Smooth Structures on Spin Manifolds in Four Dimensions

*Simon Davis*

#### **Abstract**

The estimate of the coefficient of the magnitude of the signature, which are defined by the number of positive and negative eigenvalues in the inequality representing smooth, oriented, simply connected, compact, spin four-manifolds with indefinite intersection forms can be increased until it is equal to the conjectured value. Therefore, if the intersection form is *mE*<sup>8</sup> ⊕ *n* 0 1 1 0 , the oriented, simply connected, compact, spin four-manifold will admit a smooth structure if and only if *n* ≥ <sup>3</sup> <sup>2</sup> j j *m* . The inequality is changed to *n*≥ <sup>3</sup> <sup>2</sup> j j *<sup>m</sup>* � <sup>1</sup> � <sup>1</sup> 2*i* , there is a 2*<sup>i</sup>* -fold spin covering of a non-spin manifold *M* given the demonstration of *n* ≥ <sup>3</sup> <sup>2</sup> j j *m* for oriented, compact, spin manifolds. A closer examination of the proof reveals that the lower bound for *b*<sup>+</sup> can be increased to 3|*k*| + 1, where j j *<sup>k</sup>* <sup>¼</sup> <sup>3</sup> <sup>15</sup> j j *<sup>σ</sup>* for a spin manifold, yielding *<sup>b</sup>*<sup>2</sup> <sup>≥</sup> <sup>11</sup> <sup>8</sup> j j *σ* þ 2. The projection of a spin covering to a non-spin manifold yields the lower bound *b*<sup>2</sup> ≥ <sup>11</sup> <sup>8</sup> j j *σ* , which establishes the prediction for the coefficients of intersection forms for this class of smooth, oriented, simply connected, compact four-manifolds.

**Keywords:** intersection forms, coefficients, spin manifolds, smooth structures

### **1. Introduction**

The classification of four-manifolds may be determined by the handlebody decomposition into simply connected components of a topological sum when the manifold is smooth. If it is closed, oriented, and simply connected, then it will be distinguished, within a homotopy equivalence, by an intersection form that is either definite, indefinite with odd parity, or indefinite with even parity. These manifolds also be identified by the second Betti number and the signature. The four-manifold admits a smooth structure if the intersection matrix is definite or indefinite with odd parity. Furthermore, if the

intersection form is indefinite and equals *mE*<sup>8</sup> ⊕ *n* 0 1 1 0 , it will continue to have a smooth structure if *n* ≥ <sup>3</sup> <sup>2</sup> j j *m* . In the (*b*2, *σ*) plane, the smooth structures are located above the line *b*<sup>2</sup> ≥ <sup>11</sup> <sup>8</sup> j j *<sup>σ</sup>* [1] and the nonsmooth structures are located below the line *<sup>b</sup>*<sup>2</sup> <sup>≤</sup> <sup>5</sup> <sup>4</sup> j j *σ* , with the region between the two lines undetermined. It may be shown, however, that one of the manifolds in this region does not admit a smooth structure [2]. The coefficient of 11 <sup>8</sup> also will be the maximal value for the line separating the set of manifolds with smooth and nonsmooth structures because the inequality is saturated by *K*3 [3]. Consequently, it remains to be established that all of the manifolds with an indefinite intersection form in this intermediate region do not admit a smooth structure. The condition of an indefinite intersection form is necessary because *b*<sup>2</sup> = 1 and *σ* = 1 for ℂℙ2.

By considering finite-dimensional approximations to the Seiberg-Witten map of the tensor product sections of the spinor bundle and the space of connections, the lower bound *b*<sup>2</sup> ≥ <sup>10</sup> <sup>8</sup> j j *σ* þ 2 was established for oriented, connected spin manifolds [4]. It may be increased with the use of stable homotopy groups of spheres and *Pin*2 equivariant homotopy invariants [5] to *b*<sup>2</sup> ≥ <sup>10</sup> <sup>8</sup> *σ* þ 4. The lower limit will be increased first to *<sup>b</sup>*<sup>2</sup> j j *<sup>σ</sup>* <sup>&</sup>gt; <sup>21</sup> <sup>16</sup> for spin manifolds with signature *σ* ≥ 16 as a result of a theorem on the nonexistence of smooth four-manifolds with the intersection form þ4*E* � 8 ⊕ 5*H* [6]. Then, the coefficient of <sup>11</sup> <sup>8</sup> will be found by considering precisely the form of the maps between finite-dimensional vector bundles over the four-manifold. A second proof will be derived by considering intersection products of second homology classes representable by spheres [7]. Consequences of the related <sup>3</sup> <sup>2</sup> conjecture for the embedding of surfaces with a nonvanishing second homology class in an irreducible four-manifold will be described.

#### **2. The inequality for the second Betti number and the signature**

Since the intersection matrix is symmetric and diagonalizable over ℝ, *b*<sup>+</sup> and *b*� will denote the number of positive and negative eigenvalues respectively. Then the second Betti number and the signature are *b*<sup>2</sup> = *b*<sup>+</sup> + *b*� and *σ* = *b*<sup>+</sup> � *b*� respectively. Let *<sup>k</sup>* ¼ � *<sup>σ</sup>*ð Þ *<sup>M</sup>* <sup>16</sup> and the inequality *<sup>b</sup>*<sup>þ</sup> <sup>≥</sup>3*<sup>k</sup>* ¼ � <sup>3</sup>*<sup>σ</sup>* <sup>16</sup>. When the signature is negative, and –*σ* may be replaced by |*σ*|,

$$\begin{aligned} \frac{b\_2 + \sigma}{2} &\geq \frac{3|\sigma|}{16} \\ b\_2 &\geq 2\left[ -\frac{\sigma}{2} + \frac{3|\sigma|}{16} \right] = \frac{11}{8}|\sigma|. \end{aligned} \tag{1}$$

The signature could be positive such that *b* ≥ 3 *k* is a much less stringent inequality. However, by reversing the orientation, the sign of the signature is changed, and this inequality always can be taken to imply *<sup>b</sup>*<sup>þ</sup> <sup>≥</sup> <sup>3</sup>j j *<sup>σ</sup>* 16 .

Similarly, if the orientation is chosen such that *<sup>b</sup>*<sup>þ</sup> <sup>≥</sup>2*<sup>k</sup>* <sup>þ</sup> <sup>1</sup> ¼ � *<sup>σ</sup>* <sup>8</sup> þ 1 is equivalent to *<sup>b</sup>*<sup>þ</sup> <sup>≥</sup> j j *<sup>σ</sup>* <sup>8</sup> þ 1,

$$\begin{aligned} \frac{b\_2 + \sigma}{2} &\geq \frac{|\sigma|}{8} + 1\\ b\_2 &\geq 2\left[\frac{|\sigma|}{2} + \frac{|\sigma|}{8} + 1\right] = \frac{5}{4}|\sigma| + 2. \end{aligned} \tag{2}$$

It may be demonstrated that a spin 4-manifold can admit a smooth structure when the intersection form is 4*E*<sup>8</sup> ⊕ *nH* for *n* ≥ 6 [7]. Consequently, there is no smooth

manifold with the intersection form 4*E*<sup>8</sup> ⊕ 5*H*. Since the coefficients are relatively prime, the ratio *<sup>n</sup> <sup>m</sup>* <sup>¼</sup> <sup>5</sup> <sup>4</sup> is achieved only for the intersection forms 4*kE*<sup>8</sup> ⊕ 5*kH*, *k* = 1, 2, 3, .... Given that there is a manifold *M* with the intersection form 4*E*<sup>8</sup> ⊕ 5*H*, the latter form would characterize #*kM*.

**Lemma 2.1.** Smooth, oriented, simply connected, compact 4-manifolds with a spin structure and an indefinite intersection form have second Betti numbers bounded by <sup>21</sup> <sup>16</sup> j j *<sup>σ</sup>* , which is a bound closer to the line with gradient <sup>11</sup> <sup>8</sup> for non-zero signature.

**Proof.** The line representing smooth structures must be *n* ≥ <sup>5</sup> <sup>4</sup> *m*. Then

$$\frac{b\_2}{|\sigma|} = \frac{8m + 2n}{8m} \ge \frac{8m + \frac{5}{2}m}{8m} = \frac{21}{6}.\tag{3}$$

When

$$
\frac{10}{8} + \frac{2}{|\sigma|} \le \frac{21}{16},
\tag{4}
$$

or

$$|\sigma| \ge \frac{32}{5},\tag{5}$$

this bound is better than the established value. If

$$
\frac{10}{8} + \frac{4}{|\sigma|} < \frac{21}{16},
\tag{6}
$$

or

$$|\sigma| \ge \frac{65}{5}.\tag{7}$$

Any spin manifold will have a signature divisible by 16 by Rohlin's theorem. Therefore, if it is non-zero, these inequalities will be valid.

The line with gradient <sup>21</sup> <sup>16</sup> in the geography of four-manifolds is closer to the boundary between smooth and nonsmooth structures.

Given the validity of the 11/8 conjecture, smooth connected spin four-manifolds can be regarded as the topological sums #*kK*3#ℓ*S*<sup>2</sup> � *<sup>S</sup>*<sup>2</sup> or #*k*ℂℙ<sup>2</sup> #ℓℂℙ<sup>2</sup> .

**Lemma 2.2.** All manifolds #*kK*3#ℓ*S*<sup>2</sup> � *<sup>S</sup>*<sup>2</sup> with *<sup>k</sup>* <sup>&</sup>gt; 0 have *<sup>b</sup>*<sup>2</sup> j j *<sup>σ</sup>* <sup>≥</sup> <sup>11</sup> <sup>8</sup> , with the bound saturated by *K*3. The coefficients in #*k*ℂℙ<sup>2</sup> #ℓℂℙ<sup>2</sup> must satisfy the inequalities 3 <sup>19</sup> *k*≤ℓ <sup>19</sup> <sup>3</sup> *<sup>k</sup>* for a smooth structure to exist by the <sup>11</sup><sup>Þ</sup> <sup>8</sup> conjecture.

**Proof.** Since *K*3 has an interesection form with 3 positive and 19 negative eigenvalues, *<sup>b</sup>*2(*K*3) = 22 and *<sup>σ</sup>*(*K*3) = �16. The intersection form of *<sup>S</sup>*<sup>2</sup> � *<sup>S</sup>*<sup>2</sup> , *H*, has the eigenvalues 1 and � 1, and *<sup>b</sup>*2(*S*<sup>2</sup> � *<sup>S</sup>*<sup>2</sup> ) = 2, while *<sup>σ</sup>*(*S*<sup>2</sup> � *<sup>S</sup>*<sup>2</sup> ) = 0. Then

$$\begin{aligned} b\_2 \left( \#kK \ $ \#\ell \$ ^2 \times \mathbb{S}^2 \right) &= 22k + 2\ell\\ \sigma \left( \#kK \ $ \#\ell \$ ^2 \times \mathbb{S}^2 \right) &= -16k \end{aligned} \tag{8}$$

and

$$\frac{b\_2 \left(\#kK\ $\#\ell^2 \times \mathbb{S}^2\right)}{\left|\sigma\left(\#kK\$ \#\mathbb{S}^2 \times \mathbb{S}^2\right)\right|} = \frac{\mathbf{11}}{\mathbf{8}} + \frac{\mathbf{1}}{\mathbf{8}} \frac{\ell}{k}.\tag{9}$$

The intersection matrix of #*k*ℂℙ<sup>2</sup> #ℓℂℙ<sup>2</sup> is *diag*(1, ..., 1, �1, ..,–1) with

$$\begin{split} \label{eq:SDAR} b\_2 \left( \#k \mathbb{CP}^2 \#\ell \overline{\mathbb{CP}^2} \right) &= k + \ell \\ \sigma \left( \#k \mathbb{CP}^2 \#\ell \overline{\mathbb{CP}^2} \right) &= k - \ell \end{split} \tag{10}$$

It follows that

$$\frac{b\_2 \left( \#k \mathbb{CP}^2 \#\ell \overline{\mathbb{CP}^2} \right)}{\sigma \left( \#k \mathbb{CP}^2 \#\ell \overline{\mathbb{CP}^2} \right)} = \frac{k + \ell}{|k - \ell|} = \begin{cases} 1 + \frac{2\ell}{k - \ell} & k > \ell \\ 1 + \frac{2k}{\ell - k} & \ell > k \end{cases} \tag{11}$$

either

$$\begin{aligned} 1 + \frac{2\ell}{k - \ell} &\geq \frac{11}{8}, \\ \ell &\geq \frac{3}{19}k \end{aligned} \tag{12}$$

or

$$\begin{aligned} 1 + \frac{2k}{\ell'-k} &\geq \frac{11}{8} \\ k &\geq \frac{3}{19}\ell' \end{aligned} \tag{13}$$

which may combine in the inequalities

$$\frac{3}{19}k \le \ell \le \frac{19}{3}k.\tag{14}$$

A bound may be established for simply connected complex surfaces with an even cup product form [8]. It is known that, for these manifolds, *b*<sup>2</sup> = *c*2–2 and *σ* ¼ 1 <sup>3</sup> *c*<sup>2</sup> <sup>1</sup> � 2*c*<sup>2</sup> � �, where *c*<sup>1</sup> and *c*<sup>2</sup> are the first two Chern numbers. Defining

$$b = \frac{1}{16}(8b\_2 - 11|\sigma|),\tag{15}$$

the <sup>11</sup> <sup>8</sup> conjecture is equivalent to *b* ≥ 0.When *σ* < 0,

$$\begin{split} 48b &= 3\left( 8(c\_2 - 2) + \frac{11}{8} \left( c\_2^2 - 2c\_2 \right) \right) \\ &= 11c\_1^2 + 2c\_2 - 48. \end{split} \tag{16}$$

*Smooth Structures on Spin Manifolds in Four Dimensions DOI: http://dx.doi.org/10.5772/intechopen.106368*

When *σ* > 0,

$$\begin{split} 48b &= 3\left( 8(c\_2 - 2) - 11(c\_1^2 - 2c\_2) \right) \\ &= -11c\_1^2 + 43c\_2 - 48. \end{split} \tag{17}$$

Adding the two inequalities gives *c*<sup>2</sup> ≥ <sup>32</sup> <sup>15</sup>. This inequality is satisfied for complex surfaces, since *c*<sup>2</sup> <sup>1</sup> ≥0 and *c*<sup>2</sup> ≥3. However, for negative signature,

$$2\mathbf{1}c\_1^2 + 2c\_2 - 4\mathbf{8} \ge -4\mathbf{2}.\tag{18}$$

For positive signature, *c*<sup>2</sup> <sup>1</sup> ≤ 3*c*<sup>2</sup> and

$$-11c\_1^2 + 43c\_2 - 48 \ge 10c\_2 - 48 \ge -18. \tag{19}$$

It is clear that 48*b* = 3(8*b*2–11|*σ*|) is integer. By Rohlin's theorem, the signature will be divisible by 16 and *<sup>b</sup>* also would be integer. By Eqs. (18) and (19), *<sup>b</sup>*<sup>≥</sup> � <sup>7</sup> <sup>8</sup> and *<sup>b</sup>*<sup>≥</sup> � <sup>3</sup> <sup>8</sup> respectively. Then *b* ≥ 0 and the <sup>11</sup> <sup>8</sup> conjecture is valid for simply connected complex surfaces with an even intersection form.

#### **3. Summary of the K-theoretic proof of the lesser lower bound for the second Betti number**

The Dirac operator *D* is a map from sections of spinor bundles *E*<sup>o</sup> to *E*<sup>1</sup> , *D*: *Γ*(*E*<sup>0</sup> ) ! (*E*<sup>1</sup> ), and *ind D* = *dim Ker D* – *dim Coker D*. Now consider a Whitney sum with a finite-dimensional vector bundle, such that *<sup>D</sup>* <sup>¼</sup> *<sup>L</sup>* <sup>þ</sup> *<sup>L</sup>*<sup>0</sup> : *<sup>V</sup>*<sup>0</sup> <sup>⊕</sup> <sup>Γ</sup> *<sup>E</sup>*<sup>0</sup> ! *V*<sup>1</sup> ⊕ Γ *E*<sup>1</sup> , where *L* is a finite-dimensional mapping and *L'* is an isomorphism between infinite-dimensional spaces. Then *ind D* = *dim V*<sup>0</sup> – *dim V*1. Therefore, topological information about a manifold on which the Dirac operator, arising from equations with a linearization of *N* = 2 supersymmetry on the space, deduced from the index may be evaluated through a finite-dimensional construction. When *M* is a closed spin 4-manifold, the Seiberg-Witten map is a *Pin*2-equivariant map given by <sup>ℍ</sup><sup>∞</sup> <sup>⊕</sup> *<sup>R</sup>*~<sup>∞</sup> ! <sup>ℍ</sup><sup>∞</sup> <sup>⊕</sup> *<sup>R</sup>*~<sup>∞</sup> , where *R*~ is the nontrivial one-dimensional real representation space of *Pin*2. A finite-dimensional approximation is a *Pin*2-equivariant map ℍ*<sup>c</sup>*<sup>0</sup> ⊕ *R*~*<sup>d</sup>*<sup>0</sup> ! <sup>ℍ</sup>*<sup>c</sup>*<sup>1</sup> <sup>⊕</sup> *<sup>R</sup>*~*<sup>d</sup>*<sup>1</sup> , where *<sup>c</sup>*<sup>0</sup> � *<sup>c</sup>*<sup>1</sup> ¼ � *<sup>σ</sup>*ð Þ *<sup>M</sup>* <sup>16</sup> and *d*<sup>0</sup> � *d*<sup>1</sup> ¼ *b*þð Þ *M* , in a generalized Kuranshi construction [9].

The four-dimensional spin manifold will admit a *Spin*<sup>4</sup> bundle and vector bundles *T*, *S*<sup>+</sup> , *S*� and Λ constructed from the *Spin*<sup>4</sup> � *Pin*<sup>2</sup> modules –ℍþ,þℍ,�ℍ andþℍ<sup>þ</sup> defined by the actions *<sup>q</sup>*�*aa*�<sup>1</sup> <sup>þ</sup> ,*q*þ*ϕq*�<sup>1</sup> <sup>0</sup> ,*q*þ*ωq*�<sup>1</sup> <sup>0</sup> and *<sup>q</sup>*þ*ωq*�<sup>1</sup> <sup>þ</sup> for (*q*�, *<sup>q</sup>*+, *<sup>q</sup>*0) <sup>∈</sup> *Spin*<sup>4</sup> � *Pin*<sup>2</sup> and *<sup>a</sup>*∈–ℍþ,*ϕ*<sup>∈</sup> <sup>þ</sup>ℍ,*<sup>ψ</sup>* <sup>∈</sup>–<sup>ℍ</sup> and *<sup>ω</sup>*<sup>∈</sup> <sup>þ</sup>ℍþ. If *<sup>R</sup>*<sup>~</sup> is the real one-dimensional *Pin*<sup>2</sup> module defined by multiplication by *Pin*2/*S*<sup>1</sup> , *<sup>T</sup>*<sup>~</sup> <sup>¼</sup> *<sup>T</sup>* <sup>⊗</sup> *<sup>R</sup>*~,*<sup>C</sup>* : *<sup>T</sup>* <sup>⊗</sup> *<sup>S</sup>*<sup>þ</sup> ! *<sup>S</sup>*� with ð Þ! *<sup>a</sup>*, *<sup>ϕ</sup> <sup>a</sup>ϕ*,*<sup>C</sup>* : *<sup>T</sup>* <sup>⊗</sup> *<sup>T</sup>*<sup>~</sup> ! <sup>Λ</sup><sup>~</sup> with ð Þ! *<sup>a</sup>*, *<sup>b</sup> ab*,*D*<sup>1</sup> <sup>¼</sup> *<sup>C</sup>*∇<sup>1</sup> : <sup>Γ</sup> *<sup>S</sup>*<sup>þ</sup> ð Þ! <sup>Γ</sup> *<sup>S</sup>*� ð Þ, *<sup>D</sup>*<sup>2</sup> <sup>¼</sup> *<sup>C</sup>*∇<sup>2</sup> : Γ *T*~ ! <sup>Γ</sup> <sup>Λ</sup><sup>~</sup> , *<sup>D</sup>* <sup>¼</sup> *<sup>D</sup>*<sup>1</sup> <sup>⊕</sup> *<sup>D</sup>*<sup>2</sup> : <sup>Γ</sup> *<sup>S</sup>*<sup>þ</sup> <sup>⊕</sup> *<sup>T</sup>*<sup>~</sup> ! <sup>Γ</sup> *<sup>S</sup>*� <sup>⊕</sup> <sup>Λ</sup><sup>~</sup> , *<sup>Q</sup>* : *<sup>S</sup>*<sup>þ</sup> <sup>⊕</sup> *<sup>T</sup>*<sup>~</sup> ! *<sup>S</sup>*� <sup>⊕</sup> *<sup>S</sup>*� <sup>⊕</sup> <sup>Λ</sup><sup>~</sup> with ð Þ! *<sup>ϕ</sup>*, *<sup>a</sup> <sup>a</sup>ϕi*, *<sup>ϕ</sup>i<sup>ϕ</sup>* , then *<sup>D</sup>* <sup>+</sup> *<sup>Q</sup>*: *<sup>V</sup>* ! *<sup>W</sup>* where *<sup>V</sup>* is the *<sup>L</sup>*<sup>2</sup> <sup>4</sup> completion of Γ *S*<sup>þ</sup> ⊕ *T*~ and *W* is the *L*<sup>2</sup> <sup>3</sup> completion of <sup>Γ</sup> *<sup>S</sup>*� <sup>⊕</sup> <sup>Λ</sup><sup>~</sup> [4].

Let *M* be a compact *G*-space, *E* and *F* be *G*-equivariant complex vector bundles over *M*, *BE* and *BF* be disk bundles corresponding to *E* and *F*, *SE* and *SF* be boundary sphere bundles, <sup>~</sup>*<sup>f</sup>* : *BE* ! *BF* be a G-equivariant bundle map preserving boundaries. By the Thom isomorphism theorem, *KG*(*BE, SE*) and *KG*(*BF, SF*) are generated by Thom classes *τ<sup>E</sup>* and *τF*. With ~*f* ∗ being the pullback map *KG*(*BF, SF*) ! *KG*(*BE, SE*), ~*f* ∗ *<sup>τ</sup><sup>F</sup>* <sup>¼</sup> *<sup>α</sup>*0*τE*, where *<sup>α</sup>*<sup>0</sup> <sup>∈</sup> *KG*(*M*) is the degree of <sup>~</sup>*<sup>f</sup>* ∗ . Since the restriction of the Thom classes to the zero sections are the Euler classes of E and F, P *<sup>d</sup>*ð Þ �<sup>1</sup> *<sup>d</sup>* <sup>Λ</sup>*dF* � � <sup>¼</sup> *α*0 P *<sup>d</sup>*ð Þ �<sup>1</sup> *<sup>d</sup>* <sup>Λ</sup>*dE* � �.

Contracting *M* = *pt*., *G* = *Pin*2,

$$\begin{aligned} E\_{pt.} &= V\_{\lambda, \mathbb{C}} = \left( \mathbb{H}^{k+m} + \bar{\mathbb{R}}^{\mathbb{H}} \right) \otimes \mathbb{C} \\ F\_{pt.} &= \overline{W}\_{\bar{\lambda}, \mathbb{C}} = \left( \mathbb{H}^m + \bar{\mathbb{R}}^{b\_+ + n} \right) \otimes \mathbb{C}, \end{aligned} \tag{20}$$

where *Vλ* is the subspace of *V* spanned by the eigenspace of *D\* D* with eigenvalues less than or equal to *λ,* and *W<sup>λ</sup>* is the subspace of *w* spanned by eigenspaces of *DD*\* with eigenvalues less than or equal to *λ*,*f* : *Vλ*,<sup>ℂ</sup> ! *Wλ*,<sup>ℂ</sup> is the complexification of *<sup>D</sup><sup>λ</sup>* <sup>þ</sup> *<sup>Q</sup>λ*,where *<sup>D</sup><sup>λ</sup>* <sup>þ</sup> *<sup>Q</sup><sup>λ</sup>* <sup>¼</sup> *<sup>D</sup>* <sup>þ</sup> *<sup>p</sup>λ<sup>Q</sup>* � �� � *V<sup>λ</sup>* ,*f u*ð ⊗ 1 þ *v* ⊗ *i*Þ ¼ *D<sup>λ</sup>* þ *Q<sup>λ</sup>* ð Þ*u* ⊗ 1 þ *D<sup>λ</sup>* þ *Q<sup>λ</sup>* ð Þ*v* ⊗ *i:*

Suppose that *<sup>φ</sup>* : *<sup>V</sup>* <sup>¼</sup> *ker d*<sup>∗</sup> ð Þ <sup>⊕</sup> <sup>Γ</sup> *<sup>V</sup>*<sup>þ</sup> ð Þ! <sup>Ω</sup><sup>2</sup> <sup>þ</sup> <sup>⊕</sup> <sup>Γ</sup> *<sup>V</sup>*� ð Þ¼ *<sup>W</sup>*, *<sup>φ</sup>*ð Þ¼ *<sup>v</sup> L v*ð Þþ *<sup>θ</sup>*ð Þ*<sup>v</sup>* , *<sup>L</sup>* <sup>¼</sup> *<sup>d</sup>*<sup>þ</sup> <sup>0</sup> 0 *∂* ! is linear, *θ*ð Þ¼ *a*, *ψ* ð Þ *σ ψ*ð Þ, *aψ* is quadratic, where *a* is the gauge

potential in the covariant derivative and *σ'* is an automorphism of the space *Γ*(*W*<sup>+</sup> ) and *σ*<sup>0</sup> *ψ<sup>j</sup>* � � <sup>¼</sup> *<sup>σ</sup>*<sup>0</sup> ð Þ *z* þ *jw j* ¼ *σ*<sup>0</sup> ð Þ¼� �*v* þ *jmz σ*<sup>0</sup> ð Þ *ψ* . Let *f <sup>λ</sup>* : *V* ! *W* be defined by *u* � *<sup>f</sup> <sup>λ</sup>*ð Þ¼ *<sup>v</sup> <sup>v</sup>* <sup>þ</sup> *<sup>L</sup>*�<sup>1</sup> <sup>1</sup> � *<sup>p</sup><sup>λ</sup>* � �*θ*ð Þ*<sup>v</sup>* , *L u*ð Þ¼ *L v*ð Þþ <sup>1</sup> � *<sup>p</sup><sup>λ</sup>* � �*θ*ð Þ*<sup>v</sup>* , with *<sup>p</sup>*<sup>λ</sup> being the projection of *V* and *W* onto *V<sup>λ</sup>* and *Wλ*. Defining *φ*<sup>Λ</sup> : ⊕ *<sup>λ</sup>*≤Λ*V<sup>λ</sup>* ! ⊕ *<sup>λ</sup>* <sup>≤</sup>Λ*Wλ*, *φ*Λð Þ¼ *u pλφf* �1 <sup>Λ</sup> ð Þ *u* [10].

Let *Tu*ð Þ¼ *<sup>v</sup> <sup>u</sup>* � *<sup>L</sup>*�<sup>1</sup> <sup>1</sup> � *<sup>p</sup>*<sup>Λ</sup> � �*θ*ð Þ*<sup>v</sup>* . Then

$$\begin{aligned} \left\| \left| T\_u(v\_1) - T\_u(v\_2) \right| \right\| &= \left\| \left( T\_u(v\_1) - u \right) - \left( T\_u(v\_2) - u \right) \right\| \\ &= \left\| \left| -L^{-1}(\mathbf{1} - p\_\Lambda)\theta(v\_1) + L^{-1}(\mathbf{1} - p\_\Lambda)\theta(v\_2) \right| \right\|. \end{aligned} \tag{21}$$

The eigenvalue of *L*�<sup>1</sup> (1 – *p*Λ) is <sup>1</sup> *<sup>λ</sup>* on each *Wλ*, which has the maximum value <sup>1</sup> Λ for λ > Λ. The automorphism *σ'*(*ψ*) is given by *σ*<sup>0</sup> ð Þ¼ *<sup>z</sup>*, *<sup>w</sup> <sup>i</sup>* j j *<sup>z</sup>* <sup>2</sup> �j j *<sup>w</sup>* <sup>2</sup> 2 � � � *k Re z*ð Þþ *<sup>w</sup> jIm z*ð Þ *w* , and, if *σ*^ð Þ¼ *z*, *w* ^*ρ*∘*σ*<sup>0</sup> ð Þ *<sup>z</sup>*, *<sup>w</sup>* , where ^*<sup>ρ</sup>* : *<sup>T</sup>*<sup>∗</sup> ð Þ! *<sup>X</sup> Hom W*�, *<sup>W</sup>*<sup>∓</sup> � �, ^*ρ*ð Þ¼ *v*<sup>1</sup> ∧ *v*<sup>2</sup> 1 <sup>2</sup> ½ � *<sup>ρ</sup>*ð Þ *<sup>v</sup>*<sup>1</sup> , *<sup>ρ</sup>*ð Þ *<sup>v</sup>*<sup>2</sup> , *<sup>f</sup>* <sup>1</sup> <sup>¼</sup> <sup>1</sup> <sup>2</sup> *<sup>e</sup>*<sup>1</sup> <sup>∧</sup> *<sup>e</sup>*<sup>2</sup> � *<sup>e</sup>*<sup>3</sup> <sup>∧</sup> *<sup>e</sup>*<sup>4</sup> ð Þ, *<sup>f</sup>* <sup>2</sup> <sup>¼</sup> <sup>1</sup> <sup>2</sup> *<sup>e</sup>*<sup>1</sup> <sup>∧</sup> *<sup>e</sup>*<sup>3</sup> � *<sup>e</sup>*<sup>4</sup> <sup>∧</sup> *<sup>e</sup>*<sup>2</sup> ð Þ, *<sup>f</sup>* <sup>3</sup> <sup>¼</sup> 1 <sup>2</sup> *<sup>e</sup>*<sup>1</sup> <sup>∧</sup> *<sup>e</sup>*<sup>4</sup> � *<sup>e</sup>*<sup>2</sup> <sup>∧</sup> *<sup>e</sup>*<sup>3</sup> ð Þ, where *i, j, k* correspond to *<sup>f</sup>*1, *<sup>f</sup>*2, *<sup>f</sup>* <sup>3</sup> <sup>∈</sup>Λþð Þ *<sup>M</sup>* , with <sup>Λ</sup><sup>2</sup> ð Þ¼ *M* <sup>Λ</sup>þð Þ *<sup>M</sup>* <sup>⊗</sup> <sup>Λ</sup>�<sup>1</sup> ð Þ *<sup>M</sup>* , j j *σ ψ* ^ð Þ <sup>2</sup> <sup>¼</sup> <sup>1</sup> <sup>2</sup> j j *ψ* <sup>2</sup> [3]. It follows that

$$\left\|\left(\left(\hat{\sigma}\boldsymbol{\nu}\_{1},\ a\_{1}\boldsymbol{\nu}\right)\_{1}\right) - \left(\hat{\sigma}\boldsymbol{\nu}\_{2},a\_{2}\boldsymbol{\nu}\_{2}\right)\right\| \leq \left\|\left(a\_{1},\ \boldsymbol{\nu}\_{1}\right) - \left(a\_{2},\ \boldsymbol{\nu}\_{2}\right)\right\|\tag{22}$$

if ∣*ψ*1∣<*a*1, ∣*ψ*2∣ <*a*2, and *a*1, *a*<sup>2</sup> < 1. Under these conditions, by the Banach contraction principle, *φ*�<sup>1</sup> <sup>Λ</sup> ð Þ 0 is a compact set.

*Smooth Structures on Spin Manifolds in Four Dimensions DOI: http://dx.doi.org/10.5772/intechopen.106368*

With *BV<sup>λ</sup>*,<sup>ℂ</sup> <sup>¼</sup> f g *<sup>u</sup>* <sup>⊗</sup> <sup>1</sup> <sup>þ</sup> *<sup>v</sup>* <sup>⊗</sup> *<sup>i</sup>* <sup>∈</sup>*V<sup>λ</sup>*,ℂjk k*<sup>u</sup>* , k k*<sup>v</sup>* <sup>≤</sup> *<sup>R</sup>* , *SV<sup>λ</sup>*,<sup>ℂ</sup> <sup>¼</sup> *<sup>∂</sup>BV<sup>λ</sup>*,ℂ, let *<sup>f</sup>* <sup>¼</sup> *<sup>f</sup>*∘*p*, where *p* : *W<sup>λ</sup>*,<sup>ℂ</sup>nf g0 ! *SW<sup>λ</sup>*,ℂ. Then the mapping *f* : *BV<sup>λ</sup>*,<sup>ℂ</sup> ! *BW<sup>λ</sup>*,<sup>ℂ</sup> is defined to be the cone of *<sup>f</sup>*. If *<sup>k</sup>*>0, *<sup>α</sup>* <sup>∈</sup>*R Pin* ð Þj <sup>2</sup> *<sup>ρ</sup>*ℓð Þ *<sup>F</sup> <sup>α</sup>* <sup>¼</sup> *<sup>ψ</sup>*ℓ*<sup>α</sup>* � �*ρ*ℓð Þ *<sup>E</sup>* � �⊂*Ker R Pin* ð Þ!<sup>2</sup> *R S*�<sup>1</sup> � � � � . Consider an element *<sup>α</sup>* of *Ker*(*R*(*Pin*2) ! *<sup>R</sup>*(*S*<sup>1</sup> )) satisfying P *<sup>d</sup>*ð Þ �<sup>1</sup> *<sup>d</sup>* <sup>Λ</sup>*dF* � � <sup>¼</sup> *α* P *<sup>d</sup>*ð Þ �<sup>1</sup> *<sup>d</sup>* <sup>Λ</sup>*dE* � �. Regarding E and F as *<sup>S</sup>*<sup>1</sup> modules, let *<sup>E</sup>*<sup>0</sup> <sup>¼</sup> <sup>2</sup>ð Þ *<sup>k</sup>* <sup>þ</sup> *<sup>m</sup>* <sup>ℂ</sup> <sup>⊕</sup> <sup>ℂ</sup><sup>∗</sup> ð Þ <sup>⊕</sup> *<sup>n</sup>* and *<sup>F</sup>*<sup>0</sup> <sup>¼</sup> <sup>2</sup>*<sup>m</sup>* <sup>ℂ</sup> <sup>⊕</sup> <sup>ℂ</sup><sup>∗</sup> ð Þ <sup>⊕</sup> ð Þ *<sup>b</sup>*<sup>þ</sup> <sup>þ</sup> *<sup>n</sup>* be representation spaces.

Let *<sup>ψ</sup>*<sup>ℓ</sup> be the Adams operation and *<sup>ρ</sup>*ℓð Þ *<sup>E</sup>* be the characteristic class satisfying *<sup>ψ</sup>*ℓ*τ<sup>E</sup>* <sup>¼</sup> *<sup>ρ</sup>*ℓð Þ *<sup>E</sup> <sup>τ</sup>E*. Then

$$\begin{aligned} \boldsymbol{\Psi}^{\ell} \left( \hat{\boldsymbol{f}}^{\*} \boldsymbol{\tau}\_{\mathcal{F}} \right) &= \hat{\boldsymbol{f}}^{\*} \left( \boldsymbol{\eta}^{\ell} \, \boldsymbol{\tau}\_{\mathcal{F}} \right) = \hat{\boldsymbol{f}}^{\*} \left( \boldsymbol{\rho}^{\ell} (\boldsymbol{F}) \boldsymbol{\tau}\_{\mathcal{F}} \right) \\ \boldsymbol{\eta}^{\ell} \left( \boldsymbol{a}\_{0} \boldsymbol{\tau}\_{\mathcal{E}} \right) &= \left( \boldsymbol{\eta}^{\ell} \, \boldsymbol{a}\_{0} \right) \boldsymbol{\uprho}^{\ell} \, \boldsymbol{\tau}\_{\mathcal{E}} = \boldsymbol{\eta}^{\ell} \, \boldsymbol{a}\_{0} \boldsymbol{\rho}^{\ell} (\boldsymbol{E}) \boldsymbol{\tau}\_{\mathcal{E}} \end{aligned} $$
 
$$ \begin{aligned} \boldsymbol{\tilde{f}}^{\*} \left( \boldsymbol{\rho}^{\ell} (\boldsymbol{F}) \boldsymbol{\tau}\_{\mathcal{F}} \right) &= \boldsymbol{\rho}^{\ell} (\boldsymbol{F}) \boldsymbol{\tilde{f}}^{\*} \, \boldsymbol{\tau}\_{\mathcal{F}} = \boldsymbol{\rho}^{\ell} (\boldsymbol{F}) \boldsymbol{a}\_{0} \boldsymbol{\tau}\_{\mathcal{E}} \\ \boldsymbol{\rho}^{\ell} (\boldsymbol{F}) \boldsymbol{a}\_{0} \boldsymbol{\tau}\_{\mathcal{E}} &= \left( \boldsymbol{\eta}^{\ell} \, \boldsymbol{a}\_{0} \right) \boldsymbol{\rho}^{\ell} (\boldsymbol{E}) \boldsymbol{\uprho}\_{\mathcal{E}} \end{aligned} \tag{23} $$
 
$$ \boldsymbol{\rho}^{\ell} (\boldsymbol{F}) \boldsymbol{a}\_{0} = \left( \boldsymbol{\eta}^{\ell} \, \boldsymbol{a}\_{0} \right) \boldsymbol{\rho}^{\ell} (\boldsymbol{E}) \boldsymbol{.} $$

Given that *<sup>ρ</sup>*<sup>ℓ</sup>ð Þ¼ *<sup>L</sup>* <sup>1</sup> <sup>þ</sup> ½ �þ *<sup>L</sup> <sup>L</sup>*<sup>2</sup> � � <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> *<sup>L</sup>*<sup>ℓ</sup>�<sup>1</sup> � � for a line bundle *<sup>L</sup>*, *<sup>ρ</sup>*<sup>ℓ</sup> *<sup>E</sup>*<sup>0</sup> ð Þ¼ *<sup>ρ</sup>*<sup>ℓ</sup>ð Þ <sup>ℂ</sup> *<sup>ρ</sup>*<sup>ℓ</sup> <sup>ℂ</sup><sup>∗</sup> ð Þ � �<sup>2</sup>ð Þ *<sup>k</sup>*þ*<sup>m</sup> <sup>ρ</sup>*<sup>ℓ</sup>ð Þ<sup>1</sup> *<sup>n</sup>* <sup>¼</sup> <sup>1</sup> <sup>þ</sup> *<sup>t</sup>* <sup>þ</sup> … <sup>þ</sup> *<sup>t</sup>* <sup>ℓ</sup>�<sup>1</sup> � � <sup>1</sup> <sup>þ</sup> *<sup>t</sup>* �<sup>1</sup> <sup>þ</sup> … <sup>þ</sup> *<sup>t</sup>* �ð Þ <sup>ℓ</sup>�<sup>1</sup> � � � � <sup>2</sup>*<sup>m</sup>* ℓ*<sup>n</sup>* and *<sup>ρ</sup>*<sup>ℓ</sup> *<sup>F</sup>*<sup>0</sup> ð Þ¼ <sup>1</sup> <sup>þ</sup> *<sup>t</sup>* <sup>þ</sup> … <sup>þ</sup> *<sup>t</sup>* <sup>ℓ</sup>�<sup>1</sup> � � <sup>1</sup> <sup>þ</sup> *<sup>t</sup>* �<sup>1</sup> <sup>þ</sup> … <sup>þ</sup> *<sup>t</sup>* �ð Þ <sup>ℓ</sup>�<sup>1</sup> � � � � <sup>2</sup>*<sup>m</sup>* ℓ*<sup>b</sup>*þþ*<sup>n</sup>*. Since *Ker R Pin* ð Þ!<sup>2</sup> *R S*�<sup>1</sup> � � � � <sup>¼</sup> *<sup>c</sup>* <sup>1</sup> � <sup>~</sup><sup>1</sup> � �j*c*<sup>∈</sup> <sup>ℤ</sup> � �, the trace of the degree relation gives 22*<sup>m</sup>*þ*b*þþ*<sup>n</sup>* <sup>¼</sup> <sup>2</sup>*<sup>c</sup>* 22*<sup>k</sup>*þ2*m*þ*<sup>n</sup>* for *<sup>α</sup>* <sup>¼</sup> *<sup>c</sup>* <sup>1</sup> � <sup>~</sup><sup>1</sup> � �, which is consistent with the inequality *b*<sup>þ</sup> ≥ 2*k* þ 1.

Methods have been developed for increasing the bound for *<sup>b</sup>*<sup>2</sup> j j *<sup>σ</sup>* through the inequality between *b*2(*M*) and the level number of ℂℙ<sup>2</sup>*k*�<sup>1</sup> , defined to be least *n* such that ℂℙ<sup>2</sup>*k*�<sup>1</sup> , *S<sup>n</sup>*�<sup>1</sup> � �<sup>ℤ</sup><sup>2</sup> 6¼ 0, where � *<sup>k</sup>* ¼ � *<sup>σ</sup>*ð Þ *<sup>M</sup>* <sup>16</sup> [11, 12]. The computations of level(ℂℙ<sup>2</sup>*k*�<sup>1</sup> ) yield the inequalities *level*((ℂℙ<sup>2</sup>*k*�<sup>1</sup> ) ≥ 2 *k* + *t* if *k* � *t mod* ð Þ 4 , *t* = 1, 2, 3 and the equality *level*(ℂℙ<sup>2</sup>*k*�<sup>1</sup> )=2 *k* + 3 if *k* � 0 (*mod* 4), *k* > 0 [13]. The equivalent inequalities for the second Betti number and signature would be *b*<sup>2</sup> ≥ <sup>5</sup> <sup>4</sup> j j *<sup>σ</sup>* <sup>þ</sup> <sup>2</sup>*<sup>t</sup>* for j j *<sup>σ</sup>* <sup>16</sup> � *t mod* ð Þ 4 when *t* = 1, 2, 3 and *b*<sup>2</sup> ≥ <sup>5</sup> <sup>4</sup> j j *<sup>σ</sup>* <sup>þ</sup> 6 for j j *<sup>σ</sup>* <sup>16</sup> � 0ð Þ *mod*4 ,j j *σ* >0. The Bauer-Furuta stable Seiberg-Witten invariants also yield a condition for the existence of smooth structures [14].

#### **4. Proof of the exact bound for the second Betti number**

It will be demonstrated that the previous lower bound for *b*<sup>+</sup> can be increased to a maximal value.

**Theorem 4.1.** The second Betti number and signature of a smooth, oriented, simply connected, compact, spin four-manifold with an indefinite intersection form satisfies the inequality *b*<sup>2</sup> ≥ <sup>11</sup> <sup>8</sup> ≥j j *σ* .

**Proof.** Since *c* is a non-zero integer, the trace condition also requires a stricter bound for *<sup>b</sup>*+. The Euler class of *<sup>E</sup>* will be given by that of <sup>ℍ</sup>*<sup>k</sup>*þ*<sup>m</sup>* , while the Euler class of *<sup>F</sup>* would be that of <sup>ℍ</sup>*m*, since the *<sup>R</sup>*<sup>~</sup> components do not contribute. The *Spin*<sup>4</sup> � *Pin*<sup>2</sup> actions do not alter the norm of the points in the quaternionic vector spaces. The fixed point at the origin, however, would be the source of a flow generated by a dilation, representing an invariance of the spinor equation, that must have an endpoint at ∞. Adding the point {∞} to the quaternionic plane produces a manifold that is diffeomorphic to *S*<sup>4</sup> and <sup>χ</sup>(*S*<sup>4</sup> ) = 2. Therefore, by the Thom class relation,

$$\mathfrak{Z}^m = tr(a\_{pt.})\mathfrak{Z}^{k+m} \tag{24}$$

and *tr*(*αpt*.)=2–|*k*<sup>|</sup> , except that *k* must be chosen to be nonpositive through *k* = �|*k*|, then *tr*(α*pt*.)=2|*<sup>k</sup>*<sup>|</sup> . A similar result follows from the evaluation of the K-theory characteristic classes *<sup>ρ</sup>*<sup>ℓ</sup>ð Þ *<sup>E</sup>* . Reintroducing the *<sup>R</sup>*<sup>~</sup> components in the vector bundles allows the inclusion of the factor *c*<sup>0</sup> ≥ 1 occurring in *Ker*(*R*(*Pin*2) ! *R*(*Pin*1)) in *α* ¼ 2j j *<sup>k</sup>* –<sup>1</sup> *<sup>c</sup>*<sup>0</sup> <sup>1</sup> � <sup>~</sup><sup>1</sup> .It follows that

$$\mathfrak{L}^{2^{m+b\_++n}} = \mathfrak{L}c\_0 \mathfrak{L}^{|k|-1} \mathfrak{L}^{2|k|+2m+n} = \mathfrak{L}^{|k|} \mathfrak{L}^{2|k|+2m+n} \tag{25}$$

or

$$b\_{+} \geq \mathfrak{Z}|k|\,\tag{26}$$

and

$$b\_2 \ge \frac{11}{8}|\sigma|.\tag{27}$$

Furthermore, this inequality is equivalent to *n* ≥ <sup>3</sup> <sup>2</sup> j j *m* since the second Betti number and the absolute value of the signature of a manifold with the intersection form *mE*<sup>8</sup> ⊕ *n* 0 1 1 0 are *<sup>b</sup>*<sup>2</sup> = 8|*m*|+2*<sup>n</sup>* and |*σ*| = 8|*m*|, and

$$\begin{aligned} \frac{b\_2}{|\sigma|} &= \frac{8|m| + 2n}{8|m|} \ge \frac{11}{8} \\ 8|m| + 2n &\ge 11|m| \\ n &\ge \frac{3}{2}|m|. \end{aligned} \tag{28}$$

It has been proven for cobordisms between homology three-spheres *Y*<sup>0</sup> and *Y*<sup>1</sup> with the intersection form *m*ð Þþ �*E*<sup>8</sup> *n* 0 1 1 0 that *<sup>κ</sup>*ð Þþ *<sup>Y</sup>*<sup>1</sup> *<sup>n</sup>* <sup>≥</sup>*κ*ð Þþ *<sup>Y</sup>*<sup>0</sup> *<sup>m</sup>* <sup>þ</sup> 1, where *<sup>κ</sup>* is an invariant that reduces modulo 2 to the Rohlin invariant *μ*(*Y*) [2]. When *Y*<sup>0</sup> and *Y*<sup>1</sup> are *S*<sup>3</sup> , this inequality is *n* ≥ |*m*| + 1, which is consistent with the previously derived inequality [4] for the coefficients since it would follow that

$$\begin{aligned} \frac{b\_2}{|\sigma|} &= \frac{8|m| + 2n}{8|m|} \ge \frac{5}{4} + \frac{2}{|\sigma|}\\ 8|m| + 2n &\ge 10|m| + 2\\ n &\ge |m| + 1. \end{aligned} \tag{29}$$

Nevertheless, the nonexistence of smooth spin manifolds transcending the stricter inequality, such that *<sup>n</sup>* <sup>¼</sup> <sup>3</sup> <sup>2</sup> j j *m* � 1, with a decomposition *M* ¼ *X*<sup>1</sup> ∪ *<sup>Y</sup>*1*X*<sup>2</sup> ∪ *<sup>Y</sup>*<sup>2</sup> … ∪ *Yr*�<sup>1</sup>*Xr*, where the intersection forms of *Xi*, 1 ≤ *i* ≤ *r* – 1 are 2ð Þþ �*E*<sup>8</sup> 3 0 1 1 0 � �, *Yi* is a homology three-sphere and *Xr* has the intersection form 2ð Þþ �*E*<sup>8</sup> 2 0 1 1 0 � � [15], indicates there are characteristics that cannot be preserved when under topological

sums yielding <sup>5</sup> <sup>4</sup> <sup>&</sup>lt; *<sup>b</sup>*<sup>2</sup> j j *<sup>σ</sup>* <sup>&</sup>lt; <sup>11</sup> 8 .

**Definition 4.2.** A space is Floer G-split if the *S*<sup>1</sup> action on the the *K*-theory group *<sup>K</sup>*<sup>~</sup> *<sup>G</sup>*ð Þ *<sup>M</sup>* has the form *<sup>z</sup>* ! *zr* for some positive integer *<sup>r</sup>*.

**Theorem 4.3.** The compact manifold *M* ¼ *X*<sup>1</sup> ∪ *<sup>Y</sup>*1*X*<sup>2</sup> ∪ *<sup>Y</sup>*<sup>2</sup> … ∪ *Yr*�<sup>1</sup>*Xr* with spaces *Xi*, = 1, …,*r* – 1, and *Xr* having the intersection forms *Q X*ð Þ¼*<sup>i</sup>* 2ð Þ �*E*<sup>8</sup> ∪ 3*H* and *Q X*ð Þ¼ *<sup>r</sup>* 2ð Þ �*E*<sup>8</sup> ∪ *qrH*, and *Yi* being homology three-spheres, is smooth if and only if *qr* ≥ 3.

**Proof.** When the *<sup>S</sup>*<sup>1</sup> action does not have this form, which requires *<sup>z</sup>* ! *<sup>z</sup><sup>r</sup>* with *r* ∉ ℚ, the resulting quotient produces a non-Hausdorff structure because arbitrarily near points are identified. If *<sup>r</sup>* <sup>¼</sup> *<sup>s</sup> <sup>t</sup>* ∈ ℚnℤ, then the points *e* <sup>2</sup>*πi<sup>θ</sup>* and *e* 2*π <sup>t</sup> <sup>i</sup><sup>θ</sup>* are identified, which yields an orbifold rather than a smooth manifold. A theorem proven by Manolescu states that no closed spin four-manifold has a decomposition of this type such that all homology spheres in the set {*Yi*} are Floer *G*-split [2]. It follows that a quotient of one of the homology spheres by an *<sup>S</sup>*<sup>1</sup> action with *<sup>z</sup>* ! *<sup>z</sup><sup>r</sup>* ,*r* ∉ *Z*, is not smooth. The *S*<sup>1</sup> action on the homology sphere, which is not Floer G-split, may be transferred to the manifold *Xr*, as a result of the parallelizability necessary for the existence of the spin structure, thereby proving that a spin manifold with an intersection form 2ð Þ �*E*<sup>8</sup> ⊕ 2*H* does not admit a smooth structure.

Now suppose that the spin manifold *X* has the intersection form *m*ð Þ �*E*<sup>8</sup> ⊕ *nH*, where *<sup>n</sup>* <sup>¼</sup> <sup>3</sup> <sup>2</sup> j j *<sup>m</sup>* . The analogous result to that given above for *<sup>n</sup>* <sup>¼</sup> <sup>3</sup> <sup>2</sup> j j *m* � 1 would be the decomposition

$$\begin{aligned} X &= X\_1 \cup\_{Y\_1} X\_2 \cup\_{Y\_2} \dots \cup\_{Yr-1} X\_r\\ Q(X\_i) &= 2(-E\_8) + 3H\_r i = 1, \dots, r \end{aligned} \tag{30}$$

with*Y*1, ...,*Yr*–1being homology spheres, then theinequality *κ*(*Yi* + 1)+3≥*κ*(*Yi*) + 2 + 1, or *κ*(*Yi* + 1) ≥ *κ*(*Yi*), is valid for all *j* = 1, ..., *r* – 2, and each *Yi* is Floer G-split, which requires the existence of a smooth structure on each *Xi*, *i* = 1, ..., *r*. Therefore, closed spin four-manifolds with the intersection form *mE*<sup>8</sup> <sup>⊕</sup> *nH* and *<sup>n</sup>* <sup>¼</sup> <sup>3</sup> <sup>2</sup> j j *m* admit smooth structures. For *n*≥ <sup>3</sup> <sup>2</sup> j j *m* , the inequalities for *κ*(*Yi*), *i* = 1, ..., *r*–1 continue to be valid, each of the homology spheres will be Floer *KG*-split, and there will be a smooth structure on the spin four-manifold.

Several results may be proven given the validity of the <sup>11</sup> <sup>8</sup> conjecture, including the theorem on *ξ ∙ ξ* for a characteristic second homology class *ξ* representable by *S*<sup>2</sup> for a range of values of *b*<sup>+</sup> and *b*� [7]. The following lemma is required:

Let M be a closed connected oriented four-manifold with *ξ*∈ *H*2ð Þ *M*; ℤ be a characteristic homology class representable by *S*<sup>2</sup> . Then *ξ ∙ ξ* = *σ*(*M*) + 16 *m* with *m* ≤ max ⌊*<sup>b</sup>*1�<sup>1</sup> <sup>3</sup> ⌋, ⌊*<sup>b</sup>*��*b*<sup>þ</sup> <sup>16</sup> ⌋ n o would be consistent with the <sup>11</sup> <sup>8</sup> conjecture.

The demonstration of this result is suggestive of an equivalence of the conditions with the limits of *m* being derived from geometric properties of the spin manifold, and the <sup>11</sup> <sup>8</sup> conjecture following from the ranges for *m*.

By a theorem of Kervaire and Milnor, it is known that, in a four-manifold which allows the embedding of two-spheres representing the homology class *ξ*, *ξ* ∙ *ξ* = *σ*(*M*) + 16 *m* for some *m* [16]. Since *σ*(*M*) = *b*<sup>+</sup> � *b*�, *ξ* ∙ *ξ* = *b*<sup>+</sup> � *b*� + 16 *m*. The range given in the above theorem with *ξ* ∙ *ξ* = 0 or *b*� = *b*<sup>+</sup> + 16 *m* yields the following results.

$$\begin{split} m &\leq \max\left\{ \lfloor \frac{b\_{+} - 1}{3} \rfloor, \lfloor \frac{b\_{+} + 16m - r}{16} \rfloor \right\} = \max\left\{ \lfloor \frac{b\_{+} - 1}{3} \rfloor, m + \lfloor \frac{b\_{+} - r}{16} \rfloor \right\} \\ &= \max\left\{ \lfloor \frac{b\_{+} - 1}{3} \rfloor, m \right\} \end{split} \tag{31}$$

and

$$
\delta m \le \lfloor \frac{b\_+ - 1}{3} \rfloor. \tag{32}
$$

Since spin manifolds have an even intersection form, *b*<sup>+</sup> is even. Then if *b*<sup>+</sup> = 0, 2 *or* 4 (*mod* 6) and ⌊*<sup>b</sup>*þ�<sup>1</sup> <sup>3</sup> ⌋ <sup>¼</sup> ⌊*<sup>b</sup>*<sup>þ</sup> <sup>3</sup> ⌋ � 1,⌊*<sup>b</sup>*<sup>þ</sup> <sup>3</sup> ⌋ *or*⌊*<sup>b</sup>*<sup>þ</sup> <sup>3</sup> ⌋ respectively. If *<sup>m</sup>* <sup>≤</sup> *<sup>b</sup>*<sup>þ</sup> 3 ,

$$b\_- = b\_+ + 16m \le \frac{19}{3}b\_+. \tag{33}$$

Then

$$\begin{aligned} b\_2 - \sigma &\le \frac{19}{3} (b\_2 + \sigma) \\ \frac{16}{3} b\_2 &\ge -\frac{22}{3} \sigma \\ b\_2 &\ge \frac{11}{8} |\sigma| \sigma < 0. \end{aligned} \tag{34}$$

For *σ* > 0, the roles of *b*<sup>+</sup> and *b*<sup>+</sup> are interchanged, and

$$\begin{aligned} b\_2 + \sigma &\le \frac{19}{3} (b\_2 - \sigma) \\ \frac{16}{3} b\_2 &\ge \frac{22}{3} \sigma \\ b\_2 &\ge \frac{11}{8} |\sigma| \sigma > 0. \end{aligned} \tag{35}$$

Therefore, the condition *ξ* ∙ *ξ* = 0 together with the range of *m* yielding the upper limit *<sup>b</sup>*<sup>þ</sup> <sup>3</sup> , is sufficient to prove the <sup>11</sup> <sup>8</sup> conjecture. Given that *ξ* ∙ *η* is the intersection number of *ξ* and *η* are representable by *S*<sup>2</sup> , *ξ* ∙ *ξ* would equal the sum of the eigenvalues of the intersection form of *<sup>S</sup>*<sup>2</sup> � *<sup>S</sup>*<sup>2</sup> , which equals zero.

The connected sum ℂℙ<sup>2</sup> #9ℂℙ<sup>2</sup> does not satisfy the inequalities in Eq. (14) for the coefficients *k* and ℓ. Nevertheless, it has exotic smooth structures. The nonexistence of spin structures on ℂℙ<sup>2</sup> #9ℂℙ<sup>2</sup> may be demonstrated [17, 18]. The spaces ℂℙ<sup>2</sup> #ℓℂℙ<sup>2</sup> have *<sup>b</sup>*<sup>2</sup> j j *<sup>σ</sup>* <sup>&</sup>lt; <sup>11</sup> <sup>8</sup> for ℓ ≥ 7. Furthermore, these four-manifolds have both standard and exotic smooth structures for *k* = 7, 8 and 9 [19–21].

**Proposition 4.4.** The topological sums ℂℙ<sup>2</sup> #ℓℂℙ<sup>2</sup> ,ℓ≥7, do not represent counterexamples to the inequality *<sup>b</sup>*<sup>2</sup> j j *<sup>σ</sup>* <sup>≥</sup> <sup>11</sup> <sup>8</sup> required for smooth structures on spin manifolds given the validity of the <sup>10</sup> <sup>8</sup> theorem.

**Proof.** The existence of smoooth structures on these spaces is established. From §2, the second Betti number and signature of ℂℙ<sup>2</sup> #ℓℂℙ<sup>2</sup> equal

$$\begin{aligned} b\_2 &= \mathbf{1} + \ell \\ \sigma &= \mathbf{1} - \ell. \end{aligned} \tag{36}$$

Then

$$\frac{b\_2}{|\sigma|} = \frac{\ell + 1}{\ell - 1} \; \ell \ge 2. \tag{37}$$

For ℓ = 7, 8 and 9, *<sup>b</sup>*<sup>2</sup> j j *<sup>σ</sup>* is <sup>4</sup> 3 , 9 <sup>7</sup> and <sup>5</sup> <sup>4</sup> respectively. Then

$$\begin{aligned} \frac{10}{8} &< \frac{b\_2}{|\sigma|} < \frac{11}{8} \text{ } for \, \ell = 7, 8\\ \frac{b\_2}{|\sigma|} &= \frac{10}{8} \text{ } for \, \ell = 9\\ \frac{b\_2}{|\sigma|} &< \frac{10}{8} \text{ } for \, \ell > 9. \end{aligned} \tag{38}$$

Consequently, the values ℓ ≥ 9 must be covered by the <sup>10</sup> <sup>8</sup> theorem, which will require the absence of spin structures on these manifolds.

It follows from Rohlin's theorem that the signature of a smooth, spin compact fourmanifold is divisible by 16. For ℂℙ<sup>2</sup> #ℓℂℙ<sup>2</sup> , this condition is

$$t' \equiv \mathbf{1} \pmod{\mathbf{1}}.\tag{39}$$

This congruenece condition is not satisfied by ℓ = 7 or ℓ = 8. Therefore, there will be no spin structure for these values. It follows that the connected sums for ℓ ≥ 7 will not represent a counterexample to the <sup>11</sup> <sup>8</sup> conjecture when the lower bound of <sup>10</sup> 8 suffices generally for smooth spin manifolds.

There would be a spin structure on ℂℙ<sup>2</sup> #ℓℂℙ<sup>2</sup> . Both the above proposition and the consistency of the <sup>10</sup> <sup>8</sup> theorem require the nonexistence of spin structures on ℂℙ<sup>2</sup> # 16 ð Þ *<sup>r</sup>* <sup>þ</sup> <sup>1</sup> ℂℙ<sup>2</sup> for *<sup>r</sup>* <sup>≥</sup> 1.

**Proposition 4.5.** The topological sum ℂℙ<sup>2</sup> # 16 ð Þ *<sup>r</sup>* <sup>þ</sup> <sup>1</sup> ℂℙ<sup>2</sup> is a spin manifold only if *r* = 0.

**Proof.** There exists a spin structure on a space *M* the second Stiefel-Whitney class *w*2ð Þ *M* ∈ *H*2ð Þ *M*; ℤ<sup>2</sup> is nonvanishing. The second homology group of a connected sum *M*1#*M*2, *where*dim M1 = *dim M*<sup>2</sup> = 4, is

$$H\_2(M\_1 \# M\_2) = H\_2(M\_1) \oplus H\_2(M\_2) \tag{40}$$

and. Specializing to the group ℤ2,

$$H\_2(\mathbf{M}\_1 \# \mathbf{M}\_2, \mathbb{Z}\_2) = H\_2(\mathbf{M}\_1, \mathbb{Z}\_2) \oplus H\_2(\mathbf{M}\_2, \mathbb{Z}\_2). \tag{41}$$

Then

$$H\_2\left(\mathbb{CP}^2 \# \overline{\mathbb{CP}^2}, \mathbb{Z}\_2\right) = H\_2\left(\mathbb{CP}^2, \mathbb{Z}\_2\right) \oplus H\_2\left(\overline{\mathbb{CP}^2}, \mathbb{Z}\_2\right) \tag{42}$$

Since ℂℙ<sup>2</sup> does not have a spin structure, there will be a nonvanishing generator of the second homology group ½ � *u* 6¼ ½ � 0 and

$$H\_2\left(\mathbb{CP}^2 \# \overline{\mathbb{CP}}^2, \mathbb{Z}\_2\right) = [u] \oplus -[u] \tag{43}$$

The element of the homology group must be an element of ℤ2. Therefore, it would be the image of the ½ � *u* ⊕ � ½ � *u* under the mapping

$$\rho: H\_2(\mathbb{CP}^2, \mathbb{Z}\_2) \oplus H\_2(\overline{\mathbb{CP}}^2, \mathbb{Z}\_2) \to \mathbb{Z}\_2 \tag{44}$$

This homomorphism will be defined by

$$
\rho([\mathfrak{u}\_1] \oplus [\mathfrak{u}\_2]) = [\mathfrak{u}\_1] + [\mathfrak{u}\_2] \in \mathbb{Z}\_2 \tag{45}
$$

Since

$$
\rho([u]\oplus - [u]) = [u] - [u] = [\mathbf{0}],\tag{46}
$$

*H*<sup>2</sup> ℂℙ<sup>2</sup> #ℂℙ<sup>2</sup> , ℤ<sup>2</sup> <sup>¼</sup> 0, the second Stiefel-Whitney class vanishes, and there is a � spin structure for *r* = 0.

The homology group for higher values of *r* equals

$$H\_2\left(\mathbb{CP}^2 \# (\mathbf{16r} + \mathbf{1}) \overline{\mathbb{CP}^2}\right) = H\_2\left(\mathbb{CP}^2\right) \# (\mathbf{16r} + \mathbf{1}) H\_2\left(\overline{\mathbb{CP}^2}, \mathbb{Z}\_2\right). \tag{47}$$

Given that ½ � *<sup>u</sup>* <sup>∈</sup> *<sup>H</sup>*<sup>2</sup> ℂℙ<sup>2</sup> , ℤ<sup>2</sup> and �½ � *<sup>u</sup>* <sup>∈</sup> *<sup>H</sup>*<sup>2</sup> ℂℙ<sup>2</sup> , ℤ<sup>2</sup> , the element of <sup>ℤ</sup><sup>2</sup> for the topological sum is

$$\varrho([u]\oplus(\mathbf{1}\mathfrak{G}r+\mathbf{1})(-u))=-\mathbf{1}\mathfrak{G}r[u]\tag{48}$$

Multiplication by a non-zero scalar does not affect the generator of the nontrivial second homology class, which does not vanish. Then, ℂℙ<sup>2</sup> # 16 ð Þ *<sup>r</sup>* <sup>þ</sup> <sup>1</sup> ℂℙ<sup>2</sup> is not a spin manifold for *r* > 1.

There are no counterexamples given by ℂℙ<sup>2</sup> #ℂℙ<sup>2</sup> to the lower bound of <sup>11</sup> <sup>8</sup> for *<sup>b</sup>*<sup>2</sup> j j *σ* . Topological sums with *S*<sup>4</sup> , *<sup>S</sup>*<sup>2</sup> � *<sup>S</sup>*<sup>2</sup> and *<sup>K</sup>*3 will not affect this inequality for the ratio of the second Betti number to the magnitude of the signature. No other potential counterexamples can exist for smooth, simply connected, compact spin four-manifolds.

#### **5. The local coefficients for manifolds with a spin covering**

The proof in §4 is restricted to smooth, oriented, simply connected, compact manifolds which admit spinor structures. It remains to be established if the conclusions continue to be valid for smooth non-spin four-manifolds that have a spin covering. Since the lower bound for *<sup>b</sup>*<sup>2</sup> j j *<sup>σ</sup>* has been increased to <sup>11</sup> 8 .

**Theorem 5.1.** The coefficients of the intersection form *mE*<sup>8</sup> + *nH*, where *H* ¼

0 1 1 0 satisfy the inequality *<sup>n</sup>*<sup>≥</sup> <sup>3</sup> <sup>2</sup> *<sup>m</sup>* � *<sup>ρ</sup>*ð Þþ *<sup>M</sup> <sup>ρ</sup>*ð Þ *<sup>M</sup>* <sup>2</sup>*<sup>i</sup>* , with *i* being the exponent in the order of a spin covering of the smooth, oriented, simply connected, compact manifold *M* and *ρ*(*M*) is the rank of *H*1(*M*; ℤ2).

**Proof.** Let *M* be a smooth, oriented, simply connected, compact four-manifold with the even intersection form *I* = *mE*<sup>8</sup> + *nH*, signature *σ*(*M*)=8 *m* and Euler number *e*(*M*)=2+8 *m* + 2*n*, since the first Betti number can be set equal to zero for a given intersection form. There exists a 2*<sup>i</sup>* cyclic covering *π*: *N* ! *M*, where *N* is a smooth, oriented spin manifold [22, 23]. The signature and the Euler number of the covering space are

$$\begin{aligned} \sigma(N) &= \mathfrak{Z}^i \sigma(M) = \mathfrak{8} \{ \mathfrak{2}^i m \} \equiv \mathfrak{8} r \\\\ \mathfrak{e}(N) &= \mathfrak{Z}^i \mathfrak{e}(M) = \mathfrak{Z}^{i+1} + \mathfrak{8} \{ \mathfrak{2}^i m \} + \mathfrak{2} \{ \mathfrak{2}^i n \} = \mathfrak{2} + \mathfrak{8} r + \mathfrak{2} \mathfrak{s} \end{aligned} \tag{49}$$

then *N* = *rE*8 + *sH*, where

$$r = 2^i m s = 2^i n + 2^i - 1.\tag{50}$$

Since it has been proven that *s*≥ j jþ*r* 1 for spin manifolds [4],

$$\mathcal{Z}^i n + \mathcal{Z}^i - \mathbf{1} = \mathcal{Z}^i |m| + \mathbf{1}. \tag{51}$$

and

$$m \ge |m| - \left(1 - \frac{1}{2^{i-1}}\right). \tag{52}$$

A term *b*1(*N*) can be added to *s* to give *s* þ *b*1ð Þ *N* ≥*r* þ 1. Since *<sup>b</sup>*1ð Þ *<sup>N</sup>* <sup>≤</sup> <sup>2</sup>*<sup>i</sup>* � <sup>1</sup> ð Þ *<sup>ρ</sup>*ð Þ� *<sup>M</sup>* <sup>1</sup> when *<sup>b</sup>*1(*M*) = 0, where *<sup>ρ</sup>*(*M*) is the rank of *<sup>H</sup>*1(*M*; <sup>ℤ</sup>2) [6]. Then

$$2^i n + 2^i - 1 + \left(2^i - 1\right) (\rho(M) - 1) \ge 2^i |m| + 1. \tag{53}$$

$$n \ge |m| - \rho(M) + \frac{1 + \rho(M)}{2^i} . \tag{53}$$

If *ρ*(*M*) = 1 and *N* ! *M* is a double covering with *H*1(*M*; ℤ) = ℤ<sup>2</sup> [6], the inequality *n*≥ j j *m* is valid.

With a tighter bound *n*≥ <sup>3</sup> <sup>2</sup> j j *m* for spin manifolds, a similar inequality will be found for non-spin manifolds. By the proof in §4, *s*≥ <sup>3</sup> <sup>2</sup> j j*r* for the spin covering *N*, or equivalently,

$$\begin{aligned} 2^i n + 2^i - 1 &\geq \frac{3}{2} 2^i |m| \\ n &\geq \frac{3}{2} |m| - \left(1 - \frac{1}{2^i}\right) .\end{aligned} \tag{54}$$

Since (2*<sup>i</sup>* – 1)(*ρ*(*M*) – 1) is an upper bound for *b*1(*N*) – *b*1(*M*), after *b*1(*M*) is set equal to zero,

$$2^i n + (2^i - 1)\rho(M) = \frac{3}{2} 2^i |m| \tag{55}$$

or

$$m \geq \frac{3}{2}|m| - \rho(M) + \frac{\rho(M)}{2^i}.\tag{56}$$

It has been proven that there exist nonsmoothable spin manifolds with *b*<sup>2</sup> ≥ <sup>5</sup> <sup>4</sup> j j *σ* þ 2 [24]. The strict inequality yields a contradiction with the demarcation between smooth and non-smooth structures on a spin four-manifold, which conjectured for coefficients of the intersection form generally.

Similarly, it is claimed that there are nonsmoothable non-spin manifolds with *b*<sup>2</sup> ≥ <sup>5</sup> <sup>4</sup> j j *<sup>σ</sup>* . The inequality derived for non-spin manifolds *<sup>n</sup>* <sup>≥</sup>j j *<sup>m</sup>* � <sup>1</sup> � <sup>1</sup> 2*i*�<sup>1</sup> may be translated to a bound for the second Betti number.

$$\begin{split} \frac{b\_2}{|\sigma|} &= \frac{8|m| + 2n}{8|m|}\\ &\geq 1 - \frac{1}{4} \left( 1 + \frac{1}{|m|} \left( 1 - \frac{1}{2^{i-1}} \right) \right) \\ &= \frac{5}{4} - \frac{2}{|\sigma|} \left( 1 - \frac{1}{2^{i-1}} \right) .\end{split} \tag{57}$$

This lower bound for *b*<sup>2</sup> is less than or equal to <sup>5</sup> <sup>4</sup> j j *σ* for *i* ≥ 1. The tighter inequality for the coefficients in the intersection form is equivalent to

$$\begin{split} \frac{b\_2}{|\sigma|} &\geq 1 + \frac{1}{4} \left( \frac{3}{2} - \frac{1}{|m|} \left( 1 - \frac{1}{2^i} \right) \right) \\ &= \frac{11}{8} - \frac{2}{|\sigma|} \left( 1 - \frac{1}{2^i} \right) . \end{split} \tag{58}$$

The contradiction is resolved in the inequality if the lower bound for *<sup>b</sup>*<sup>2</sup> j j *<sup>σ</sup>* can be increased. Then, the nonsmooth manifolds can exist in the region <sup>5</sup> <sup>4</sup> j j *<sup>σ</sup>* <sup>þ</sup> <sup>2</sup><sup>≤</sup> *<sup>b</sup>*<sup>2</sup> <sup>≤</sup> <sup>11</sup> <sup>8</sup> j j *σ* when a spin structure exists and <sup>5</sup> <sup>4</sup> j j *<sup>σ</sup>* � 2 1 � <sup>1</sup> 2*i*�<sup>1</sup> <sup>≤</sup>*b*<sup>2</sup> <sup>≤</sup> <sup>11</sup> <sup>8</sup> j j *<sup>σ</sup>* � 2 1 � <sup>1</sup> 2*i* when there is no spin structure.

A lower bound for *b*<sup>2</sup> also can be derived for smooth, oriented non-spin manifolds by the following set of equations

$$\begin{aligned} \dim\_{\mathbb{Z}\_2} \mathcal{H}^2(\mathcal{M}; \mathbb{Z}\_2) &= b\_2(\mathcal{M}) + \mathbf{2}t \\ t &= \dim\_{\mathbb{Z}\_2} (\operatorname{Tor}\_2 \mathcal{H}\_1(\mathcal{M}; \mathbb{Z}\_2 \times \mathbb{Z}\_2)), \end{aligned} \tag{59}$$

$$b\_1(\mathbf{N}; \mathbb{Z}\_2) \le 2^i b\_1(\mathbf{M}; \mathbb{Z}\_2) - 2^i + \mathbf{1},\tag{60}$$

and

*Smooth Structures on Spin Manifolds in Four Dimensions DOI: http://dx.doi.org/10.5772/intechopen.106368*

$$\begin{split} b\_{2}(\mathcal{N};\mathbb{Z}\_{2}) &= 2^{i}e(\mathcal{M}) - 2 + 2b\_{1}(\mathcal{N};\mathbb{Z}\_{2}) \\ &\leq 2^{i}(2 - 2b\_{1}(\mathcal{M};\mathbb{Z}\_{2}) + b\_{2}(\mathcal{M};\mathbb{Z}\_{2})) + 2\left(2^{i}b\_{1}(\mathcal{M};\mathbb{Z}\_{2}) - 2^{i} + 1\right) \\ &= 2^{i+1} - 2^{i+1}b\_{1}(\mathcal{M};\mathbb{Z}\_{2}) + 2^{i}b\_{2}(\mathcal{M};\mathbb{Z}\_{2}) + 2^{i+1}b\_{1}(\mathcal{M};\mathbb{Z}\_{2}) - 2^{i+1} + 2 \\ &= 2^{i}b\_{2}(\mathcal{M};\mathbb{Z}\_{2}) + 2, \end{split} \tag{61}$$

and, since the degree of the spin covering of *M* is even, *t* equals one [6]. It follows that

$$b\_2(N) + 2 \le 2^i (b\_2(M) + 2) + 2 = 2^i b\_2(M) + 2^{i+1} + 2 \tag{62}$$

or

$$b\_2(N) \le 2^i b\_2(M) + 2^{i+1}.\tag{63}$$

By the strong <sup>10</sup> <sup>8</sup> inequality for spin manifolds,

$$b\_2(N) \ge \frac{5}{4} |\sigma(N)| + 2 = \frac{5}{4} 2^i |\sigma(M)| + 2. \tag{64}$$

and

$$\begin{split} b\_2(M) &\geq \frac{1}{2^i} \left( \frac{5}{4} 2^i |\sigma(M)| + 2 - 2^{i+1} \right) \\ &= \frac{5}{4} |\sigma(M)| - \left( 2 - \frac{1}{2^{i-1}} \right) .\end{split} \tag{65}$$

The tighter inequality derived in §4 for spin manifolds yields

$$b\_2(N) \ge \frac{11}{8} |\sigma(N)| \ge \frac{11}{8} 2^i |\sigma(M)|\tag{66}$$

and

$$\begin{split} b\_2(M) &\geq \frac{1}{2^i} \left( \frac{\mathbf{11}}{8} 2^i |\sigma(M)| - 2^{i+1} \right) \\ &= \frac{\mathbf{11}}{8} |\sigma(M)| - 2. \end{split} \tag{67}$$

The range for *b*<sup>2</sup> is narrower for *i* ≥ 1 and there is a region below <sup>11</sup> <sup>8</sup> j j *σ* for which the existence of smooth structures remains to be established.

**Theorem 5.2.** An oriented, simply connected, compact four-manifold with an indefinite intersection form and a spin covering space has a smooth structure only if *b*2 j j *<sup>σ</sup>* <sup>≥</sup> <sup>11</sup> 8 .

**Proof.** Consider an oriented, simply connected, compact, four-dimensional manifold *M* and the 2*<sup>i</sup>* -fold spin covering *<sup>N</sup>* ! *<sup>M</sup>*. From the equation <sup>~</sup>*<sup>f</sup>* <sup>∗</sup> *τ<sup>F</sup>* <sup>¼</sup> *<sup>α</sup>*0*τE*, where *α*<sup>0</sup> is the degree of the pull-back map ~*f* ∗ from *KG*(*BF, SF*) to *KG*(*BE, SE*), the trace of the relation *<sup>ρ</sup>*<sup>ℓ</sup> <sup>~</sup>*<sup>f</sup>* <sup>∗</sup> *τ<sup>F</sup>* <sup>¼</sup> *<sup>ψ</sup>*<sup>ℓ</sup>*α*<sup>0</sup> *<sup>ρ</sup>*<sup>ℓ</sup>ð Þ *<sup>E</sup> <sup>τ</sup><sup>E</sup>* when projected to an *<sup>S</sup>*<sup>1</sup> module in the subspace *E*' and *F*<sup>0</sup> , introduces a factor of 2*<sup>i</sup>* in the pull-back of the kernel of the map from *Pin*<sup>2</sup> to *S*<sup>1</sup> , *<sup>c</sup>* **<sup>1</sup>** � <sup>~</sup>**<sup>1</sup>** *c*<sup>∈</sup> <sup>ℤ</sup> <sup>2</sup>*<sup>i</sup>* , and an overall factor of 2<sup>2</sup>*<sup>i</sup>* . Then,

$$a = \mathfrak{Z}^{\acute{i}} \mathfrak{Z}^{|k|} \tag{68}$$

and

$$|b\_+(N) \ge 2|k| + k + 2^i = 3|k| + 2^i. \tag{69}$$

The lower bound for the second Betti number of the spin covering *N* would be

$$b\_2(N) = \frac{\mathbf{11}}{\mathbf{8}} |\sigma(N)| + \mathbf{2}^{i+1}.\tag{70}$$

By Eq. (63),

$$\begin{aligned} \mathfrak{L}^i b\_2(M) + \mathfrak{L}^{i+1} &= \frac{\mathfrak{11}}{8} |\sigma(N)| + \mathfrak{2}^{i+1} = \frac{\mathfrak{11}}{8} \mathfrak{2}^i |\sigma(N)| + \mathfrak{2}^{i+1} \\ b\_2(M) &= \frac{\mathfrak{11}}{8} |\sigma(M)|. \end{aligned} \tag{71}$$

Therefore, the theoretically predicted inequality for the second Betti number of smooth, oriented four-manifolds has been derived.

#### **6. Lower bound for the genus of a surface embedded in a four-manifold**

The genus of an embedded surface Σ in a four-manifold *M* may be given the lower bound

$$g(\Sigma) > \left| \frac{2^r + 1}{2^{r+1}} \sum^2 - \frac{2^{r+1}}{2^r - 1} \sigma(M) \right| - \frac{2^{r-1}}{2^r - 1} b\_2(M). \tag{72}$$

where Σ<sup>2</sup> is the intersection product of the second cohomology class Σ [19]. If the <sup>11</sup> 8 conjecture is true, the bound can be increased to

$$g(\Sigma) > \frac{11}{8} \left| \frac{2^r + 1}{2^{r+1}} \Sigma^2 - \frac{2^{r+1}}{2^r - 1} \sigma(M) \right| - \frac{2^{r-1}}{2^r - 1} b\_2(M). \tag{73}$$

For an algebraic surface Σ*<sup>d</sup>* of even degree d embedded in ℂℙ2, with 2 *d*, *g* P *d* � �≥ <sup>11</sup> <sup>32</sup> *<sup>d</sup>*<sup>2</sup> � � � <sup>19</sup> <sup>9</sup> [25].

The Thom conjecture for curves of algebraic curves of degree *d* states that *g* P *d* � �≥ <sup>1</sup> <sup>2</sup> *<sup>d</sup>*<sup>2</sup> � <sup>3</sup> <sup>2</sup> <sup>þ</sup> 1. The replacement of <sup>11</sup> <sup>32</sup> by <sup>1</sup> <sup>2</sup> in the lower bound for the genus, requires the substitution of 2 for <sup>11</sup> <sup>8</sup> as the lower limit for *<sup>b</sup>*<sup>2</sup> j j *<sup>σ</sup>* . Consider the intersection form *mE*<sup>8</sup> ⊕ *nH*. Since

$$\begin{aligned} b\_2(mE\_8 \oplus nH) &= 8m + 2n\\ \sigma(mE\_8 \oplus nH) &= 8m, \end{aligned} \tag{74}$$

the inequality

$$\frac{8m+2n}{8m} \ge 2\tag{75}$$

is equivalent to *n* ≥ 4 *m*. This very tight bound is not expected to be valid for a large class of smooth four-manifolds.

The <sup>3</sup> <sup>2</sup> conjecture for irreducible simply connected four-manifolds is *χ* ≥ <sup>3</sup> <sup>2</sup> j j *σ* , where *χ* ¼ 2 þ *rank Q*ð Þ¼ 2 þ *b*<sup>2</sup> is the Euler characteristic. It follows that

$$b\_2 \ge \frac{3}{2}|\sigma| - 2.\tag{76}$$

Substituting the new coefficient into the lower bound for the genus of the embedded surface in an irreducible manifold,

$$\log\left(\sum\right) \ge \left(\frac{3}{2} - \frac{2}{|\sigma|}\right) \left|\frac{\mathcal{Z}^r + 1}{3 \cdot \mathcal{Z}^{r+1}} \sum\right.\\ - \left. \frac{\mathcal{Z}^{r-1}}{\mathcal{Z}^r - 1} \sigma(M) \right| - \left. \frac{\mathcal{Z}^{r-1}}{\mathcal{Z}^r - 1} b\_2(M), \tag{77}$$

where 2*<sup>r</sup>* <sup>P</sup>k½ �. Therefore, the genus of an algebraic curve of degree *<sup>d</sup>* embedded in an irreducible, simply connected manifold would satisfy the inequality *g* P *d* � �≥ <sup>3</sup> <sup>8</sup> � <sup>1</sup> 2j j *σ* � �*d*<sup>2</sup> <sup>þ</sup> *<sup>γ</sup>*1*<sup>d</sup>* <sup>þ</sup> *<sup>γ</sup>*<sup>0</sup> with *<sup>γ</sup>*<sup>0</sup> and *<sup>γ</sup>*<sup>1</sup> being constants.

#### **7. Conclusion**

The classification of four-manifolds has been reduced to the definite signature with odd intersection forms that are diagonal, *n***1** + *m*(�**1**) or indefinite signature with even intersection forms *mE*<sup>8</sup> + *nH*, where *E*<sup>8</sup> is the exceptional Lie group Cartan matrix and *H* is the matrix 0 1 1 0 � �, which is the form of *<sup>S</sup>*<sup>2</sup> � *<sup>S</sup>*<sup>2</sup> . All known smooth, oriented four-manifolds with an even intersection form are known to have coefficients satisfying *n*≥ <sup>3</sup> <sup>2</sup> j j *<sup>m</sup>* or equivalently, a second Betti number satisfying *<sup>b</sup>*<sup>2</sup> <sup>≥</sup> <sup>11</sup> <sup>8</sup> j j *σ* . The oriented, spin geometries in four dimensions have been demonstrated to admit a smooth structure only if *n* ≥ |*m*| + 1 or *b*<sup>2</sup> ≥ <sup>5</sup> <sup>4</sup> j j *σ* þ 2. The proof has been extended to non-spin manifolds with the inequality *<sup>n</sup>*<sup>≥</sup> j j *<sup>m</sup>* � <sup>1</sup> � <sup>1</sup> 2*i* � � and *<sup>b</sup>*<sup>2</sup> <sup>≥</sup> <sup>5</sup> <sup>4</sup> j j *<sup>σ</sup>* � <sup>2</sup> � <sup>1</sup> 2*i*�<sup>1</sup> � �. It is found here that the lower bound for the signed Betti number *b*<sup>+</sup> is larger than 2| *<sup>k</sup>*| + 1, where j j *<sup>k</sup>* <sup>¼</sup> <sup>3</sup> <sup>16</sup> j j *σ* . Considering a cyclic covering of a non-spin manifold and introducing the degree into the proof for the spin manifold, the inequality *b*<sup>2</sup> ≥ <sup>11</sup> <sup>8</sup> j j *σ* . This increase in the lower bound for the second Betti number allows the existence of nonsmoothable manifolds with *b*<sup>2</sup> ≥ <sup>5</sup> <sup>4</sup> j j *σ* within a strict demarcation between the regions for smooth and nonsmooth structures.

The existence of smooth, compact simply-connected manifolds with <sup>5</sup> <sup>4</sup> <sup>&</sup>lt; *<sup>b</sup>*<sup>2</sup> j j *<sup>σ</sup>* <sup>&</sup>lt; <sup>11</sup> 8 present potential counterexamples to the <sup>11</sup> <sup>8</sup> . The topological sums ℂℙ<sup>2</sup> #ℂℙ<sup>2</sup> are included within these limits. It is proven in §4 that these spaces cannot have spin structures by Rohlin's theorem. Amongst the connected sums ℂℙ<sup>2</sup> #ℓℂℙ<sup>2</sup> , only those values of ℓ congruent to 1 modulo 16 would satisfy the condition on the divisibility of the signature by 16. The absence of spin structures on ℂℙ<sup>2</sup> # 16 ð Þ *<sup>r</sup>* <sup>þ</sup> <sup>1</sup> ℂℙ<sup>2</sup> for <sup>ℓ</sup> <sup>≥</sup> 1 is established through the computation of the second Stiefel-Whitney class. Therefore, ℂℙ<sup>2</sup> #ℂℙ<sup>2</sup> is the unique spin manifold in this set, which is necessary for consistency of the <sup>10</sup> <sup>8</sup> theorem. The conclusion on the nonexistence of oriented, compact simply

connected four-manifolds, having both a smooth structure and a spin geometry, continues to be valid for topological sums of *S*<sup>4</sup> , *<sup>S</sup>*<sup>2</sup> � *<sup>S</sup>*<sup>2</sup> and *<sup>K</sup>*3 and complex algebraic surfaces, since the condition of the existence of a spin structure requires generally an increased minimum value of <sup>11</sup> <sup>8</sup> for *<sup>b</sup>*<sup>2</sup> j j *σ* .

### **Classification:**

MSC: 57N13; 57R19

## **Author details**

Simon Davis Research Foundation of Southern California, La Jolla, CA, USA

\*Address all correspondence to: sbvdavis@outlook.com

© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Smooth Structures on Spin Manifolds in Four Dimensions DOI: http://dx.doi.org/10.5772/intechopen.106368*

### **References**

[1] Matsumoto Y. On the bounding genus of homology 3-spheres. Journal of the Faculty of Science of the University of Tokyo. Section I, Mathematics. 1982;**29**: 287-318

[2] Manolescu C. On the intersection forms of spin four-manifolds with boundary. Mathematische Annalen. 2014;**359**:695-728

[3] Eguchi T, Gilkey PB, Hanson AJ. Gravitation, gauge theories and differential geometry. Physics Reports. 1980;**66**:213-293

[4] Furuta M. Monopole equation and <sup>11</sup> 8 conjecture. Mathematical Research Letters. 2001;**8**:279-291

[5] Hopkins M, Lin J, Shi X, Xu Z. Intersection forms of spin 4-manifolds and the pin(2)-Equivariant Mahowald invariant. Communications of the American Mathematical Society. 2022;**2**

[6] Furuta M, Kametani Y, Matsue H. Spin 4-manifolds with signature=32. Mathematical Research Letters. 2001;**8**: 293-301

[7] Kikuchi K. Representing positive homology classes of ℂℙ<sup>2</sup> #2ℂℙ<sup>2</sup> and ℂℙ<sup>2</sup> #3ℂℙ<sup>2</sup> . Proceedings of the American Mathematical Society. 1993;**117**:861-869

[8] Barth W, Hulek K, Peters C, Van den Ven A. Compact Complex Surfaces. Heidelberg: Springer; 2004

[9] Furuta M. Finite-dimensional approximations in geometry. In: Li T, editor. Proceedings of the International Congress on Mathematics 2002. Vol. III. Beijing: World Scientific Publishing, Singapore; August 20-28, 2002, 2002. pp. 395-403

[10] Lawson B. The Theory of Gauge Fields in Four Dimensions, CMBS Regional Conference in Mathematics. Providence: American Mathematical Society; 1985

[11] Minami N. G-join theorem - an Unbased G-Freudenthal theorem, Homotopy representations, and a Borsuk-Ulam theorem. Nagoya, Japan: Nagoya Institute of Technology

[12] Schmidt B. Ein Kriterium für die Existenz äquivarianter Abbildungen zwishcen reellen Darstellungssphären der Gruppe Pin(2). University of Bielefeld: Diplomarbeit; 1997

[13] Furuta M, Kametani Y. The Seiberg-Witten Equations and Equivariant e-Invariants. Tokyo, Japan: University of Tokyo; 2001

[14] Furuta M, Kametani Y, Matsue H, Minami N. Homotpy theoretical considerations of the Bauer-Furuta stable Seiberg-Witten invariants. Geom. & Topology Monographs. 2007;**10**: 155-166

[15] Freedman MH, Taylor L. Λ-splitting of 4-manifolds. Topology. 1977;**16**: 181-184

[16] Kervaire M, Milnor JW. On 2-spheres in 4-manifolds. Proceedings of the National Academy of Sciences of the United States of America. 1981;**47**:1861-1657

[17] Goto R. Unobstructed deformations of generalized complex structures induced by C∞ logarithmic Symplectic structures and logarithmic Poisson structures. In: Futaki A, Miyaoka R, Tang Z, Zhang W, editors. 10th China-Japan Conference 2014. Geometry and Topology of Manifolds. Tokyo: Springer; 2016. pp. 159-184

[18] Kirby RC. *T*<sup>3</sup> *Lie* and ℂℙ<sup>2</sup> #9 �ℂℙ<sup>2</sup> , the Topology of 4-Manifolds. Vol. 1374. Heidelberg: Lecture Notes in Mathematics; Springer-Verlag; 1989

[19] Park J. Simply connected symplectic 4-manifolds with *b*<sup>þ</sup> <sup>2</sup> <sup>¼</sup> 1 and *<sup>c</sup>*<sup>2</sup> <sup>1</sup> ¼ 2. Inventiones Mathematicae. 2005;**159**: 657-667

[20] Kotschick D. On manifolds homeomorphic to ℂℙ<sup>2</sup> #ℂℙ<sup>2</sup> . Inventiones Mathematicae. 1989;**95**:591-600

[21] Donaldson SK. Irrationality and the h-Cobordism conjecture. Journal of Differential Geometry. 1987;**26**:141-168

[22] Lee R, Li T-J. Intersection forms of non-spin four manifolds. Mathematische Annalen. 2001;**319**:311-318

[23] Bohr C. On the signature of even 4 manifolds. Mathematical Proceedings of the Cambridge Philosophical Society. 2002;**132**:453-469

[24] Nakamura N. Pin�(2)-monopole equations and intersection forms with local coefficients of 4-manifolds. Mathematische Annalen. 2013;**357**: 915-939

[25] Kotschick D, Matic G. Embedded surfaces in four-manifolds, branched covers, and SO(3)-invariants. Mathematical Proceedings of the Cambridge Philosophical Society. 1995; **117**:275-286

#### **Chapter 5**
