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MINIMAL EULER CHARACTERISTICS OF

4-MANIFOLDS WITH 3-MANIFOLD GROUPS

HONGBIN SUN AND ZHONGZI WANG

Abstract. Let π = π1(M) for a compact 3-manifold M , and letχ4, p and q∗ be the invariants of Hausmann-Weinberger, Kotschickand Hillman respectively. For certain class of compact 3-manifoldsM , including all those not containing two-sided RP 2, we determineχ4(π). We address when does χ4(π) = p(π), when does χ4(π) =q∗(π), and answer a question raised by Hillman.

Contents

1. Introduction 1

2. Decomposition of 3-manifolds by S2 and two-sided RP 2 5

3. Betti numbers of 3-manifold groups 7

4. An upper bound of χ4(π) 11

5. A lower bound for χ4(π) 14

6. p(π) = χ4(π) for 3-manifolds without closed non-orientableprime factor 21

7. q∗(π) = χ4(π) for 3-manifold groups containing no 2-torsions 24

References 26

1. Introduction

Since any finitely presented group G can be realized as the fundamentalgroup of some closed 4-manifold X and it is impossible to classify finite

2010 Mathematics Subject Classification. Primary 57N65; Secondary 57M05,57N13.

Key words and phrases. 3-manifolds, 4-manifolds, fundamental group, EulerCharacteristic.

The first author is partially supported by Simons Collaboration Grant 615229.1

http://arxiv.org/abs/2103.10273v2

2 HONGBIN SUN AND ZHONGZI WANG

presentable groups, it is impossible to classify closed 4-manifolds. Evenjust to classify 4-manifolds with fundamental groups in a special familyof groups (that can be classified), in general it is still a very difficultproblem. However, studying some basic topological invaraints of 4-manifolds with given fundamental group, e.g. Euler characteristicsand signatures, is an interesting and important topic, see [HW], [JK],[Ko1], [Ec], [Hi], [KL], [KLPT] and the references therein.

For any finitely presented group G, denote βi(G) to be the real coeffi-cient i-th betti number of K(G, 1), the classifying space of G. Denoteχ(W ) to be the Euler characteristic of a finite CW complex W , anddenote σ(X) to be the signature of a closed oriented 4-manifold X .

In 1985 Hausmann and Weinberger [HW] introduced the 4-manifoldEuler characteristic for a finitely presented group G, defined by

χ4(G) = inf{χ(X)|X is a closed orientable 4-manifold and π1(X) ∼= G}.

There are also some variations of χ4(G). In 1994 Kotschick [Ko1] in-troduced

p(G) = inf{χ(X)−|σ(X)||X is a closed orientable 4-manifold and π1(X) ∼= G}.

In 2002 Hillman [Hi] introduced

q∗(G) = inf{χ(X)|X is a closed 4-manifold and π1(X) ∼= G}.

It is known that

χ4(G) ≥ p(G), χ4(G) ≥ q∗(G) (1.0).

Note both χ4(G) and q∗(G) are denote as q(G) in their original def-initions, see [HW] and [Hi]. Moreover, q∗(G) is originally defined forPD4-complexes in [Hi], and Theorem 1.4 is also valid when q∗ is definedfor PD4-complexes, see Remark 7.3 (2).

Kotschick made a useful observation to estimate the lower bound ofχ4(G) and p(G) when β4(G) = 0 [Ko1, Theorems 2.8 and 4.2].

More variations and generalizations of χ4(G) can be found in [Hi],[Ko2], [BK], [KL], and [AH].

An important family of finitely presented groups which can be classifiedare groups of compact 3-manifolds, which include all cyclic groups, freegroups, surface groups and knot groups. Many studies have been madefor 4-manifolds with 3-manifold groups, see [JK], [Ko1], [Hi], [KL],[KLPT] and the references therein.

We survey known results on these invariants of 3-manifold groups inthe following theorem.

EULER CHARACTERISTICS OF 4-MANIFOLDS WITH 3-MANIFOLD GROUPS3

Theorem 1.0 Suppose M is a compact 3-manifold and π1(M) = π.

(1) χ4(π) = p(π) = 2 when M is closed, orientable and aspherical,[Ko1, Proposition 5.6].

(2) χ4(π) = 2 − 2q when M is closed and orientable, where q is themaximal rank of free group in the free product decomposition of π, [Hi,page 61-62], [KL, Theorem 3.3].

(3) χ4(π) = 0, when π is the group of a knot complement in the 3-sphereS3, [Hi, Corollary 3.12.3], [KL, Theorem 3.4].

(4) Suppose M is a closed aspherical 3-manifold and π1(X) = π fora closed 4-manifold X. If X and M have the same orientablity, thenχ(X) > 0 and 1 ≤ q∗(π) ≤ 2, [Hi, Theorem 3.13 and Corollary 3.13.1].

Hillman asked [Hi, page 63] whether results in (4) above can be ex-tended to all closed 3-manifold groups without torsion and withoutfree Z-factors.

In this paper, we try to determine χ4(π) for fundamental groups ofcompact 3-manifolds in general. We also get some results relatedto p(π) and q∗(π). It turns out that, to deal with non-orientable3-manifolds, the problem becomes more difficult and some new ap-proaches are needed.

For the undefined terminologies, see [He] and [Ha2] about 3-manifolds,see [Kir] about 4-manifolds, see [Ha1] about algebraic topology.

Conventions: (1) All homology and cohomology groups have real coef-ficients, and all Betti-numbers are taken with respect to real coefficient(co)homology groups, unless otherwise indicated.

(2) Since each 3-manifold group can be realized by a 3-manifold withoutS2 boundary components, below we assume each 3-manifold containsno S2 boundary component unless otherwise indicated. This assump-tion does not affect the correctness of our results, see Remark 7.3 (3).

The Kneser-Milnor theorem claims that each compact 3-manifold has aprime decomposition, whose prime factors are unique up to homeomor-phism and permutation ([He]). Then M has a prime decomposition

M = (#mi=1Mi)#(#n

j=1Nj)#(#pl=1Ql)#(#q

e=1Se) (1.1).

Here each prime factor may or may not be orientable and belongs toone of the following categories (see Section 2 for more detail):

(i) each Mi is a closed prime 3-manifold with |π1(Mi)| = ∞ and isnot an S2- or RP 2-bundle over S1;

(ii) each Nj is a closed prime 3-manifold with |π1(Nj)| < ∞;


(iii) each Ql is a prime 3-manifold and ∂Ql is non-empty;(iv) each Se is an S2- or RP 2-bundle over S1.

Note in (ii) each Nj is orientable, and in (iv) each RP 2-bundle over S1

is homeomorphic to RP 2×S1. Also each orientable 3-manifold containsno 2-sided RP 2, and each 3-manifold containing RP 2 has 2-torsion inits fundamental group.

We made the following conjecture.

Conjecture 1.1. Suppose M is a compact 3-manifold with prime de-composition described as in (1.1) and (i)-(iv). Let π = π1(M), then

χ4(π) = p(π) = 2− 2(p+ q) + χ(∂M).

If Conjecture 1.1 is true, it implies that any closed orientable 4-manifoldX realizing χ4(π) has zero signature.

We state our works on χ4(π) and related invariants in the followingthree theorems. Theorem 1.2 is more difficult to prove and is themain result of this paper. Theorems 1.2 and 1.3 provide evident forConjecture 1.1.

Theorem 1.2. Suppose M is a compact 3-manifold with prime decom-position described as in (1.1) and (i)-(iv). Let π = π1(M), then

χ4(π) = 2− 2(p+ q) + χ(∂M) (1.2)

if each Mi in (i) and each Ql in (iii) contains no two-sided RP 2.

In particular, (1.2) holds when M contains no two-sided RP 2.


p(π) = χ4(π) = 2− 2(p+ q) + χ(∂M) (1.3)

if each closed 3-manifold Mi in (i) is orientable.

In particular, (1.3) holds when M is orientable.

The third one confirms the question asked by Hillman above, and infact we prove a stronger result.


q∗(π) = χ4(π) = 2− 2(p+ q) + χ(∂M) (1.4)

if π contains no 2-torsions. Furthermore if p = q = 0, then q∗(π) = 2.


Note q∗(Z2) = 1 is realized by RP 4. So the equality q∗(π) = χ4(π) inTheorem 1.4 is not true for 3-manifold groups in general.

Remark 1.5. (1) Among those conditions posed on Theorems 1.2,1.3, 1.4, no one is stronger than another except one case: the conditionposed in Theorem 1.4 is stronger than that posed in Theorems 1.2.

Also note in Theorem 1.2, those Mi in (i) might be non-orientable andSe in (iv) may contain two-sided RP 2; in Theorem 1.3, those Ql in (iii)and Se in (iv) may contain two-sided RP 2.

(2) If M is closed and orientable, then (1.2) in Theorem 1.2 becomesχ4(π) = 2 − 2q, since p = χ(∂M) = 0 in this case. If M is a knotcomplement in S3, then (1.2) becomes χ4(π) = 0, since p = 1 andq = χ(∂M) = 0. These cases are presented in Theorem 1.0 (2) and (3),proved by Hillman and by Kirk-Livingston. If M is closed, orientableand aspherical, then (1.3) in Theorem 1.3 becomes p(π) = χ4(π) = 2,which is presented in Theorem 1.0 (1), proved by Kotschick.

(3) By Theorem 1.4, if π1(M) contains no 2-torsions, then q∗(π) can berealized by closed orientable 4-manifolds. The condition that π1(M)contains no 2-torsions implies that all prime factors of M with finitefundamental groups are lens spaces L(p, q) with odd p.

(4) There are (potentially) different versions of χ4(π) by working inthe categories of topological and smooth 4-manifolds, as discussed in[Ko1, page 91], and we work in the topological category in this paper.In particular, our lower bounds of χ4(π) in Sections 5, 6 and 7 workin the topological category. On the other hand, our construction inSection 4 produces smooth 4-manifolds, which provides upper boundsof χ4(π) in both topological and smooth categories. So if Conjecture1.1 is true, then χ4(π) in the topological category and in the smoothcategory will be same for 3-manifold groups.

The content of the paper is as following: In Section 2, we will studydecompositions of compact 3-manifolds by S2 and two-sided RP 2. InSection 3, we will compute the second and fourth Betti numbers ofcompact 3-manifold groups. In Section 4, Proposition 4.1 provides theupper bound of χ4(π). Theorems 1.2, 1.3 and 1.4 are proved in Sections5, 6 and 7 respectively.

2. Decomposition of 3-manifolds by S2 and two-sided RP 2

In this section, we first review the prime decomposition of 3-manifoldsand a decomposition theorem of non-orientable prime 3-manifolds by


two-sided RP 2, then we study the topology of resulting pieces. Theresult for non-orientable prime 3-manifolds should be well-known, butwe can not find it in literature.

In this paper, we assume all 3-manifolds are compact and connected.The Kneser-Milnor decomposition (prime decomposition) of compact3-manifolds is a fundamental result in 3-manifold topology. Althoughthe result might be more well-known for orientable 3-manifolds, it alsoholds for non-orientable 3-manifolds (see Theorem 3.15 of [He]).

Theorem 2.1. Let M be a compact connected 3-manifold, then M isa connected sum of finitely many prime 3-manifolds.

The following description of orientable prime 3-manifolds is well known:Any orientable prime 3-manifold either has finite fundamental group,or is homeomorphic to S2×S1, or is aspherical (by the sphere theorem).

However, we can not find such a description of non-orientable prime3-manifolds in literature, so we will prove a similar result here.

At first, the only prime but reducible non-orientable 3-manifold isthe twisted S2-bundle over S1. Since any irreducible non-orientable3-manifold does not contain one-sided RP 2, we will decompose suchmanifolds by two-sided RP 2. The following theorem can be proved bythe theory of normal surfaces, as the proof of the existence of primedecompositon, which appears as Exercise 1.4 of [Ha2].

Theorem 2.2. Let M be a non-orientable irreducible 3-manifold, thenthere is a finite collection P of embedded two-sided RP 2 in M , suchthat for any component N of M \ P, any embedded two-sided RP 2 inN is parallel to its boundary.

For any component N of M \ P in the above decomposition, we provethe following lemma.

Lemma 2.3. Suppose N is a non-orientable 3 manifold satisfying

(i) any 2-sphere in N bounds a 3-ball;(ii) any projective plane in N is boundary parallel.

Let N → N be the orientable double cover of N , and let N = N∪(∪B3i ),

where each S2 boundary component of N is capped with a 3-ball. ThenN is irreducible.

Proof. Let τ : N → N be the nontrivial deck transformation of thedouble cover p : N → N , and let τ : N → N be an extension of τ to N


as a Z2-action. Then each 3-ball in the decomposition N = N ∪ (∪B3i )

gives a unique fixed point of τ .

We prove the statement by contradiction and suppose that N is not ir-reducible. Since τ acts on N , by Meeks-Simon-Yau’s equivariant spheretheorem [MSY], there exists a τ -invariant essential 2-sphere or a τ -

invariant pair of essential 2-spheres S ⊂ N . Then there are two cases:(1) S ∩ Fix(τ ) = ∅ or (2) S ∩ Fix(τ ) 6= ∅.

Case (1): Suppose first S ∩ Fix(τ ) = ∅. Then we may assume thatS ⊂ N . Clearly, S is τ -equivariant and essential in N . There aretwo cases: p(S) is a 2-sphere (if S is a pair of 2-spheres) or p(S) isa projective plane (if S is a 2-sphere). If p(S) is a 2-sphere, thenit contradicts with assumption (i). If p(S) is a projective plane, byassumption (ii), p(S) must be boundary parallel. Thus, S is boundary

parallel in N , which cannot be essential in N .

Case (2): Then we suppose S ∩ Fix(τ ) 6= ∅. We first prove that τ |Sis orientation preserving. Otherwise, τ |S is orientation reversing andis not fixed-point free. Then Fix(τ ) ∩ S is one-dimensional, which isimpossible. Let U be a τ -equivariant regular neighborhood of S suchthat Fix(τ ) ∩ (U \ S) = ∅. Since both S and N are orientable, ∂Uhas two 2-sphere components, say S+ and S−. Since τ is orientationreversing on N and is orientation preserving on S, it reverses the normalvector field of S. Thus, τ exchanges two components of U \ S, andτ(S+) = S−, so τ(S+) ∩ S+ = ∅ holds. Let S± = S+ ∪ S−, then wehave that S± ∩ Fix(τ ) = ∅. Clearly S± is also essential. Applying theargument of case (1) to S±, we reach a contradiction. �

3. Betti numbers of 3-manifold groups

In this section, we prove the following result on Betti numbers of 3-manifold groups.

Proposition 3.1. Let M be a compact 3-manifold, and let its primedecomposition be

M = (#mi=1Mi)#(#n


e=1Se)

as in (1.1). Here each Mi is a closed prime 3-manifold with |π(Mi)| =∞ and is not an S2- or RP 2-bundle, each Nj is closed and has finitefundamental group, each Ql is prime and has non-empty ∂Ql, each Se

is an S2- or RP 2-bundle over S1. Then we have

(1) β4(K(π, 1)) = 0;


(2) β2(K(π, 1)) = Σmi=1β2(Mi) + Σp

l=1β2(Ql).

To prove Proposition 3.1, we will first work on each prime factor ofM and prove the following proposition. It turns out that it is moredifficult to prove this proposition for non-orientable 3-manifolds.

Proposition 3.2. Let M be a prime 3-manifold that is not an S2- orRP 2-bundle over S1, and let π = π1(M). Then we have

(1) β4(K(π, 1)) = 0;(2) β2(K(π, 1)) = β2(M).

Proof. Case I. M is an orientable 3-manifold.

If M has finite fundamental group, since no boundary component ofM is S2, M must be closed. Then the result holds since β2(M) = 0(by Poincare duality) and any finite group has zero betti numbers inall positive dimensions (for example, see Theorem 6.5.8 of [We]).

If M has infinite fundamental group, since M is not an S2-bundle, itis an irreducible 3-manifold. By the sphere theorem and the Hurewicztheorem, M is aspherical and it is a model ofK(π, 1). So β4(K(π, 1)) =β4(M) = 0 and β2(K(π, 1)) = β2(M).

Case II. M is a non-orientable 3-manifold.

If M has finite fundamental group, let M be the orientable doublecover of M , and let M be obtained by capping off all S2 boundarycomponents of M . Since π1(M) is finite, M must be a closed man-

ifold with H2(M) = 0. Let S1, · · · , Sk be all oriented S2 boundary

components of M , then H2(M) is generated by [S1], · · · , [Sk], modulorelation [S1] + · · ·+ [Sk] = 0. Let τ : M → M be the nontrivial decktransformation, since no boundary component of M is S2, we haveτ∗[Si] = −[Si] for each i. Since τ is an orientation reversing fixed-pointfree homeomorphism on M , by the Lefschetz fixed-point theorem, we

must have k = 2. So χ(M) = χ(∂M )2

= 2 and χ(M) = χ(M)2

= 1. Thenwe have b2(M) = 0, and the result holds as in Case I.

If M does not contain any embedded RP 2 and has infinite π1, thenby Lemma 2.3, the orientable double cover M of M is irreducible. Soboth M and M are aspherical, thus β4(K(π, 1)) = β4(M) = 0 andβ2(K(π, 1)) = β2(M) hold.

Now we assume that M contains embedded RP 2 and has infinite fun-damental group. We can also assume that all embedded RP 2 in M are


two-sided. Otherwise, if M contains a one-sided RP 2, the primeness ofM implies that M is homeomorphic to RP 3, which is orientable.

So it remains to consider the following two cases.

Case II.1. Any embedded RP 2 in M is boundary parallel, thus Msatisfies the assumption of Lemma 2.3. We denote this manifold by Nso that we can use the notations in Lemma 2.3.

Let the RP 2-boundary components of N be denoted by ∪ki=1RP 2

i . Notethat all of these RP 2’s are π1-injective in N . For each RP 2

i , we attachan RP∞

i to N via the standard embedding RP 2i → RP∞

i (to the 2-skeleton), and get

K = N ∪∪k

i=1RP 2

i

(∪ki=1RP∞

i ).

We claim that K is a model of K(π1(N), 1).

By Van Kampen theorem, π1(K) is isomorphic to π1(N), so we onlyneed to prove that higher homotopy groups of K are trivial. We usenotations N and N as in Lemma 2.3, which are the orientable doublecover of N and the capping off of N , respectively. Then the doublecover K of K corresponding to π1(N) < π1(N) is homeomorphic to

N ∪∪k

i=1S2

i

(∪ki=1S

∞

i ),

where each sphere boundary component S2i of N is pasted with an S∞

i

via the standard embedding S2i → S∞

i (to the 2-skeleton). We can

rewrite K asN ∪∪k

i=1B3

i

(∪ki=1S

∞

i ),

where each B3i in N (used to cap off N) is pasted with S∞

i via anembedding B3

i → S∞

i . Since both S∞ and B3 are contractible, K is

homotopic equivalent to N . By Lemma 2.3, N is irreducible. Sinceπ1(N) is infinite, N is an aspherical manifold. So both K and K areaspherical, thus K is a model of K(π1(N), 1).

Then we use the following M-V sequence to compute the second andfourth betti numbers of K:

· · · → ⊕ki=1Hn(RP 2

i ) → Hn(N)⊕ (⊕ki=1Hn(RP∞

i )) → Hn(K) → · · · .

Recall that all homology groups have real coefficients. Since βn(RP 2) =βn(RP∞) = 0 for all n > 0, we have βn(π1(N)) = βn(K) = βn(N) forall n ≥ 1. In particular, β4(π1(N)) = 0 and β2(π1(N)) = β2(N) hold.

Case II.2. M contains embedded two-sided RP 2 that is not boundaryparallel. By Theorem 2.2, there is a finite collection P of two-sidedRP 2’s in M , such that any RP 2 in M \ P is boundary parallel.


Then M has a space of graph structure and π1(M) has a space of groupstructure. Under this graph of space structure of M , the vertex spacesare components of M \ P, denoted by N1, · · · , Nk; the edge spaces arecomponents of P, denoted by RP 2

1 , · · · , RP 2l .

Then we can construct a K(π1(M), 1) space as following. For eachvertex space Ni, we take a K(π1(Ni), 1) space of π1(Ni). For eachedge space RP 2

j , we take a copy of RP∞ and denote it by RP∞

j . For

any RP 2j , it is adjacent to two vertex spaces Nj1, Nj2 of M . Then we

take maps RP∞

j → K(π1(Nj1), 1) and RP∞

j → K(π1(Nj2), 1) induced

by π1(RP 2j ) → π1(Nj1) and π1(RP 2

j ) → π1(Nj2) respectively. Thesepasting maps between K(G, 1)’s together give rise to a space K. ByTheorem 1B.11 of [Ha1], K is a K(π1(M), 1) space.

So we can use the following M-V sequences to compute betti numbersof M and K respectively:

· · · → ⊕lj=1Hn(RP 2

j ) → ⊕ki=1Hn(Ni) → Hn(M) → · · · ,

· · · → ⊕lj=1Hn(RP∞

j ) → ⊕ki=1Hn(K(π1(Ni), 1)) → Hn(K) → · · · .

We have the fact that βn(RP 2) = βn(RP∞) = 0 for all n > 0. Theresults in Case II.1 and the finite fundamental group case imply thatβn(Ni) = βn(K(π1(Ni), 1)) for all n. So for any n ≥ 2, we have

βn(π1(M)) = βn(K) =

k∑

i=1

βn(K(π1(Ni), 1)) =

k∑

i=1

βn(Ni) = βn(M).

In particular, β4(π1(M)) = 0 and β2(π1(M)) = β2(M) hold. �

Now we are ready to prove Proposition 3.1.

Proof of Proposition 3.1. Suppose the 3-manifold M has prime decom-position

M = (#mi=1Mi)#(#n


e=1Se)

as in (1.1). Here each Mi is a closed prime 3-manifold with |π1(Mi)| =∞ and is not an S2- or RP 2-bundle, each Nj is closed and has finitefundamental group, each Ql is prime and has non-empty ∂Ql, each Se

is an S2- or RP 2-bundle over S1.

Among the q prime factors of M that are S2- or RP 2-bundles, wesuppose q1 of them are S2-bundles and q2 of them are RP 2-bundles.Then a K(π1(M), 1) space can be taken to be

(∨mi=1K(π1(Mi), 1)) ∨ (∨n

j=1K(π1(Nj), 1)) ∨ (∨pl=1K(π1(Ql), 1))∨

(∨q1S1) ∨ (∨q2RP∞ × S1).


So we have

β4(π1(M)) =m∑

i=1

β4(π1(Mi)) +n

∑

j=1

β4(π1(Nj)) +

p∑

l=1

β4(π1(Ql)) = 0.

Here the last equality follows from Proposition 3.2 (1). We also have

β2(π1(M)) =

m∑

i=1

β2(π1(Mi)) +

n∑

j=1

β2(π1(Nj)) +

p∑

l=1

β2(π1(Ql))

=m∑

i=1

β2(Mi) +n

∑

j=1

β2(Nj) +

p∑

l=1

β2(Ql) =m∑

i=1

β2(Mi) +

p∑

l=1

β2(Ql).

Here the second equality follows from Proposition 3.2 (2) and the thirdequality follows from the fact that π1(Nj) is finite. �

Corollary 3.3. Let M be a compact 3-manifold whose fundamentalgroup π contains no 2-torsions. Then

(1) β4(K(π, 1), Z2) = 0;

(2) β2(K(π, 1), Z2) = Σmi=1β2(Mi, Z2) + Σp

l=1β2(Ql, Z2).

Proof. Still we have

M = (#mi=1Mi)#(#n


e=1Se)

as in (1.1) (i)-(iv). Since π contains no 2-torsions, we conclude thateach Mi and Qi contains no 2-sided projective plane, each Se is an S2-bundle over S1, and the fundamental group of each Nj has odd order.As we discussed in the proof of Proposition 3.2, each Mi and Ql isaspherical and

K(π1(M), 1) = (∨mi=1Mi) ∨ (∨n

j=1K(π1(Nj), 1)) ∨ (∨pl=1Ql) ∨ (∨qS1).

Since π1(Nj) is odd, βk(K(π1(Nj), 1), Z2) = 0 for all k ≥ 1 (for exam-ple, see Theorem 6.5.8 of [We]). Then the conclusion follows. �

4. An upper bound of χ4(π)

We prove the following proposition in this section, which gives theupper bound of χ4(π) in Theorems 1.2, 1.3 and 1.4.

Proposition 4.1. Suppose M is a compact 3-manifold with a primedecomposition described as in (1.1) and (i)-(iv). Let π = π1(M), then

χ4(π) ≤ 2− 2(p+ q) + χ(∂M).


Proof. For a compact 3-manifold M associated with decomposition(1.1) and (i)-(iv), to prove Proposition 4.1, we need only to constructa closed orientable 4-manifold M∗ with π1(M

∗) = π1(M) and

χ(M∗) = 2 + χ(∂M)− 2(p+ q).

Construction of M∗: Suppose M is a compact 3-manifold and ∂M hask components {S1, ..., Sk}.

Case 1: M is orientable and ∂M 6= ∅. Let

M∗ = (M × S1) ∪ (∪iSi ×D2),

where each component Si × S1 of ∂(M × S1) is identified with ∂(Si ×D2) = Si × S1 canonically. By Van Kampen theorem, we can verifythat π1(M

∗) = π1(M).

Case 2: M is non-orientable and ∂M 6= ∅. Let p : M → M bethe orientable double cover of M with a fixed-point free orientationreversing involution τ : M → M such that M/τ = M . Let r : S1 →S1 be the orientation reversing involution on S1. Now we have anorientation preserving fixed-point free involution τ × r : M × S1 →M ×S1. Then M ×S1/τ × r is an orientable 4-manifold, which indeedis a twisted product M×S1. Each boundary component of M×S1 iseither Sj × S1 if Sj is orientable, or Sj×S1 otherwise. Then we closethese components canonically by Sj ×D2 or Sj×D2, depending on Sj

is orientable or not. So we still get a closed orientable 4-manifold M∗

with π1(M∗) = π1(M).

Case 3: M is closed. let B3 be a 3-ball in M , we denote by M =M \ intB3. Then ∂M = S2, and M∗ is defined.

Final Construction: For the compact 3-manifold M with prime decom-position given in (1.1), denote the connected sum of closed 3-manifoldpieces by P = (#m

i=1Mi)#(#nj=1Nj). We suppose that there are q1

prime factors that are S2-bundles and there are q2 prime factors thatare RP 2-bundles, with q1 + q2 = q. Then we define

M∗ = P ∗#(#pl=1Q

∗

l )#(#q1e=1(S

3 × S1))#(#q2f=1(RP 3 × S1)) (4.1).

Then M∗ is a closed orientable 4-manifold, and we check the desiredconditions in the following.

Lemma 4.2.

π1(M∗) ∼= π1(M).


Proof. Since π1(P ) ∼= π1(P ), π1(S3×S1) ∼= π1(S

2×S1) ∼= π1(S2×S1) ∼=

Z, and π1(RP 2 × S1) ∼= π1(RP 3 × S1) ∼= Z⊕ Z2, we have

π1(M∗) ∼= π1(P

∗)∗(∗pi=1π1(Q∗

i ))∗(∗q1e=1π1(S

3 × S1))∗(∗q2f=1π1(RP 3 × S1))

∼= π1(P ) ∗ (∗pi=1π1(Qi)) ∗ (∗q1e=1Z) ∗ (∗

q2f=1Z⊕ Z2) ∼= π1(M).

�

To calculate χ, we list the following elementary facts.

Lemma 4.3. [Ha1, Exercises 20, 21, 22, p157]

(i) If X and Y are finite CW complexes, then χ(X×Y ) = χ(X)×χ(Y ).

(ii) If a finite CW complex X is a union of sub-complexes A and B,then χ(X) = χ(A) + χ(B)− χ(A ∩B).

(iii) If p : M → M is a covering map of degree p, then χ(M) = pχ(M).

We compute the Euler characteristic of M∗ in the following lemma.

Lemma 4.4.

χ(M∗) = 2 + χ(∂M)− 2(p+ q).

Proof. First note by Lemma 4.3 (ii), for any 4-manifolds N1 and N2 wehave χ(N1#N2) = χ(N1) + χ(N2)− 2, and inductively we have

χ(M∗) = χ(P ∗)+Σpl=1χ(Q

∗

l )+q1χ(S3×S1)+q2χ(RP 3×S1)−2(p+q) (4.2).

Since χ(S3 × S1) = χ(RP 3 × S1) = 0 (by Lemma 4.3 (i)), we have

χ(M∗) = χ(P ∗) + Σpl=1χ(Q

∗

l )− 2(p+ q) (4.3).

Let Q be either P or some Ql. If Q is orientable, by Lemma 4.3 (ii) wehave

χ(Q∗) = χ(Q× S1) +∑

χ(Si ×D2)−∑

χ(Si × S1)

= 0 +∑

χ(Si)− 0 =∑

χ(Si) = χ(∂Q).

If Q is non-orientable, and some Si is non-orientable, since Q×S1 andSi×S1 are doubly covered by M × S1 and Si × S1 respectively, westill have χ(Q×S1) = 0 and χ(Si×S1) = 0 by Lemma 4.3 (iii). Sinceχ(Si×D2) = χ(Si), so we still have

χ(Q∗) = χ(∂Q).

Since χ(Q∗) = χ(∂Q), we get χ(P ) = χ(S2) = 2 and Σpl=1χ(∂Ql) =

χ(∂M). By plugging into (4.3), we have

χ(M∗) = 2 + χ(∂M)− 2(p+ q).


In the proof above (especially equation (4.2)), we assumed that P isnot empty. If P is empty, then the −(p + q) term in (4.2) should be−(p+ q − 1), and the result is still valid. Alternatively, if P is empty,we can also take P = S3 and the result follows. �

The proof of Proposition 4.1 is done. �

5. A lower bound for χ4(π)

We first prove the following result on χ4(G) for a general group G.

Proposition 5.1. Let G be a finitely presented group, and let G < Gbe an index-2 subgroup such that b4(G) = 0. Then we have

χ4(G) ≥ 1− b1(G) + b2(G) + |2(b1(G)− b2(G))− (b1(G)− b2(G))− 1|.

Proof. Let X be a closed orientable 4-manifold with π1(X) ∼= G, thenwe have a map i : X → K(G, 1) that induces an isomorphism on π1.

Let p : X → X and q : K(G, 1) → K(G, 1) be the double covers of Xand K(G, 1) corresponding to G < G respectively, we get the followingcommutative diagram:

Xj> K(G, 1)

X

p

∨i> K(G, 1)

q

∨

Since π1(X) ∼= G, we have b1(X) = b1(G). By Poincare duality, wehave b3(X) = b1(X) = b1(G). So we get

χ(X) = 2− 2b1(G) + b2(X), χ(X) = 1− b1(G) +1

2b2(X) (5.1).

Then we need to bound b2(X) from below to prove this proposition.

Since j : X → K(G, 1) induces an isomorphism on π1, j∗ : H1(K(G, 1)) →

H1(X) is an isomorphism. Since K(G, 1) can be obtained by attaching

cells to X with dimension at least 3 (up to homotopy equivalence),j∗ : H2(K(G, 1)) → H2(X) is injective.

Let τX : X → X and τK : K(X, 1) → K(X, 1) be the nontrivial decktransformations of p : X → X and q : K(G, 1) → K(G, 1) respec-

tively. Since q : K(G, 1) → K(G, 1) is a double cover, for each n,q∗ : Hn(K(G, 1)) → Hn(K(G, 1)) is injective, and we have

q∗(Hn(K(G, 1))) = (Hn(K(G, 1)))τ∗

K


(by [Ha1, Proposition 3G.1]). Here (Hn(K(G, 1)))τ∗

K denotes the sub-

space ofHn(K(G, 1)) consisting of elements fixed by τ ∗K : Hn(K(G, 1)) →Hn(K(G, 1)).

Since τ ∗K : Hn(K(G, 1)) → Hn(K(G, 1)) gives a Z2-action, we have

Hn(K(G, 1)) = Hn(K(G, 1))+ ⊕Hn(K(G, 1))−,

where Hn(K(G, 1))+ and Hn(K(G, 1))− denote the eigenspaces of τ ∗Kcorresponding to eigenvalues 1 and −1 respectively. Similarly, by con-sidering the Z2-action on Hn(X) given by τ ∗X and the eigenspaces cor-responding to 1 and −1, we have

Hn(X) = Hn(X)+ ⊕Hn(X)−.

Then we have

Hn(K(G, 1))+ = Hn(K(G, 1))τ∗

K = q∗(Hn(K(G, 1))).

Since q∗ : Hn(K(G, 1)) → Hn(K(G, 1)) is injective, we have

dimHn(K(G, 1))+ = dim q∗(Hn(K(G, 1))) = dimHn(K(G, 1) = bn(G)

and

dimHn(K(G, 1))− = dimHn(K(G, 1))−dimHn(K(G, 1))+ = bn(G)−bn(G).

Since j∗ : H1(K(G, 1)) → H1(X) is an isomorphism, we have

dim H1(X)+ = dim j∗(H1(K(G, 1))+) = dim H1(K(G, 1))+ = b1(G)

and

dimH1(X)− = dim j∗(H1(K(G, 1))−) = dimH1(K(G, 1))− = b1(G)−b1(G).

So we get

tr(τ ∗X : H1(X) → H1(X)) = 2b1(G)− b1(G) (5.2).

Since τX : X → X is an orientation preserving homeomorphism, for anyα ∈ H1(X) and β ∈ H3(X), we have α∪β = τ ∗X(α)∪τ

∗

X(β) ∈ H4(X) ∼=R. By Poincare duality, we have dim H3(X)+ = dim H1(X)+ = b1(G)and dim H3(X)− = dim H1(X)− = b1(G)− b1(G), so we get

tr(τ ∗X : H3(X) → H3(X)) = 2b1(G)− b1(G) (5.3).

By the assumption that β4(G) = 0, the restrictions of the cup productof H∗(K(G, 1)) on H2(K(G, 1))+×H2(K(G, 1))+ and H2(K(G, 1))−×H2(K(G, 1))+ are trivial. InH2, we have j∗(H2(K(G, 1))+) ⊂ H2(X)+


and j∗(H2(K(G, 1))−) ⊂ H2(X)−. So the restrictions of the cup prod-

uct of H∗(X) on j∗(H2(K(G, 1))+)× j∗(H2(K(G, 1))+) andj∗(H2(K(G, 1))−)× j∗(H2(K(G, 1))−) are also trivial.

By Poincare duality, the cup product H2(X)×H2(X) → H4(X) ∼= R

is a non-singular bilinear form.

Now we state two lemmas.

Lemma 5.2. The restrictions of the cup product of H∗(X) on bothH2(X)+ ×H2(X)+ and H2(X)− ×H2(X)− are non-degenerate.

Proof. For any non-zero elements α ∈ H2(X)+ and β ∈ H2(X)−, thenτ ∗X(α) = α and τ ∗X(β) = −β, hence

τ ∗X(α ∪ β) = −α ∪ β.

However since τX is orientation preserving, we have

τ ∗X(α ∪ β) = α ∪ β

which implies that α ∪ β = 0.

Since the cup product H2(X)×H2(X) → H4(X) ∼= R is a non-singular

bilinear form, for any nonzero α ∈ H2(X)+, there must be an non-zeroelement γ ∈ H2(X)+ such that γ ∪α 6= 0. So the restriction of the cup

product of H∗(X) on H2(X)+×H2(X)+ is non-degenerate. The sameargument works for the case H2(X)−. �

The next one is well-known.

Lemma 5.3. Let V be a vector space of dimension n over a field Fand let q be a non-degenerated quadratic form on V . If q vanishes ona sub-space W of dimension m, then n ≥ 2m.

Proof. Choose a basis of v1, ..., vm of the W , then extend it to a basisv1, ..., vm, vm+1, ..., vn of V , then q has representative matrix

A =

(

0m×m Bm×(n−m)

BT C

)

n×n

For an n× n matrix with nonzero determinant, if its upper left m×msub-matrix is 0, then n ≥ 2m must hold. �

Since j∗ : H2(K(G, 1)) → H2(X) is injective, by Lemma 5.2 andLemma 5.3 we have

dimH2(X)+ ≥ 2 dim j∗(H2(K(G, 1))+) = 2 dimH2(K(G, 1))+ = 2b2(G)


and

dim H2(X)− ≥ 2 dim H2(K(G, 1))− = 2b2(G)− 2b2(G).

The above inequalities imply that

dimH2(X)+ = 2b2(G)+∆+, dimH2(X)− = 2b2(G)−2b2(G)+∆− (5.4)

for some non-negative integers ∆+,∆−. So we have

tr(τ ∗X : H2(X) → H2(X)) = 4b2(G)− 2b2(G) + (∆+ −∆−) (5.5).

Since τX : X → X has no fixed-point, by the Lefschetz fixed-pointtheorem and equations (5.2), (5.3), (5.5), we have

0 =

4∑

i=0

(−1)i · tr(τ ∗X : H i(X) → H i(X))

= 1− (2b1(G)− b1(G)) + (4b2(G)− 2b2(G) + (∆+ −∆−))− (2b1(G)− b1(G)) + 1

= 2− 4(b1(G)− b2(G)) + 2(b1(G)− b2(G)) + (∆+ −∆−).

Thus

∆+ −∆− = 4(b1(G)− b2(G))− 2(b1(G)− b2(G))− 2 (5.6).

By equation (5.1), we have

χ(X) = 1− b1(G) +1

2(dim H2(X)+ + dim H2(X)−)

= 1− b1(G) + b2(G) +1

2(∆+ +∆−) by (5.4)

≥ 1− b1(G) + b2(G) +1

2|∆+ −∆−|

= 1− b1(G) + b2(G) + |2(b1(G)− b2(G))− (b1(G)− b2(G))− 1| by (5.6).

�

Now we are ready to prove the lower bound of χ4(π) in Theorem 1.2.The case of orientable 3-manifolds will be covered in Section 6, so itsuffices to prove the non-orientable 3-manifold case.

Proposition 5.4. Suppose M is a non-orientable compact 3-manifoldwith prime decomposition described as in (1.1) and (i)-(iv). Let p andq be the number of prime factors of M in (iii) and (iv) respectively. Ifeach Mi in (i) and each Ql in (iii) contains no two-sided RP 2, then

χ4(π1(M)) ≥ 2− 2(p+ q) + χ(∂M).


Proof. We need a finer description of the prime decomposition of M :

M = (#ah=1M

co∞h )#(#b

i=1Mcn∞i )#(#c

j=1Mcfj )#(#d

k=1Mbok )#

(#el=1M

bnl )#(#f

m=1MS2

×S1

m )#(#gn=1M

nfn ).

Here the summands satisfy the following conditions:

(1) Each M co∞h is a closed orientable prime 3-manifold with |π1| =

∞ and is not an S2-bundle.(2) Each M cn∞

i is a closed non-orientable prime 3-manifold with|π1| = ∞ and is not an S2- or RP 2-bundle.

(3) Each M cfj is a closed prime 3-manifold with |π1| < ∞ (which

must be orientable).(4) Each M bo

k is an orientable prime 3-manifold with boundary.(5) Each M bn

l is a non-orientable prime 3-manifold with boundary.

(6) Each MS2×S1

m is homeomorphic to S2 × S1.(7) Each Mnf

n is a non-orientable S2- or RP 2-bundle over S1.

Moreover, each M cn∞i in (2) and each M bn

l in (5) contains no two-sidedRP 2. Since M is non-orientable, we must have b+ e+ g ≥ 1. We alsohave

d+ e = p, f + g = q (5.7).

Let G = π1(M). Since b1(Mcfj ) = 0 and b1(M

S2×S1

m ) = b1(Mnfn ) = 1,

we have

b1(G) =a

∑

h=1

b1(Mco∞h )+

b∑

i=1

b1(Mcn∞i )+

d∑

k=1

b1(Mbok )+

e∑

l=1

b1(Mbnl )+f+g (5.8).

By Proposition 3.1 (2), we have

b2(G) =a

∑

h=1

b2(Mco∞h )+

b∑

i=1

b2(Mcn∞i )+

d∑

k=1

b2(Mbok )+

e∑

l=1

b2(Mbnl ) (5.9).

Let M be the orientable double cover of M , then we have

M = (#ah=1M

co∞h )#2#(#b

i=1Mcn∞i )#(#c

j=1Mcfj )#2#(#d

k=1Mbok )#2#

(#el=1M

bnl )#(#f

m=1MS2

×S1

m )#2#(#gn=1M

nfn )#(#b+e+g−1S2 × S1).

For any involved manifold N in the above, N#2 denotes N#N . Foreach involved non-orientable N , N denotes its orientable double cover,then N must be prime since either N contains no two-sided RP 2 (by

Lemma 2.3) or N = S2×S1. When taking the orientable double coverM of M , each orientable prime summand lifts to two copies of the samesummand, and each non-orientable prime summand lifts to one copy


of its orientable double cover. There are (a+ b+ c+ d+ e+ f + g− 1)decomposition spheres in the prime decomposition of M , and they lifttwo 2(a + b + c + d + e + f + g − 1) decomposition spheres in M . In

the above expression of M , other than the last a few terms of S2 ×S1,there are (2a + b+ 2c+ 2d+ e+ 2f + g) pieces, so there must be

2(a+b+c+d+e+f+g−1)−(2a+b+2c+2d+e+2f+g−1) = b+e+g−1

extra copies of S2 × S1 in the prime decomposition of M .

Let G = π1(M), then it is an index-2 subgroup of G, and β4(G) = 0holds by Proposition 3.1 (1). Then we have

b1(G) = 2a

∑

h=1

b1(Mco∞h ) +

b∑

i=1

b1(Mcn∞i ) + 2

d∑

k=1

b1(Mbok )

+e

∑

l=1

b1(Mbnl ) + b+ e + 2f + 2g − 1 (5.10).

By Proposition 3.1 (2), we have

b2(G) = 2

a∑

h=1

b2(Mco∞h )+

b∑

i=1

b2(Mcn∞i )+2

d∑

k=1

b2(Mbok )+

e∑

l=1

b2(Mbnl ) (5.11).

For any compact connected 3-manifold N , we denote b(N) = b2(N)−b1(N). Since χ(N) = b0(N)− b1(N) + b2(N)− b3(N), we have

b(N) = b2(N)− b1(N) = χ(N)− b0(N) + b3(N) =χ(∂N)

2− 1+ b3(N).

Then b(N) can be computed as in the following cases:

• If N is closed orientable, then b(N) = 0.• If N is closed non-orientable, then b(N) = −1.

• If N has boundary, then b(N) = χ(∂N)2

− 1.

We also have

χ(∂M) =

d∑

k=1

χ(∂M bok ) +

e∑

l=1

χ(∂M bne ),

χ(∂M) = 2d

∑

k=1

χ(∂M bok ) +

e∑

l=1

χ(∂M bne ) (5.12).


Now we apply Proposition 5.1 to get

χ4(G) ≥ 1− b1(G) + b2(G) +∣

∣

∣2(b1(G) − b2(G)) − (b1(G)− b2(G))− 1

∣

∣

∣

= 1− (2a

∑

h=1

b1(Mco∞

h) +

b∑

i=1

b1(Mcn∞

i) + 2

d∑

k=1

b1(Mbo

k) +

e∑

l=1

b1(Mbn

l) + b+ e+ 2f + 2g − 1)

+ (2a

∑

h=1

b2(Mco∞

h) +

b∑

i=1

b2(Mcn∞

i) + 2

d∑

k=1

b2(Mbo

k) +

e∑

l=1

b2(Mbn

l))

+∣

∣

∣2(

a∑

h=1

b1(Mco∞

h) +

b∑

i=1

b1(Mcn∞

i) +

d∑

k=1

b1(Mbo

k) +

e∑

l=1

b1(Mbn

l) + f + g)

− 2(

a∑

h=1

b2(Mco∞

h ) +

b∑

i=1

b2(Mcn∞

i ) +

d∑

k=1

b2(Mbo

k ) +

e∑

l=1

b2(Mbn

l ))

− (2

a∑

h=1

b1(Mco∞

h ) +

b∑

i=1

b1(Mcn∞

i ) + 2

d∑

k=1

b1(Mbo

k ) +

e∑

l=1

b1(Mbn

l ) + b+ e+ 2f + 2g − 1)

+ (2

a∑

h=1

b2(Mco∞

h ) +

b∑

i=1

b2(Mcn∞

i ) + 2

d∑

k=1

b2(Mbo

k ) +

e∑

l=1

b2(Mbn

l ))− 1∣

∣

∣

= 1 + 2a

∑

h=1

b(M co∞

h) +

b∑

i=1

b(M cn∞

i) + 2

d∑

k=1

b(M bo

k) +

e∑

l=1

b(M bn

l)− (b+ e+ 2f + 2g − 1)

+∣

∣

∣− 2

a∑

h=1

b(M co∞

h)− 2

b∑

i=1

b(M cn∞

i)− 2

d∑

k=1

b(M bo

k)− 2

e∑

l=1

b(M bn

l)

+ 2

a∑

h=1

b(M co∞

h ) +

b∑

i=1

b(M cn∞

i ) + 2

d∑

k=1

b(M bo

k ) +

e∑

l=1

b(M bn

l )− b− e

∣

∣

∣

= 1 + 2

a∑

h=1

0 +

b∑

i=1

0 + 2

d∑

k=1

(χ(∂M bo

k)

2− 1) +

e∑

l=1

(χ(∂M bn

l)

2− 1)− (b+ e+ 2f + 2g − 1)

+∣

∣

∣− 2

a∑

h=1

0− 2

b∑

i=1

(−1)− 2

d∑

k=1

(χ(∂M bo

k)

2− 1)− 2

e∑

l=1

(χ(∂M bn

l)

2− 1)

+ 2a

∑

h=1

0 +b

∑

i=1

0 + 2d

∑

k=1

(χ(∂M bo

k)

2− 1) +

e∑

l=1

(χ(∂M bn

l)

2− 1)− b− e

∣

∣

∣

= 2 +χ(∂M)

2− (b+ 2d+ 2e+ 2f + 2g) +

∣

∣

∣− χ(∂M) +

χ(∂M)

2+ b

∣

∣

∣

= 2 + χ(∂M)− (b+ 2d+ 2e+ 2f + 2g) + |b|

= 2 + χ(∂M)− 2(d+ e)− 2(f + g)

= 2 + χ(∂M)− 2(p+ q).

Here the second equation follows from equations (5.8), (5.9), (5.10),(5.11). The third equation follows from the definition of b(N). The


fourth equation follows from the computations of b(N). The fifth equa-tion follows from equation (5.12). The last equation follows from equa-tion (5.7). �

Remark 5.5. In the proof of Proposition 5.4, if we allow some Mi

in (i) or some Ql in (iii) contains two-sided RP 2, then we also geta lower bound of χ4(π), which depends on the number of two-sidedRP 2 and does not match the upper bound of χ4(π) in Section 4. Thecomputation in this case is more complicated and we omit it.

Proof of Theorem 1.2. Let M be a compact 3-manifold with prime de-composition described as in (1.1) and (i)-(iv), such that any primefactor that is not an RP 2-bundle does not contain 2-sided RP 2. Thenwe have

2− 2(p+ q) + χ(∂M) ≤ χ4(M) ≤ 2− (p+ q) + χ(∂M),

and the result follows. Here the first and second inequalities followfrom Propositions 5.4 and 4.1 respectively. �

6. p(π) = χ4(π) for 3-manifolds without closed

non-orientable prime factor

The proof of Theorem 1.3 follows from Proposition 4.1, Proposition 3.1and the next two propositions.

Proposition 6.1. [Ko1, Theorem 4.2] If β4(π) = 0, then

p(π) ≥ 2− 2β1(π) + 2β2(π).

Proposition 6.2. SupposeM is a compact 3-manifold andM = P#Q#Ssatisfying the following.

(i) P = (#mi=1Mi)#(#n

j=1Nj), such that each Mi is a closed prime 3-

manifold with |π1(Mi)| = ∞ and not an S2- or RP 2-bundle overS1, and the number of non-orientable Mi is k ≥ 0; each Nj is aclosed prime 3-manifold with finite fundamental group;

(ii) Q = #pl=1Ql and each Ql is a prime 3-manifold with non-empty

∂Ql;(iii) S = #q

e=1Se and each Si is homeomorphic to an S2- or RP 2

bundle over S1.

Let π = π1(M), then

2− 2β1(π) + 2β2(π) = 2− 2(k + p+ q) + χ(∂M).


Proof. Since each Nj is closed and orientable, and the number of non-orientable Mi is k ≥ 0, the number of non-orientable prime factor in Pis k ≥ 0. By M-V sequence we have

β2(P ) = Σmi=1β2(Mi) + δk, (6.1)

where δk is 0 if k = 0, and is k − 1 if k > 0.

So rewrite Proposition 3.1(2)

β2(K(π, 1)) = Σmi=1β2(Mi) + Σp

l=1β2(Ql).

as the following formula

β2(K(π, 1)) + δk = β2(P ) + Σpl=1β2(Ql) (6.2).

Still by Euler’s formula, for any compact 3-manifold Y , we have

χ(Y ) = 1− β1(Y ) + β2(Y )− β3(Y ) (6.3).

When a 3-manifold Y has non-empty boundary, we have β3(Y ) = 0and 2χ(Y ) = χ(∂Y ). Applying these facts to Ql and (6.3), we canrewrite (6.3) as

χ(∂Ql) = 2− 2β1(Ql) + 2β2(Ql) (6.4).

When Y is a closed 3-manifold, we have χ(Y ) = 0, and moreover0 ≤ β3(Y ) ≤ 1 (which equals 1 if and only if Y is orientable). Applyingthese facts to P and (6.4), we can rewrite (6.3) as

1− β1(P ) + β2(P )− β3(P ) = 0 (6.5).

Note that M is the connected sum of P and Qi and Si, so

β1(π) = β1(M) = β1(P ) + Σpl=1β1(Ql) + Σq

e=1β1(Se)

= β1(P ) + Σpl=1β1(Ql) + q (6.6).


Now we have,

2− 2β1(π) + 2β2(π)

= 2− 2(β1(P ) + Σpl=1β1(Ql) + q) + 2β2(π)

=− 2q + (2− 2β1(P ))− Σpl=12β1(Ql) + 2β2(π)

=− 2q + (2β3(P )− 2β2(P )) + Σpl=1(χ(∂Ql)− 2− 2β2(Ql)) + 2β2(π)

=− 2(p+ q) + 2β3(P )− 2β2(P )− 2Σpl=1β2(Ql) + Σp

l=1χ(∂Ql) + 2β2(π)

=− 2(p+ q) + 2β3(P )− 2β2(P )− 2Σpl=1β2(Ql) + χ(∂M) + 2β2(π)

=− 2(p+ q) + χ(∂M) + 2β3(P )− 2(β2(P ) + Σpl=1β2(Ql)) + 2β2(π)

=− 2(p+ q) + χ(∂M) + 2β3(P )− 2(β2(K(π1, 1)) + δk) + 2β2(π)

=− 2(p+ q) + χ(∂M) + 2(β3(P )− δk).

Here the first equation follows from equation (6.6), the third equationfollows from equations (6.4) and (6.5), the fifth equation follows fromΣp

l=1χ(∂Ql) = χ(∂M), and the seventh equation follows from (6.2).

If k = 0, then β3(P ) = 1 (since P is orientable) and δk = 0; If k > 0,then β3(P ) = 0 (since P is non-orientable), and δk = k− 1. So in bothcases we have β3(P )− δk = 1− k. Then we have

2− 2β1(π) + 2β2(π) = 2− 2(k + p+ q) + χ(∂M) (6.7).

�

Proof of Theorem 1.3. Suppose M is a compact 3-manifold with primedecomposition described as in (1.1) and (i)-(iv), and the number ofnon-orientable Mi is k ≥ 0. Let π = π1(M). Then we have

2− 2(p+ q) + χ(∂M) ≥ χ4(π) ≥ p(π)

≥ 2− 2β1(π) + 2β2(π) = 2− 2(k + p + q) + χ(∂M).

Here the first inequality follows from Proposition 4.1, the second in-equality follows from (1.0), the third inequality follows from from Propo-sition 3.1 and Proposition 6.1, and the last equality follows from Propo-sition 6.2.

When k = 0, we get

χ4(π) = p(π) = 2− 2(p+ q) + χ(∂M).

�


7. q∗(π) = χ4(π) for 3-manifold groups containing no

2-torsions

In this section, we work on (co)homology groups and betti numberswith Z2-coefficients. The proof of Theorem 1.4 follows from Proposition4.1, Corollary 3.3 and the next two propositions.

Proposition 7.1. Let M be a compact 3-manifold whose fundamentalgroup π containing no 2-torsions, and let X be a closed 4-manifold withπ1(X) = π. Then we have

β2(X,Z2) ≥ 2β2(K(π, 1), Z2).

Proof. Since M is a compact 3-manifold whose fundamental group πcontains no 2-torsions, H4(K(π, 1);Z2) is trivial according to Corollary3.3 (1). So the cup product

H2(K(π, 1), Z2)×H2(K(π, 1), Z2) → H4(K(π, 1), Z2)

is trivial. Denote m = β2(K(π, 1), Z2) and n = β2(X,Z2).

Since X is a 4-manifold with π1(X) ∼= π1(M), we can construct aK(π, 1) from X by adding n-cells with n ≥ 3. So we have an inclu-sion i : X → K(π, 1) such that the induced map i∗ : H2(X,Z2) →H2(K(π, 1), Z2) is surjective. Then by algebraic duality, the inducedmap i∗ : H2(K(π, 1), Z2) → H2(X,Z2) is injective.

Then i∗(H2(K(π, 1), Z2)) ⊂ H2(X,Z2) is an m-dimensional subspaceof an n-dimensional space, and the restriction of the cup product ofH∗(X,Z2) on this subspace is trivial. By Poincare duality,

H2(X,Z2)×H2(X,Z2) → H4(X,Z2) ∼= Z2

is a non-singular bilinear form.

By Lemma 5.3, we must have β2(X,Z2) ≥ 2β2(K(π, 1), Z2). �

Suppose M is a compact 3-manifold such that π1(M) contains no 2-torsions. Then we can rewrite M as

M = P#Q#S (7.0)

satisfying the following.

(i) P = (#mi=1Mi)#(#n

j=1Nj), eachMi is a closed aspherical 3-manifold,and each Nj is a lens space with odd order group;

(ii) Q = #pl=1Ql, each Ql is an aspherical 3-manifold with ∂Ql 6= ∅;

(iii) S = #qe=1Se, each Si is homeomorphic to an S2-bundle over S1.


Proposition 7.2. Suppose M is a compact 3-manifold such that π1(M)contains no 2-torsions, and M = P#Q#S satisfying (i)-(iii). Letπ = π1(M), then

q∗(π) ≥ 2− 2(p+ q) + χ(∂M).

Proof. Suppose X is a closed 4-manifold such that π1(X) ∼= π1(M).

By M-V sequence we have

β2(P, Z2) = Σmi=1β2(Mi, Z2), (7.1)

So rewrite Corollary 3.3 (2)

β2(K(π, 1), Z2) = Σmi=1β2(Mi, Z2) + Σp

l=1β2(Ql, Z2).

as the following formula

β2(K(π, 1), Z2) = β2(P, Z2) + Σpj=1β2(Qj , Z2) (7.2).

The formulas (6.3)–(6.6) still hold if we replace R-coefficient by Z2-coefficient. In the remainder of this proof, we always use Z2-coefficientand we omit Z2. We will still call those formulas by (6.3)–(6.6).

Since π1(X) = π1(M), we have β1(X) = β1(M) and

χ(X) = 2− 2β1(X) + β2(X) = 2− 2β1(M) + β2(X) (7.3).

We will prove

χ(X) ≥ 2− 2(p+ q) + χ(∂M).

Now we have

χ(X) = 2− 2(β1(P ) + Σpl=1β1(Ql) + q) + β2(X)

= −2q + (2− 2β1(P ))− Σpl=12β1(Ql) + β2(X)

= −2q + (2− 2β2(P )) + Σpl=1(χ(∂Ql)− 2− 2β2(Ql)) + β2(X)

= −2(p + q) + 2− 2β2(P )− 2Σpl=1β2(Ql) + Σp

l=1χ(∂Ql) + β2(X)

= −2(p + q) + 2− 2β2(P )− 2Σpl=1β2(Ql) + χ(∂M) + β2(X)

= −2(p + q) + χ(∂M) + 2− 2(β2(P ) + Σpl=1β2(Ql)) + β2(X)

= −2(p + q) + χ(∂M) + 2− 2β2(K(π1, 1) + β2(X)

= −2(p + q) + χ(∂M) + 2 + (β2(X)− 2β2(K(π1, 1))).

Here the first equation follows from equations (7.3) and (6.6), the thirdequation follows from equations (6.4) and (6.5), the fifth equation fol-lows from Σp

i=1χ(∂Qi) = χ(∂M), and the seventh equation follows from(7.2).


By Proposition 7.1, β2(X) ≥ 2β2(K(π, 1)) holds. So we have

χ(X) ≥ 2− 2(q + p) + χ(∂M).

We finished the proof of Proposition 7.2. �

Proof of Theorem 1.4. Suppose M is a compact 3-manifold with primedecomposition described as in (1.1) and (i)-(iv) and π = π1(M) con-tains no 2-torsion. Then we have

2− 2(p+ q) + χ(∂M) ≥ χ4(π) ≥ q∗(π) ≥ 2− 2(p+ q) + χ(∂M).

Here the first inequality follows from Proposition 4.1, the second in-equality follows from (1.0), and the third inequality follows from fromProposition 7.2. So we get

χ4(π) = q∗(π) = 2− 2(p+ q) + χ(∂M).

. �

Remark 7.3. (1) The proofs of Proposition 6.2 and Proposition 7.2look similar, but different in fact: Firstly, Proposition 6.2 proves anequality and Proposition 7.2 proves an inequality. More substantially,equation (7.1) claims the additivity of Z2-coefficient β2 under connectedsum, and equation (6.1) claims that this not true in Z-coefficient. Thisdifference matches the different conditions posed in Theorems 1.3 and1.4.

(2) If we assume thatX is a PD4 complex, then Proposition 7.2 and theequation (7.3) still hold. So Theorem 1.4 still hold when q∗ is definedfor PD4 complexes.

(3) Our assumption that M has no S2 boundary component does notaffect the correctness of Theorems 1.2, 1.3 and 1.4 for 3-manifolds withS2 boundary components. If we cap off an S2 boundary component ofM to get M ′, then we have M = M ′#B3. This B3 prime factor makesp increase by 1 and makes χ(∂M) increase by 2 in formulas (1.2) (1.3)(1.4), so they cancel with each other.

Acknowledgments. We thank Professor Jonathan Hillman, ProfessorDieter Kotchick, Professor Yang Su and Professor Shicheng Wang fortheir comments on literatures and known results in this topic.

References

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Department of Mathematics, Rutgers University - New Brunswick,

Hill Center, Busch Campus, Piscataway, NJ 08854, USA

Email address : [email protected]

Department of Mathematics Science, Tsinghua University, Beijing,

100080, CHINA

Email address : [email protected]

http://pi.math.cornell.edu/~hatcher/

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