =Hg∪ϕ−Hg (Hg :handlebody, ϕ∈ Mg) Mg :mapping class group S³ =H_g∪ιg −H_g (ι_g ∈ Mg : 90^◦-rotation on each handle)

⇒ Mg3[ϕ]7−→[Mϕ=Hg∪ϕ−Hg]∈M(3)

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Theorem (Reidemeister-Singer)

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(a

g

Mg

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/R.S. stabilization=M(3)

alternative description:

fix a Heegaard embeddingΣg⊂S³=Hg∪Σg −Hg

Ng ={ϕ∈ Mg;ϕextends to diffeo. ofHg} (⊂ Mg) N⁰_g ={ϕ∈ Mg;ϕextends to diffeo. of−H_g} (⊂ Mg)

M(3) = lim

g→∞N⁰_g\Mg/N_g

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Homology3-spheres and the Torelli group (3) restriction to the Torelli group :

Ig= Ker(Mg →Sp(2g,Z))

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Proposition

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limg→∞Ig/∼=H(3) (Ig3ϕ7−→Wϕ =Hg∪ιgϕ−Hg∈H(3)) where ϕ∼ψ⇔ιgϕ=ιgψ∈N⁰_g\Mg/Ng

two filtrations ofIg:

Ig =Mg(1)⊃ Mg(2)⊃ · · · (Johnson filtration)

Ig =Ig(1)⊃ Ig(2) = [Ig(1),Ig(1)]⊃ · · · (lower central series) Ig(k)⊂ Mg(k) for anyk

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Mg(2) = Ker(τ1 :Ig

first Johnson hom.

→ ∧³H/H) (H =H1(Σg;Z)) Mg(k+ 1) = Ker(τ_k :Mg(k)Johnson hom.

→ hg(k))

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Theorem (Johnson)

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Ig(2)finite index

⊂ Mg(2) =Kg =hDehn twists on BSCCi Ig(1)/Ig(2) =H₁(Ig)∼=∧³H/H⊕2-torsion

BCSS=bounding simple closed curve

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Two filtrations ofIg induces those ofH(3)andQH(3):

H(3) =H(3)¹ ⊃H(3)²⊃H(3)³ ⊃ · · · H(3) =H(3)₁ ⊃H(3)₂⊃H(3)₃ ⊃ · · ·

QH(3) =QH(3)¹ ⊃QH(3)² ⊃QH(3)³⊃ · · · QH(3) =QH(3)₁ ⊃QH(3)₂ ⊃QH(3)₃⊃ · · · where

H(3)^k= lim

g→∞Mg(k)/∼ ⊃ H(3)k = lim

g→∞Ig(k)/∼

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Casson invariant and the first MMM class (1)

Casson invariant (1985):

λ:H(3)→Z

(i)λ≡ Rohlin homomorphism:H(3)→Z/2 (mod2) (ii)λ= 1

2 “alg. number” of{irred. rep.:π1W →SU(2)}/conj.

(iii)λ(−W) =−λ(W), additive w.r.t. connected sum (iv)W ⊃K (knot)⇒λ(W_1/n(K)) =λ(W) +n1

2∆¯⁰⁰_K(1)

Extensions by Walker (to rational homology3-spheres) and Lescop (to all3-manifolds)

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Consider the mapping

λ^∗ :Ig→Z defined byλ^∗(ϕ) =λ(Wϕ) NOT a homomorphism, but its restriction toKg

λ^∗:Kg→Z can be shown to be a homomorphism!

What is it?

Answer: secondary class associated to the fact: the first MMM-class vanishes in the Torelli group e₁ = 0∈H²(Ig;Q)

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e₁ ∈H²(Mg;Z)

geometric meaning: signature of surface bundles over surfaces

⇒ e₁= 0∈H²(Ig;Q)because signature of any fiber bundle F →E→B vanishes ifπ₁B acts onH_∗(F;Q)trivially (Chern-Hirzebruch-Serre)

There aretwocanonical cocycles representinge1: pull back of −3c₁ ∈Z²(Sp(2g,Z);Q)

image of ∧²(∧³H_Q/H_Q)^Sp∼=Q→ ∧²˜k∈Z²(Mg;Q) under (˜k, ρ0) :Mg → ∧³H_Q/H_QoSp(2g,Z)

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⇒3c1+∧²˜k=d^∃u1, u1 ∈C¹(Mg;Q) {

c1|Ig = 0

∧²k˜|_Kg = 0 ⇒d(u₁|_Kg) = 0

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Theorem (M.)

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H¹(Kg;Q)^M^g ∼=Q(g≥2)generated byd₁:= [u₁|_Kg] Furthermored1(BSCC map of type(h, g−h)) =ct. h(g−h)

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Theorem (M.)

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λ^∗= 1

24d1+ ¯τ2 : Kg →Q

where

¯ τ2 =Kg

τ2

→hg(2)certain quotient

→ Q

Furthermore

λ^∗ = 1

24d₁ onMg(3)

so that we may say thatd1 is thecoreof the Casson invariant

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Difference between two filtrations of the Torelli group (1) (recall) two filtrations ofIg:

Ig =Mg(1)⊃ Mg(2) =Kg ⊃ Mg(3)· · · (Johnson filtration)

Ig =Ig(1)⊃ Ig(2) = [Ig(1),Ig(1)]⊃ Ig(3)· · · (lower central series) Ig(k)⊂ Mg(k)for anyk

Johnson showedMg(2)/Ig(2)⊗Q= 0and he asked [Mg(k) :Ig(k)]<∞?

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Theorem (M. 1988)

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The index ofIg(3) = [[Ig,Ig],Ig]inMg(3)is infinite This was proved by showing that

d16= 0 onMg(3)whereasd1 = 0onIg(3)

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Difference between two filtrations of the Torelli group (2)

alternatively

τ₁ :Ig → ∧³H/H

over∼=Q(Ig/Ig(2))⊗Q

τ₁^∗ :H²(∧³H_Q/H_Q)^Sp ∼=Q→H²(Ig;Q)

is trivial because the image ise₁which vanishes on the Torelli group. Then by considering the Sullivan1-minimal model ofIg

and by taking its dual, we can conclude(Ig(2)/Ig(3))^Sp 6= 0 whereas we know that(Mg(2)/Mg(3))^Sp= 0

Hain determinedτ₁^∗completely and obtained

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Theorem (Hain 1997)

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⊕_∞

k=1(Ig(k)/Ig(k+ 1))⊗Q∼=Free Lieh∧³H/Hi/quad. relation

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Difference between two filtrations of the Torelli group (3)

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Theorem (Hain)

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⊕_∞

k=1(Ig(k)/Ig(k+ 1))⊗Q→⊕_∞

k=1(Mg(k)/Mg(k+ 1))⊗Qis surjective so that the latter is generated by the degree1part It is a mystery whether an analogue of the above difference would appear forAutFnor not, namely comparison of

Andreadakis filtration vs l.c.s. filtration ofIA_n Andreadakis Conjecture(overQ)

works of Satoh, recently Bartholdi, Massuyeau-Sakasai

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Finite type invariants and the Torelli group (1)

Ohtsuki’s (finite type) invariants:

λ_k:H(3)→Q (k= 1,2, . . .) and Ohtsuki filtration based on the LMO-invariant

QH(3) =QH(3)(3)⊃QH(3)(6) ⊃QH(3)(9) ⊃ · · ·

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A(∅) =commutative algebra generated by vertex oriented connected trivalent graphs/AS+IHX

degree=half the number of vertices

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Theorem (Garoufalidis-Ohtsuki+Le-Murakami-Ohtsuki)

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There exists an isomorphism

GrmA(∅)∼=QH(3)_(3m)/QH(3)_(3m+1)

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Theorem (Garoufalidis-Levine)

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There exists a mapping

Ig(2m)/Ig(2m+ 1)⊗Q→GrmA^conn(∅) which is surjective forg≥5m+ 1

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In particular

(Ig(2m)/Ig(2m+ 1))^Sp⊗Q gives rise to invariants for

QH(3)_(3m)/QH(3)_(3m+1) The casem= 1:

Gr1A(∅)∼=QH(3)(3)/QH(3)(4) ∼=Q (Ig(2)/Ig(3))^Sp⊗Q∼= Gr1A^conn(∅)∼=Q given by

theta graph, Casson invariantλandd₁:Kg →Q

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Open problems (1)

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Problem (Hain)

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Is the following sequence exact?

Q→

⊕∞ k=1

(Ig(k)/Ig(k+ 1))⊗Q

⊕∞ k=1

(Mg(k)/Mg(k+ 1))⊗Q

In other words, the difference (overQ) between Johnson and l.c.s filtrations ofIg is only theCassoninvariant?

How about Ohtsuki invariants?

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Problem

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Compare the following three filtrations ofQH(3)

QH(3) =QH(3)¹ ⊃QH(3)² ⊃QH(3)³⊃ · · · (Mg(k)/∼) QH(3) =QH(3)1 ⊃QH(3)2 ⊃QH(3)3⊃ · · · (Ig(k)/∼) QH(3) =QH(3)(3)⊃QH(3)(6)⊃QH(3)(9) ⊃ · · · (Ohtsuki) M.:QH(3) =QH(3)²

any homology3-sphere can be obtained by pastingKg

Pitsch:QH(3) =QH(3)³

any homology3-sphere can be obtained by pastingMg(3)

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Casson invariant appears in

QH(3)2/QH(3)3 andQH(3)(3)/QH(3)(6)

The following two problems are modified ones from the original after comments by M. Sato and S. Tsuji

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Problem

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Determine whether the Ohtsuki invariant

λ_k:H(3)∼=Ig/∼(=Kg/∼=Mg(3)/∼)→Q can be described explicitly in terms ofd₁and Johnson homomorphisms. If so, give the formula. In particular determine whether its restriction to some deep subgroup Mg(m_k)(depending onk) becomes ahomomorphismor not

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Open problems (4) k= 1(Casson invariant):

λ1 is a homomorphism onKg but NOT onMg(1) =Ig

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Problem (special case of Hain’s problem)

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By Garoufalidis-Levine, the Ohtsuki invariantλkrestricted to Ig(2k)gives a homomorphism

λk:Ig(2k)/Ig(2k+ 1)→Q

For eachk≥2, determine whether it is described as a quotient of the higher Johnson homomorphism

Ig(2k)/Ig(2k+ 1)^⊗Q Mg(2k)/Mg(2k+ 1)^τ^g⊂^(2k)hg(2k) or not

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Extending the above picture to a wider context (1)

H²(Mg;Q)3e17→0∈H²(Ig;Q) ⇒ λ:H(3)→Z More precisely

Kg → Mg→ Mg/Kg

∼Q

=U_QoSp(2g,Z) (U =∧³H/H)

H¹(Mg;Q) = 0→H¹(Kg;Q)^M^g ∼=Q→

H²(U_QoSp(2g,Z);Q)∼=Q²→H²(Mg;Q)∼=Q

the difference of two natural cocycles fore₁ ⇒ H¹(Kg;Q)^M^g ∼=Q ⇒ Casson invariant

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extendingMg ⇒ Hg,1 ande₁ ⇒˜t_2k+1, ultimate goal:

H²(H^top_g,1;Q)3˜t2k+1 7→0∈H²(H^smooth_g,1 ;Q) ⇒ νk:Θ³ →Q

Garoufalidis-Levine (based on Goussarov and Habiro)

H^smoothg,1 ={homology cylinder overΣ_g,1}/smooth H-cobordism H^smooth0,1 =Θ³ =H(3)/smooth H-cobordism ^central⊂ H^smoothg,1

Θ³→ H^smoothg,1 → Hg,1 =H^smoothg,1 /Θ³ (central extension)

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Extending the above picture to a wider context (3) exact sequence:

0→H¹(Hg,1;Q)→H¹(H^smoothg,1 ;Q)→H¹(Θ³;Q)

∼= Hom(Θ³,Q)∼=Q^N→H²(Hg,1;Q)→H²(H^smooth_g,1 ;Q)

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Theorem (Furuta, Fintushel-Stern)

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Θ³ has infinite rank

⇒Θ³/torsion⊂Q^∞(becauseΘ³ is countable)

⇒Hom(Θ³,Q)∼=Q^N(direct product of countably manyQ) so there exist (uncountably) many homomorphisms

Θ³→Q

but explicitly known one(s): Frφyshov and Ozsv ´ath-Szab ´o

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Problem

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How is the huge groupH¹(Θ³;Q)∼=Q^Ndivided into Coker

(

H¹(Hg,1;Q)→H¹(H^smooth_g,1 ;Q) )

and Ker

(

H²(Hg,1;Q)→H²(H^smoothg,1 ;Q) )

?

Cokeris non-trivial⇔

∃homomorphismΘ³ →Q(6= 0)which extends toH^smoothg,1 →Q many works onHg,1by

Sakasai, Habiro, Massuyeau, Cha-Friedl-Kim,...

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Extending the above picture to a wider context (5) Mal’cev completion ofπ1Σg,1: · · · →N_d→ · · · →N1 =H_Q

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Theorem (Garoufalidis-Levine )

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∃ρ˜_∞:H^smooth_g,1 →lim←−^d^→∞Aut₀N_d (symplectic auto. groups) each factorρ˜_d:H^smooth_g,1 →Aut₀N_dis surjective overZ

candidates forKer: constructed a homomorphism

˜

ρ:Hg,1 → (

∧³H_Q⊕∏^∞

k=1

S^2k+1H_Q )

oSp(2g,Z)

and defined

(∧²S^2k+1H_Q)^Sp∼=Q317→˜t_2k+1 ∈H²(Hg,1;Q)

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Extending the above picture to a wider context (6) replacingHg,1 with more geometric object

(2008, after a comment by Orr):

H^top_g,1 ={homology cylinder overΣg,1}/topological H-cobordism

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Theorem (Freedman)

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Any homology3-sphere bounds acontractibletopological4-mfd

It follows that H^smooth_g,1 → H^top_g,1 factors throughHg,1

Θ³ → H^smoothg,1 → Hg,1 → H^top_g,1

and the homomorphismsρ˜_∞,ρ˜are actually defined onH^top_g,1

⇒ ˜t2k+1∈H²(H^top_g,1;Q)

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Extending the above picture to a wider context (7) how aboutCoker?

hg,1 =symplectic derivation Lie algebra ofL(H_Q) extremely rich and mysterious structure

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Theorem (Massuyeau-Sakasai)

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(i) Hg,1 homo.

→ Hˆ1(h⁺_g,1)oSp(2g,Z)with dense image

(ii)H₁(Hg,1;Q)⊃Q(sharp contrast:Mgis perfect (g≥3))

⇒Hˆ_c¹(ˆh_g,1)⊂H¹(H^smooth_g,1 ;Q)but this part comes from H¹(H^top_g,1;Q)so that it vanishes in theCoker

At present, there is no information about Coker

(

H¹(Hg,1;Q)→H¹(H^smoothg,1 ;Q) )

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Extending the above picture (8)

back toKer

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Theorem (Sakasai-Suzuki-M.)

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∃ρ˜^∗_∞:H_c^∗(ˆh⁺_∞_,1)^Sp⊗H^∗(Sp(2∞,Z))→H^∗(H^top_g,1;Q)

⇒ H_c²(ˆh_∞,1)→H²(H^top_g,1;Q)→H²(Hg,1;Q)

H_c²(ˆh_∞,1)3t2k+1 (Lie algebra version)7→˜t2k+1 ∈H²(Hg,1;Q) Conant-Kassabov-Vogtmann defined more classes on the LHS, but, at present, onlyt₃,t₅,t₇ are known to be non-trivial...

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Prospect (1)

only known homomorphism(s) (Frφyshov and Ozsv ´ath-Szab ´o) Θ³→Z

candidate: Neumann-Siebenmann, Fukumoto-Furuta-Ue, Saveliev

ν :=

∑7 i=0

(−1)ⁱ⁽ⁱ⁺¹⁾² rankHFⁱ (instanton Floer homology) recall:

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Theorem (Taubes)

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∑₇

i=0(−1)ⁱrankHFⁱ= 2λ (Casson invariant)

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Theorem (Manolescu)

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The Rohlin homomorphismΘ³ →Z/2does not split

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Prospect (2)

geometric meaning of the classes˜t_2k+1∈H²(H^top_g,1;Q):

Intersection numbersof higher and higherMasseyproducts (using works of Kitano, Garoufalidis-Levine)

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Conjecture

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The homomorphismH¹(Θ³;Q)∼=Q^N→H²(Hg,1;Q)induced by

0→Θ³ → H^smooth_g,1 → Hg,1→1

is highly non-trivial (possibly injective) and its image contains the classes˜t_2k+1∈H²(Hg,1;Q) ⇒

˜t_2k+16= 0∈H²(H^top_g,1;Q) and

˜t_2k+1= 0∈H²(H^smoothg,1 ;Q)

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Prospect (3)

If Conjecture is true⇒obtain homomorphisms

ν_k:Θ³→Q (k= 1,2, . . .) homology cobordism invariants