Smooth Structures on Spin Manifolds in Four Dimensions

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 mE8⊕n0110, the oriented, simply connected, compact, spin four-manifold will admit a smooth structure if and only if n≥32m. The inequality is changed to n≥32m−1−12i, there is a 2i-fold spin covering of a non-spin manifold M given the demonstration of n≥32m for oriented, compact, spin manifolds. A closer examination of the proof reveals that the lower bound for b+ can be increased to 3|k| + 1, where k=315σ for a spin manifold, yielding b2≥118σ+2. The projection of a spin covering to a non-spin manifold yields the lower bound b2≥118σ, which establishes the prediction for the coefficients of intersection forms for this class of smooth, oriented, simply connected, compact four-manifolds.

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 ${\mathit{mE}}_8\oplus n\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$, it will continue to have a smooth structure if $n\ge \frac{3}{2}\left|m\right|$. In the (b₂, σ) plane, the smooth structures are located above the line $b_2\ge \frac{11}{8}\left|\sigma \right|$ [1] and the nonsmooth structures are located below the line $b_2\le \frac{5}{4}\left|\sigma \right|$, 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 $\frac{11}{8}$ 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 K3 [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 b₂ = 1 and σ = 1 for $\mathrm{ℂ}{\mathrm{\mathbb{P}}}^2$.

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 $b_2\ge \frac{10}{8}\left|\sigma \right|+2$ was established for oriented, connected spin manifolds [4]. It may be increased with the use of stable homotopy groups of spheres and Pin₂-equivariant homotopy invariants [5] to $b_2\ge \frac{10}{8}\sigma +4$. The lower limit will be increased first to $\frac{b_2}{\left|\sigma \right|}>\frac{21}{16}$ for spin manifolds with signature σ ≥ 16 as a result of a theorem on the nonexistence of smooth four-manifolds with the intersection form $+4E-8\oplus 5H$ [6]. Then, the coefficient of $\frac{11}{8}$ 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 $\frac{3}{2}$ 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 $\mathrm{ℝ}$, b₊ and b₋ will denote the number of positive and negative eigenvalues respectively. Then the second Betti number and the signature are b₂ = b₊ + b₋ and σ = b₊ − b₋ respectively. Let $k=-\frac{\sigma \left(M\right)}{16}$ and the inequality $b_{+}\ge 3k=-\frac{3\sigma }{16}$. When the signature is negative, and –σ may be replaced by |σ|,

$$\begin{array}{l}\frac{b_2+\sigma }{2}\ge \frac{3\left|\sigma \right|}{16}\\ b_2\ge 2\left[-\frac{\sigma }{2}+\frac{3\left|\sigma \right|}{16}\right]=\frac{11}{8}\left|\sigma \right|.\end{array}$$E1

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 $b_{+}\ge \frac{3\left|\sigma \right|}{16}$.

Similarly, if the orientation is chosen such that $b_{+}\ge 2k+1=-\frac{\sigma }{8}+1$ is equivalent to $b_{+}\ge \frac{\left|\sigma \right|}{8}+1$,

$$\begin{array}{l}\frac{b_2+\sigma }{2}\ge \frac{\left|\sigma \right|}{8}+1\\ b_2\ge 2\left[\frac{\left|\sigma \right|}{2}+\frac{\left|\sigma \right|}{8}+1\right]=\frac{5}{4}\left|\sigma \right|+2.\end{array}$$E2

It may be demonstrated that a spin 4-manifold can admit a smooth structure when the intersection form is $4E_8\oplus \mathit{nH}$ for n ≥ 6 [7]. Consequently, there is no smooth manifold with the intersection form $4E_8\oplus 5H$. Since the coefficients are relatively prime, the ratio $\frac{n}{m}=\frac{5}{4}$ is achieved only for the intersection forms $4{\mathit{kE}}_8\oplus 5\mathit{kH}$, k = 1, 2, 3, .... Given that there is a manifold M with the intersection form $4E_8\oplus 5H$, 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 $\frac{21}{16}\left|\sigma \right|$, which is a bound closer to the line with gradient $\frac{11}{8}$ for non-zero signature.

Proof. The line representing smooth structures must be $n\ge \frac{5}{4}m$. Then

$$\frac{b_2}{\left|\sigma \right|}=\frac{8m+2n}{8m}\ge \frac{8m+\frac{5}{2}m}{8m}=\frac{21}{6}.$$E3

When

$$\frac{10}{8}+\frac{2}{\left|\sigma \right|}\le \frac{21}{16},$$E4

or

$$\left|\sigma \right|\ge \frac{32}{5},$$E5

this bound is better than the established value. If

$$\frac{10}{8}+\frac{4}{\left|\sigma \right|}<\frac{21}{16},$$E6

or

$$\left|\sigma \right|\ge \frac{65}{5}.$$E7

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 $\frac{21}{16}$ 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 $\#\mathit{kK}3\#\mathrm{\ell }{S}^2\times S^2$ or $\#k{\mathrm{ℂ\mathbb{P}}}^2\#\mathrm{\ell }\overline{{\mathrm{ℂ\mathbb{P}}}^2}$.

Lemma 2.2. All manifolds $\#\mathit{kK}3\#\mathrm{\ell }{S}^2\times S^2$ with k > 0 have $\frac{b_2}{\left|\sigma \right|}\ge \frac{11}{8}$, with the bound saturated by K3. The coefficients in $\#k{\mathrm{ℂ\mathbb{P}}}^2\#\mathrm{\ell }\overline{{\mathrm{ℂ\mathbb{P}}}^2}$ must satisfy the inequalities $\frac{3}{19}k\le \mathrm{\ell }\frac{19}{3}k$ for a smooth structure to exist by the $\frac{11)}{8}$ conjecture.

Proof. Since K3 has an interesection form with 3 positive and 19 negative eigenvalues, b₂(K3) = 22 and σ(K3) = −16. The intersection form of S² × S², H, has the eigenvalues 1 and − 1, and b₂(S² × S²) = 2, while σ(S² × S²) = 0. Then

$$\begin{array}{l}{b}_2\left(\#\mathit{kK}3\#\mathrm{\ell }{S}^2\times S^2\right)=22k+2\mathrm{\ell }\\ \sigma \left(\#\mathit{kK}3\#\mathrm{\ell }{S}^2\times S^2\right)=-16k\end{array}$$E8

and

$$\frac{b_2\left(\#\mathit{kK}3\#\mathrm{\ell }{S}^2\times S^2\right)}{\left|\sigma \left(\#\mathit{kK}3\#S^2\times S^2\right)\right|}=\frac{11}{8}+\frac{1}{8}\frac{\mathrm{\ell }}{k}.$$E9

The intersection matrix of $\#k{\mathrm{ℂ}\mathbb{O}\mathrm{\mathbb{P}}}^2\#\mathrm{\ell }\overline{{\mathrm{ℂ\mathbb{P}}}^2}$ is diag(1, ..., 1, −1, ..,–1) with

$$\begin{array}{l}{b}_2\left(\#k{\mathrm{ℂ\mathbb{P}}}^2\#\mathrm{\ell }\overline{{\mathrm{ℂ\mathbb{P}}}^2}\right)=k+\mathrm{\ell }\\ \sigma \left(\#k{\mathrm{ℂ\mathbb{P}}}^2\#\mathrm{\ell }\overline{{\mathrm{ℂ\mathbb{P}}}^2}\right)=k-\mathrm{\ell }.\end{array}$$E10

It follows that

$$\frac{b_2\left(\#k{\mathrm{ℂ\mathbb{P}}}^2\#\mathrm{\ell }\overline{{\mathrm{ℂ\mathbb{P}}}^2}\right)}{\sigma \left(\#k{\mathrm{ℂ\mathbb{P}}}^2\#\mathrm{\ell }\overline{{\mathrm{ℂ\mathbb{P}}}^2}\right)}=\frac{k+\mathrm{\ell }}{\left|k-\mathrm{\ell }\right|}=\left\{\begin{array}{ll}1+\frac{2\mathrm{\ell }}{k-\mathrm{\ell }}& k>\mathrm{\ell }\\ 1+\frac{2k}{\mathrm{\ell }-k}& \mathrm{\ell }>k\end{array}\right.$$E11

either

$$\begin{array}{c}1+\frac{2\mathrm{\ell }}{k-\mathrm{\ell }}\ge \frac{11}{8},\\ \mathrm{\ell }\ge \frac{3}{19}k\end{array}$$E12

or

$$\begin{array}{c}1+\frac{2k}{\mathrm{\ell }-k}\ge \frac{11}{8}\\ k\ge \frac{3}{19}\mathrm{\ell }\end{array}$$E13

which may combine in the inequalities

$$\frac{3}{19}k\le \mathrm{\ell }\le \frac{19}{3}k.$$E14

A bound may be established for simply connected complex surfaces with an even cup product form [8]. It is known that, for these manifolds, b₂ = c₂–2 and $\sigma =\frac{1}{3}\left(c_1^2-2c_2\right)$, where c₁ and c₂ are the first two Chern numbers. Defining

$$b=\frac{1}{16}\left(8b_2-11|\sigma |\right),$$E15

the $\frac{11}{8}$ conjecture is equivalent to b ≥ 0.When σ < 0,

$$\begin{array}{c}48b=3\left(8\left(c_2-2\right)+\frac{11}{8}\left(c_2^2-2c_2\right)\right)\\ =11c_1^2+2c_2-48.\end{array}$$E16

When σ > 0,

$$\begin{array}{c}48b=3\left(8\left(c_2-2\right)-11\left(c_1^2-2c_2\right)\right)\\ =-11c_1^2+43c_2-48.\end{array}$$E17

Adding the two inequalities gives $c_2\ge \frac{32}{15}$. This inequality is satisfied for complex surfaces, since $c_1^2\ge 0$ and $c_2\ge 3$. However, for negative signature,

$$11c_1^2+2c_2-48\ge -42.$$E18

For positive signature, $c_1^2\le 3c_2$ and

$$-11c_1^2+43c_2-48\ge 10c_2-48\ge -18.$$E19

It is clear that 48b = 3(8b₂–11|σ|) is integer. By Rohlin’s theorem, the signature will be divisible by 16 and b also would be integer. By Eqs. (18) and (19), $b\ge -\frac{7}{8}$ and $b\ge -\frac{3}{8}$ respectively. Then b ≥ 0 and the $\frac{11}{8}$ 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_o to E¹, D: Γ(E⁰) → (E¹), and ind D = dim Ker D – dim Coker D. Now consider a Whitney sum with a finite-dimensional vector bundle, such that $D=L+L^{\prime }:V^0\oplus \mathrm{\Gamma }\left(E^0\right)\to V^1\oplus \mathrm{\Gamma }\left(E^1\right)$, where L is a finite-dimensional mapping and L’ is an isomorphism between infinite-dimensional spaces. Then ind D = dim V₀ – dim V₁. 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 Pin₂-equivariant map given by ${\mathrm{ℍ}}^{\infty }\oplus {\stackrel{\sim }{R}}^{\infty }\to {\mathrm{ℍ}}^{\infty }\oplus {\stackrel{\sim }{R}}^{\infty }$, where $\stackrel{\sim }{R}$ is the nontrivial one-dimensional real representation space of Pin₂. A finite-dimensional approximation is a Pin₂-equivariant map ${\mathrm{ℍ}}^{c_0}\oplus {\stackrel{\sim }{R}}^{d_0}\to {\mathrm{ℍ}}^{c_1}\oplus {\stackrel{\sim }{R}}^{d_1}$, where $c_0-c_1=-\frac{\sigma \left(M\right)}{16}$ and $d_0-d_1=b^{+}\left(M\right)$, in a generalized Kuranshi construction [9].

The four-dimensional spin manifold will admit a Spin₄ bundle and vector bundles T, S⁺, S⁻ and Λ constructed from the Spin₄ × Pin₂ modules $–{\mathrm{ℍ}}_{+}{,}_{+}\mathrm{ℍ}{,}_{-}\mathrm{ℍ}{\text{and}}_{+}{\mathrm{ℍ}}_{+}$ defined by the actions $q_{-}{\mathit{aa}}_{+}^{-1},q_{+}\varphi q_0^{-1},q_{+}\omega q_0^{-1}$ and $q_{+}\omega q_{+}^{-1}$ for (q₋, q₊, q₀) ∈ Spin₄ × Pin₂ and $a{\in }_{–}{\mathrm{ℍ}}_{+},\varphi {\in }_{+}\mathrm{ℍ},\psi \in –\mathrm{ℍ}\text{and}\omega {\in }_{+}{\mathrm{ℍ}}_{+}$. If $\stackrel{\sim }{R}$ is the real one-dimensional Pin₂ module defined by multiplication by Pin₂/S¹, $\stackrel{\sim }{T}=T\otimes \stackrel{\sim }{R},C:T\otimes S^{+}\to S^{-}$ with $\left(a,\varphi \right)\to \mathit{a\varphi },\overline{C}:T\otimes \stackrel{\sim }{T}\to \stackrel{\sim }{\mathrm{\Lambda }}$ with $\left(a,b\right)\to \overline{a}b,D_1=C{\nabla }_1:\mathrm{\Gamma }\left(S^{+}\right)\to \mathrm{\Gamma }\left(S^{-}\right)$, $D_2=\overline{C}{\nabla }_2:\mathrm{\Gamma }\left(\stackrel{\sim }{T}\right)\to \mathrm{\Gamma }\left(\stackrel{\sim }{\mathrm{\Lambda }}\right)$, $D=D_1\oplus D_2:\mathrm{\Gamma }\left(S^{+}\oplus \stackrel{\sim }{T}\right)\to \mathrm{\Gamma }\left(S^{-}\oplus \stackrel{\sim }{\mathrm{\Lambda }}\right)$, $Q:S^{+}\oplus \stackrel{\sim }{T}\to S^{-}\oplus S^{-}\oplus \stackrel{\sim }{\mathrm{\Lambda }}$ with $\left(\varphi ,a\right)\to \left(\mathit{a\varphi i},\mathit{\varphi i}\overline{\varphi }\right)$, then D + Q: V → W where V is the $L_4^2$ completion of $\mathrm{\Gamma }\left(S^{+}\oplus \stackrel{\sim }{T}\right)$ and W is the $L_3^2$ completion of $\mathrm{\Gamma }\left(S^{-}\oplus \stackrel{\sim }{\mathrm{\Lambda }}\right)$ [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, $\stackrel{\sim }{f}:\mathit{BE}\to \mathit{BF}$ be a G-equivariant bundle map preserving boundaries. By the Thom isomorphism theorem, K_G(BE, SE) and K_G(BF, SF) are generated by Thom classes τ_E and τ_F. With ${\stackrel{\sim }{f}}^{\ast }$ being the pullback map K_G(BF, SF) → K_G(BE, SE), ${\stackrel{\sim }{f}}^{\ast }{\tau }_F={\alpha }_0{\tau }_E$, where α₀ ∈ K_G(M) is the degree of ${\stackrel{\sim }{f}}^{\ast }$. Since the restriction of the Thom classes to the zero sections are the Euler classes of E and F, ${\sum }_d{\left(-1\right)}^d\left[{\mathrm{\Lambda }}^{d}F\right]={\alpha }_0{\sum }_d{\left(-1\right)}^d\left[{\mathrm{\Lambda }}^{d}E\right]$.

Contracting M = pt., G = Pin₂,

$$\begin{array}{l}{E}_{\mathit{pt}.}=V_{\lambda ,\mathrm{ℂ}}=\left({\mathrm{ℍ}}^{k+m}+{\stackrel{\sim }{R}}^n\right)\otimes \mathrm{ℂ}\\ F_{\mathit{pt}.}={\overline{W}}_{\lambda ,\mathrm{ℂ}}=\left({\mathrm{ℍ}}^m+{\stackrel{\sim }{R}}^{b_{+}+n}\right)\otimes \mathrm{ℂ},\end{array}$$E20

where V_λ is the subspace of V spanned by the eigenspace of D* D with eigenvalues less than or equal to λ, and W_λ is the subspace of w spanned by eigenspaces of DD* with eigenvalues less than or equal to $\lambda ,f:V_{\lambda ,\mathrm{ℂ}}\to {\overline{W}}_{\lambda ,\mathrm{ℂ}}$ is the complexification of $D_{\lambda }+Q_{\lambda }$,where $D_{\lambda }+Q_{\lambda }={\left.\left(D+p_{\lambda }Q\right)\right|}_{V_{\lambda }},f\left(u\otimes 1+v\otimes i\right)=\left(D_{\lambda }+Q_{\lambda }\right)u\otimes 1+\left(D_{\lambda }+Q_{\lambda }\right)v\otimes i.$

Suppose that $\phi :V=\mathit{ker}\left(d^{\ast }\right)\oplus \mathrm{\Gamma }\left(V^{+}\right)\to {\mathrm{\Omega }}_{+}^2\oplus \mathrm{\Gamma }\left(V^{-}\right)=W$, $\phi \left(v\right)=L\left(v\right)+\theta \left(v\right)$, $L=\left(\begin{array}{cc}{d}^{+}& 0\\ 0& \partial \end{array}\right)$ is linear, $\theta \left(a,\psi \right)=\left(\sigma \left(\psi \right),\mathit{a\psi }\right)$ is quadratic, where a is the gauge potential in the covariant derivative and σ’ is an automorphism of the space Γ(W⁺) and ${\sigma }^{\prime }\left({\psi }_j\right)={\sigma }^{\prime }\left(z+\mathit{jw}\right)j={\sigma }^{\prime }\left(-\overline{v}+j\overline{m}z\right)=-{\sigma }^{\prime }\left(\psi \right)$. Let $f_{\lambda }:V\to W$ be defined by $u\equiv f_{\lambda }\left(v\right)=v+L^{-1}\left(1-p_{\lambda }\right)\theta \left(v\right)$, $L\left(u\right)=L\left(v\right)+\left(1-p_{\lambda }\right)\theta \left(v\right)$, with p_λ being the projection of V and W onto V_λ and W_λ. Defining ${\phi }_{\mathrm{\Lambda }}:{\oplus }_{\lambda \le \mathrm{\Lambda }}{V}_{\lambda }\to {\oplus }_{\lambda \le \mathrm{\Lambda }}{W}_{\lambda }$, ${\phi }_{\mathrm{\Lambda }}\left(u\right)=p_{\lambda }\phi f_{\mathrm{\Lambda }}^{-1}\left(u\right)$ [10].

Let $T_u\left(v\right)=u-L^{-1}\left(1-p_{\mathrm{\Lambda }}\right)\theta \left(v\right)$. Then

$$\begin{array}{c}‖T_u\left(v_1\right)-T_u\left(v_2\right)‖=‖\left(T_u\left(v_1\right)-u\right)-\left(T_u\left(v_2\right)-u\right)‖\\ =‖-L^{-1}\left(1-p_{\mathrm{\Lambda }}\right)\theta \left(v_1\right)+L^{-1}\left(1-p_{\mathrm{\Lambda }}\right)\theta \left(v_2\right)‖.\end{array}$$E21

The eigenvalue of L⁻¹(1 – p_Λ) is $\frac{1}{\lambda }$ on each W_λ, which has the maximum value $\frac{1}{\mathrm{\Lambda }}$ for λ > Λ. The automorphism σ’($\psi$) is given by ${\sigma }^{\prime }\left(z,w\right)=i\left(\frac{{\left|z\right|}^2-{\left|w\right|}^2}{2}\right)-k\mathit{Re}\left(z\overline{w}\right)+j\mathit{Im}\left(z\overline{w}\right)$, and, if $\widehat{\sigma }\left(z,w\right)=\widehat{\rho }\circ {\sigma }^{\prime }\left(z,w\right)$, where $\widehat{\rho }:T^{\ast }\left(X\right)\to \mathit{Hom}\left(W^{\pm },W^{\mp }\right)$, $\widehat{\rho }\left(v_1\wedge v_2\right)=\frac{1}{2}\left[\rho \left(v_1\right),\rho \left(v_2\right)\right]$, $f_1=\frac{1}{2}\left(e^1\wedge e^2\pm e^3\wedge e^4\right)$, $f_2=\frac{1}{2}\left(e^1\wedge e^3\pm e^4\wedge e^2\right)$, $f_3=\frac{1}{2}\left(e^1\wedge e^4\pm e^2\wedge e^3\right)$, where i, j, k correspond to f₁, f₂, $f_3\in {\mathrm{\Lambda }}^{+}\left(M\right)$, with ${\mathrm{\Lambda }}^2\left(M\right)={\mathrm{\Lambda }}^{+}\left(M\right)\otimes {\mathrm{\Lambda }}^{-1}\left(M\right)$, ${\left|\widehat{\sigma }\left(\psi \right)\right|}^2=\frac{1}{2}{\left|\psi \right|}^2$ [3]. It follows that

$$‖{\left(\widehat{\sigma }{\psi }_1,a_1\psi \right)}_1)-(\widehat{\sigma }{\psi }_2,a_2{\psi }_2)‖\le ‖\left(a_1,{\psi }_1\right)-(a_2,{\psi }_2)‖$$E22

if $\mid {\psi }_1\mid <a_1$, $\mid {\psi }_2\mid <a_2$, and a₁, a₂ < 1. Under these conditions, by the Banach contraction principle, ${\phi }_{\mathrm{\Lambda }}^{-1}\left(0\right)$ is a compact set.

With ${\mathit{BV}}_{\lambda ,\mathrm{ℂ}}=\left\{\left.u\otimes 1+v\otimes i\in V_{\lambda ,\mathrm{ℂ}}\right|‖u‖,‖v‖\le R\right\}$, ${\mathit{SV}}_{\lambda ,\mathrm{ℂ}}=\partial {\mathit{BV}}_{\lambda ,\mathrm{ℂ}}$, let $\overline{f}=f\circ p$, where $p:{\overline{W}}_{\lambda ,\mathrm{ℂ}}\backslash \left\{0\right\}\to S{\overline{W}}_{\lambda ,\mathrm{ℂ}}$. Then the mapping $\overline{f}:{\mathit{BV}}_{\lambda ,\mathrm{ℂ}}\to B{\overline{W}}_{\lambda ,\mathrm{ℂ}}$ is defined to be the cone of $\overline{f}$. If $k>0,\left\{\left.\alpha \in R\left({\mathit{Pin}}_2\right)\right|{\rho }^{\mathrm{\ell }}\left(F\right)\alpha =\left({\psi }^{\mathrm{\ell }}\alpha \right){\rho }^{\mathrm{\ell }}\left(E\right)\right\}\subset \mathit{Ker}\left(R\left({\mathit{Pin}}_2\right)\to R\left(S^{-1}\right)\right)$. Consider an element α of Ker(R(Pin₂) → R(S¹)) satisfying ${\sum }_d{\left(-1\right)}^d\left[{\mathrm{\Lambda }}^{d}F\right]=\alpha {\sum }_d{\left(-1\right)}^d\left[{\mathrm{\Lambda }}^{d}E\right]$. Regarding E and F as S¹ modules, let $E^{\prime }=2\left(k+m\right)\left(\mathrm{ℂ}\oplus {\mathrm{ℂ}}^{\ast }\right)\oplus n$ and $F^{\prime }=2m\left(\mathrm{ℂ}\oplus {\mathrm{ℂ}}^{\ast }\right)\oplus \left(b_{+}+n\right)$ be representation spaces.

Let ${\psi }^{\mathrm{\ell }}$ be the Adams operation and ${\rho }^{\mathrm{\ell }}\left(E\right)$ be the characteristic class satisfying ${\psi }^{\mathrm{\ell }}{\tau }_E={\rho }^{\mathrm{\ell }}\left(E\right){\tau }_E$. Then

$$\begin{array}{c}{\psi }^{\mathrm{\ell }}\left({\stackrel{\sim }{f}}^{\ast }{\tau }_F\right)={\stackrel{\sim }{f}}^{\ast }\left({\psi }^{\mathrm{\ell }}{\tau }_F\right)={\stackrel{\sim }{f}}^{\ast }\left({\rho }^{\mathrm{\ell }}\left(F\right){\tau }_F\right)\\ {\psi }^{\mathrm{\ell }}\left({\alpha }_0{\tau }_E\right)=\left({\psi }^{\mathrm{\ell }}{\alpha }_0\right){\psi }^{\mathrm{\ell }}{\tau }_E={\psi }^{\mathrm{\ell }}{\alpha }_0{\rho }^{\mathrm{\ell }}\left(E\right){\tau }_E\\ {\stackrel{\sim }{f}}^{\ast }\left({\rho }^{\mathrm{\ell }}\left(F\right){\tau }_F\right)={\rho }^{\mathrm{\ell }}\left(F\right){\stackrel{\sim }{f}}^{\ast }{\tau }_F={\rho }^{\mathrm{\ell }}\left(F\right){\alpha }_0{\tau }_E\\ {\rho }^{\mathrm{\ell }}\left(F\right){\alpha }_0{\tau }_E=\left({\psi }^{\mathrm{\ell }}{\alpha }_0\right){\rho }^{\mathrm{\ell }}\left(E\right){\tau }_E\\ {\rho }^{\mathrm{\ell }}\left(F\right){\alpha }_0=\left({\psi }^{\mathrm{\ell }}{\alpha }_0\right){\rho }^{\mathrm{\ell }}\left(E\right).\end{array}$$E23

Given that ${\rho }^{\mathrm{\ell }}\left(L\right)=1+\left[L\right]+\left[L^2\right]+\cdots +\left[L^{\mathrm{\ell }-1}\right]$ for a line bundle L, ${\rho }^{\mathrm{\ell }}\left(E^{\prime }\right)={\left\{{\rho }^{\mathrm{\ell }}\left(\mathrm{ℂ}\right){\rho }^{\mathrm{\ell }}\left({\mathrm{ℂ}}^{\ast }\right)\right\}}^{2\left(k+m\right)}{\rho }^{\mathrm{\ell }}{\left(1\right)}^n={\left\{\left(1+t+\dots +t^{\mathrm{\ell }-1}\right)\left(1+t^{-1}+\dots +t^{-\left(\mathrm{\ell }-1\right)}\right)\right\}}^{2m}{\mathrm{\ell }}^n$ and ${\rho }^{\mathrm{\ell }}\left(F^{\prime }\right)={\left\{\left(1+t+\dots +t^{\mathrm{\ell }-1}\right)\left(1+t^{-1}+\dots +t^{-\left(\mathrm{\ell }-1\right)}\right)\right\}}^{2m}{\mathrm{\ell }}^{b_{+}+n}$. Since $\mathit{Ker}\left(R\left({\mathit{Pin}}_2\right)\to R\left(S^{-1}\right)\right)=\left\{c\left(1-\stackrel{\sim }{1}\right)|c\in \mathrm{\mathbb{Z}}\right\}$, the trace of the degree relation gives $2^{2m+b_{+}+n}=2c{2}^{2k+2m+n}$ for $\alpha =c\left(1-\stackrel{\sim }{1}\right)$, which is consistent with the inequality $b_{+}\ge 2k+1$.

Methods have been developed for increasing the bound for $\frac{b_2}{\left|\sigma \right|}$ through the inequality between b₂(M) and the level number of $\mathrm{ℂ}{\mathrm{\mathbb{P}}}^{2k-1}$, defined to be least n such that ${\left[\mathrm{ℂ}{\mathrm{\mathbb{P}}}^{2k-1},S^{n-1}\right]}^{{\mathrm{\mathbb{Z}}}_2}\ne \cancel{0}$, where $k=-\frac{\sigma \left(M\right)}{16}$ [11, 12]. The computations of level($\mathrm{ℂ}{\mathrm{\mathbb{P}}}^{2k-1}$) yield the inequalities level(($\mathrm{ℂ}{\mathrm{\mathbb{P}}}^{2k-1}$) ≥ 2 k + t if $k\equiv t\left(\mathit{mod}4\right)$, t = 1, 2, 3 and the equality level($\mathrm{ℂ}{\mathrm{\mathbb{P}}}^{2k-1}$) = 2 k + 3 if k ≡ 0 (mod 4), k > 0 [13]. The equivalent inequalities for the second Betti number and signature would be $b_2\ge \frac{5}{4}\left|\sigma \right|+2t$ for $\frac{\left|\sigma \right|}{16}\equiv t\left(\mathit{mod}4\right)$ when t = 1, 2, 3 and $b_2\ge \frac{5}{4}\left|\sigma \right|+6$ for $\frac{\left|\sigma \right|}{16}\equiv 0\left(\mathit{mod}4\right),\left|\sigma \right|>0$. 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₊ 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 $b_2\ge \frac{11}{8}\ge \left|\sigma \right|$.

Proof. Since c is a non-zero integer, the trace condition also requires a stricter bound for b₊. The Euler class of E will be given by that of ${\overline{\mathrm{ℍ}}}^{k+m}$, while the Euler class of F would be that of ${\overline{\mathrm{ℍ}}}^m$, since the $\stackrel{\sim }{R}$ components do not contribute. The Spin₄ × Pin₂ 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 S⁴ and _χ(S⁴) = 2. Therefore, by the Thom class relation,

$$2^m=\mathit{tr}\left({\alpha }_{\mathit{pt}.}\right)2^{k+m}$$E24

and tr(α_pt.) = 2^–|k|, except that k must be chosen to be nonpositive through k = −|k|, then tr(α_pt.) = 2^|k|. A similar result follows from the evaluation of the K-theory characteristic classes ${\rho }^{\mathrm{\ell }}\left(E\right)$. Reintroducing the $\stackrel{\sim }{R}$ components in the vector bundles allows the inclusion of the factor c₀ ≥ 1 occurring in Ker(R(Pin₂) → R(Pin₁)) in $\alpha =2^{\left|k\right|–1}{c}_0\left(1-\stackrel{\sim }{1}\right)$.It follows that

$$2^{2m+b_{+}+n}=2c_0{2}^{\left|k\right|-1}{2}^{2\left|k\right|+2m+n}=2^{\left|k\right|}{2}^{2\left|k\right|+2m+n}$$E25

or

$$b_{+}\ge 3\left|k\right|$$E26

and

$$b_2\ge \frac{11}{8}\left|\sigma \right|.$$E27

Furthermore, this inequality is equivalent to $n\ge \frac{3}{2}\left|m\right|$ since the second Betti number and the absolute value of the signature of a manifold with the intersection form ${\mathit{mE}}_8\oplus n\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$ are b₂ = 8|m| + 2n and |σ| = 8|m|, and

$$\begin{array}{c}\frac{b_2}{\left|\sigma \right|}=\frac{8\left|m\right|+2n}{8\left|m\right|}\ge \frac{11}{8}\\ 8\left|m\right|+2n\ge 11\left|m\right|\\ n\ge \frac{3}{2}\left|m\right|.\end{array}$$E28

It has been proven for cobordisms between homology three-spheres Y₀ and Y₁ with the intersection form $m\left(-E_8\right)+n\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$ that $\kappa \left(Y_1\right)+n\ge \kappa \left(Y_0\right)+m+1$, where κ is an invariant that reduces modulo 2 to the Rohlin invariant μ(Y) [2]. When Y₀ and Y₁ are S³, this inequality is n ≥ |m| + 1, which is consistent with the previously derived inequality [4] for the coefficients since it would follow that

$$\begin{array}{l}\frac{b_2}{\left|\sigma \right|}=\frac{8\left|m\right|+2n}{8\left|m\right|}\ge \frac{5}{4}+\frac{2}{\left|\sigma \right|}\\ 8\left|m\right|+2n\ge 10\left|m\right|+2\\ n\ge \left|m\right|+1.\end{array}$$E29

Nevertheless, the nonexistence of smooth spin manifolds transcending the stricter inequality, such that $n=\frac{3}{2}\left|m\right|-1$, with a decomposition $M=X_1{\cup }_{Y_1}{X}_2{\cup }_{Y_2}\dots {\cup }_{\mathit{Yr}-1}{X}_r$, where the intersection forms of X_i, 1 ≤ i ≤ r – 1 are $2\left(-E_8\right)+3\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$, Y_i is a homology three-sphere and X_r has the intersection form $2\left(-E_8\right)+2\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$ [15], indicates there are characteristics that cannot be preserved when under topological sums yielding $\frac{5}{4}<\frac{b_2}{\left|\sigma \right|}<\frac{11}{8}$.

Definition 4.2. A space is Floer G-split if the S¹ action on the the K-theory group ${\stackrel{\sim }{K}}_G\left(M\right)$ has the form $z\to z^r$ for some positive integer r.

Theorem 4.3. The compact manifold $M=X_1{\cup }_{Y_1}{X}_2{\cup }_{Y_2}\dots {\cup }_{Y_{r-1}}{X}_r$ with spaces X_i, = 1, …, r – 1, and X_r having the intersection forms $Q\left(X_i\right)=2\left(-E_8\right)\cup 3H$ and $Q\left(X_r\right)=2\left(-E_8\right)\cup q_{r}H$, and Y_i being homology three-spheres, is smooth if and only if q_r ≥ 3.

Proof. When the S¹ action does not have this form, which requires $z\to z^r$ with $r\notin \mathrm{\mathbb{Q}}$, the resulting quotient produces a non-Hausdorff structure because arbitrarily near points are identified. If $r=\frac{s}{t}\in \mathrm{\mathbb{Q}}\backslash \mathrm{\mathbb{Z}}$, then the points e^2πiθ and $e^{\frac{2\pi }{t}\mathit{i\theta }}$ 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 {Y_i} are Floer G-split [2]. It follows that a quotient of one of the homology spheres by an S¹ action with $z\to z^r,r\notin Z$, is not smooth. The S¹ action on the homology sphere, which is not Floer G-split, may be transferred to the manifold X_r, as a result of the parallelizability necessary for the existence of the spin structure, thereby proving that a spin manifold with an intersection form $2\left(-E_8\right)\oplus 2H$ does not admit a smooth structure.

Now suppose that the spin manifold X has the intersection form $m\left(-E_8\right)\oplus \mathit{nH}$, where $n=\frac{3}{2}\left|m\right|$. The analogous result to that given above for $n=\frac{3}{2}\left|m\right|-1$ would be the decomposition

$$\begin{array}{l}X=X_1{\cup }_{Y_1}{X}_2{\cup }_{Y_2}\dots {\cup }_{\mathit{Yr}-1}{X}_r\\ Q\left(X_i\right)=2\left(-E_8\right)+3H,i=1,\dots ,r\end{array}$$E30

with Y₁, ..., Y_r–1 being homology spheres, then the inequality κ(Y_i + 1) + 3 ≥ κ(Y_i) + 2 + 1, or κ(Y_i + 1) ≥ κ(Y_i), is valid for all j = 1, ..., r – 2, and each Y_i is Floer G-split, which requires the existence of a smooth structure on each X_i, i = 1, ..., r. Therefore, closed spin four-manifolds with the intersection form ${\mathit{mE}}_8\oplus \mathit{nH}$ and $n=\frac{3}{2}\left|m\right|$ admit smooth structures. For $n\ge \frac{3}{2}\left|m\right|$, the inequalities for κ(Y_i), i = 1, ..., r–1 continue to be valid, each of the homology spheres will be Floer K_G-split, and there will be a smooth structure on the spin four-manifold.

Several results may be proven given the validity of the $\frac{11}{8}$ conjecture, including the theorem on ξ ∙ ξ for a characteristic second homology class ξ representable by S² for a range of values of b₊ and b₋ [7]. The following lemma is required:

Let M be a closed connected oriented four-manifold with $\xi \in H_2\left(M;\mathrm{\mathbb{Z}}\right)$ be a characteristic homology class representable by S². Then ξ ∙ ξ = σ(M) + 16 m with $m\le max\left\{⌊\frac{b_1-1}{3}⌋,⌊\frac{b_{-}-b_{+}}{16}⌋\right\}$ would be consistent with the $\frac{11}{8}$ 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 $\frac{11}{8}$ 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₊ − b₋, ξ ∙ ξ = b₊ − b₋ + 16 m. The range given in the above theorem with ξ ∙ ξ = 0 or b₋ = b₊ + 16 m yields the following results.

$$\begin{array}{c}m\le \mathit{max}\left\{⌊\frac{b_{+}-1}{3}⌋,⌊\frac{b_{+}+16m-r}{16}⌋\right\}=\mathit{max}\left\{⌊\frac{b_{+}-1}{3}⌋,m+⌊\frac{b_{+}-r}{16}⌋\right\}\\ =\mathit{max}\left\{⌊\frac{b_{+}-1}{3}⌋,m\right\}\end{array}$$E31

and

$$m\le ⌊\frac{b_{+}-1}{3}⌋.$$E32

Since spin manifolds have an even intersection form, b₊ is even. Then if b₊ = 0, 2 or 4 (mod 6) and $⌊\frac{b_{+}-1}{3}⌋=⌊\frac{b_{+}}{3}⌋-1,⌊\frac{b_{+}}{3}⌋\mathit{\text{or}}⌊\frac{b_{+}}{3}⌋$ respectively. If $m\le \frac{b_{+}}{3}$,

$$b_{-}=b_{+}+16m\le \frac{19}{3}{b}_{+}.$$E33

Then

$$\begin{array}{c}{b}_2-\sigma \le \frac{19}{3}\left(b_2+\sigma \right)\\ \frac{16}{3}{b}_2\ge -\frac{22}{3}\sigma \\ b_2\ge \frac{11}{8}\left|\sigma \right|\sigma <0.\end{array}$$E34

For σ > 0, the roles of b₊ and b₊ are interchanged, and

$$\begin{array}{c}{b}_2+\sigma \le \frac{19}{3}\left(b_2-\sigma \right)\\ \frac{16}{3}{b}_2\ge \frac{22}{3}\sigma \\ b_2\ge \frac{11}{8}\left|\sigma \right|\sigma >0.\end{array}$$E35

Therefore, the condition ξ ∙ ξ = 0 together with the range of m yielding the upper limit $\frac{b_{+}}{3}$, is sufficient to prove the $\frac{11}{8}$ conjecture. Given that ξ ∙ η is the intersection number of ξ and η are representable by S², ξ ∙ ξ would equal the sum of the eigenvalues of the intersection form of S² × S², which equals zero.

The connected sum ${\mathrm{ℂ\mathbb{P}}}^2\#9\overline{{\mathrm{ℂ\mathbb{P}}}^2}$ does not satisfy the inequalities in Eq. (14) for the coefficients k and $\mathrm{\ell }$. Nevertheless, it has exotic smooth structures. The nonexistence of spin structures on ${\mathrm{ℂ\mathbb{P}}}^2\#9\overline{{\mathrm{ℂ\mathbb{P}}}^2}$ may be demonstrated [17, 18]. The spaces ${\mathrm{ℂ\mathbb{P}}}^2\#\mathrm{\ell }\overline{{\mathrm{ℂ\mathbb{P}}}^2}$ have $\frac{b_2}{\left|\sigma \right|}<\frac{11}{8}$ for $\mathrm{\ell }$ ≥ 7. Furthermore, these four-manifolds have both standard and exotic smooth structures for k = 7, 8 and 9 [19, 20, 21].

Proposition 4.4. The topological sums ${\mathrm{ℂ\mathbb{P}}}^2\#\mathrm{\ell }\overline{{\mathrm{ℂ\mathbb{P}}}^2},\mathrm{\ell }\ge 7$, do not represent counterexamples to the inequality $\frac{b_2}{\left|\sigma \right|}\ge \frac{11}{8}$ required for smooth structures on spin manifolds given the validity of the $\frac{10}{8}$ theorem.

Proof. The existence of smoooth structures on these spaces is established. From §2, the second Betti number and signature of ${\mathrm{ℂ\mathbb{P}}}^2\#\mathrm{\ell }\overline{{\mathrm{ℂ\mathbb{P}}}^2}$ equal

$$\begin{array}{l}{b}_2=1+\mathrm{\ell }\\ \sigma =1-\mathrm{\ell }.\end{array}$$E36

Then

$$\frac{b_2}{\left|\sigma \right|}=\frac{\mathrm{\ell }+1}{\mathrm{\ell }-1}\mathrm{\ell }\ge 2.$$E37

For $\mathrm{\ell }$ = 7, 8 and 9, $\frac{b_2}{\left|\sigma \right|}$ is $\frac{4}{3},\frac{9}{7}$ and $\frac{5}{4}$ respectively. Then

$$\begin{array}{l}\frac{10}{8}<\frac{b_2}{\left|\sigma \right|}<\frac{11}{8}\mathit{\text{for}}\mathrm{\ell }=7,8\\ \frac{b_2}{\left|\sigma \right|}=\frac{10}{8}\mathit{\text{for}}\mathrm{\ell }=9\\ \frac{b_2}{\left|\sigma \right|}<\frac{10}{8}\mathit{\text{for}}\mathrm{\ell }>9.\end{array}$$E38

Consequently, the values $\mathrm{\ell }$ ≥ 9 must be covered by the $\frac{10}{8}$ 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 four-manifold is divisible by 16. For ${\mathrm{ℂ\mathbb{P}}}^2\#\mathrm{\ell }\overline{{\mathrm{ℂ\mathbb{P}}}^2}$, this condition is

$$\mathrm{\ell }\equiv 1\left(\mathit{mod}16\right).$$E39

This congruenece condition is not satisfied by $\mathrm{\ell }$ = 7 or $\mathrm{\ell }$ = 8. Therefore, there will be no spin structure for these values. It follows that the connected sums for $\mathrm{\ell }$ ≥ 7 will not represent a counterexample to the $\frac{11}{8}$ conjecture when the lower bound of $\frac{10}{8}$ suffices generally for smooth spin manifolds.

There would be a spin structure on ${\mathrm{ℂ\mathbb{P}}}^2\#\mathrm{\ell }\overline{{\mathrm{ℂ\mathbb{P}}}^2}$. Both the above proposition and the consistency of the $\frac{10}{8}$ theorem require the nonexistence of spin structures on ${\mathrm{ℂ\mathbb{P}}}^2\#\left(16r+1\right)\overline{{\mathrm{ℂ\mathbb{P}}}^2}$ for r ≥ 1.

Proposition 4.5. The topological sum ${\mathrm{ℂ\mathbb{P}}}^2\#\left(16r+1\right)\overline{{\mathrm{ℂ\mathbb{P}}}^2}$ is a spin manifold only if r = 0.

Proof. There exists a spin structure on a space M the second Stiefel-Whitney class $w_2\left(M\right)\in H_2\left(M;{\mathrm{\mathbb{Z}}}_2\right)$ is nonvanishing. The second homology group of a connected sum M₁#M₂, wheredim M₁ = dim M₂ = 4, is

$$H_2\left(M_1\#M_2\right)=H_2\left(M_1\right)\oplus H_2\left(M_2\right)$$E40

and. Specializing to the group ${\mathrm{\mathbb{Z}}}_2$,

$$H_2\left(M_1\#M_2,{\mathrm{\mathbb{Z}}}_2\right)=H_2\left(M_1,{\mathrm{\mathbb{Z}}}_2\right)\oplus H_2\left(M_2,{\mathrm{\mathbb{Z}}}_2\right).$$E41

Then

$$H_2\left({\mathrm{ℂ\mathbb{P}}}^2\#\overline{{\mathrm{ℂ\mathbb{P}}}^2},{\mathrm{\mathbb{Z}}}_2\right)=H_2\left({\mathrm{ℂ\mathbb{P}}}^2,{\mathrm{\mathbb{Z}}}_2\right)\oplus H_2\left(\overline{{\mathrm{ℂ\mathbb{P}}}^2},{\mathrm{\mathbb{Z}}}_2\right)$$E42

Since ${\mathrm{ℂ\mathbb{P}}}^2$ does not have a spin structure, there will be a nonvanishing generator of the second homology group $\left[u\right]\ne \left[0\right]$ and

$$H_2\left({\mathrm{ℂ\mathbb{P}}}^2\#{\overline{\mathrm{ℂ\mathbb{P}}}}^2,{\mathrm{\mathbb{Z}}}_2\right)=\left[u\right]\oplus -\left[u\right]$$E43

The element of the homology group must be an element of ${\mathrm{\mathbb{Z}}}_2$. Therefore, it would be the image of the $\left[u\right]\oplus -\left[u\right]$ under the mapping

$$\phi :H_2\left({\mathrm{ℂ\mathbb{P}}}^2,{\mathrm{\mathbb{Z}}}_2\right)\oplus H_2\left({\overline{\mathrm{ℂ\mathbb{P}}}}^2,{\mathrm{\mathbb{Z}}}_2\right)\to {\mathrm{\mathbb{Z}}}_2$$E44

This homomorphism will be defined by

$$\phi \left(\left[u_1\right]\oplus \left[u_2\right]\right)=\left[u_1\right]+\left[u_2\right]\in {\mathrm{\mathbb{Z}}}_2$$E45

Since

$$\phi \left(\left[u\right]\oplus -\left[u\right]\right)=\left[u\right]-\left[u\right]=\left[0\right],$$E46

$H_2\left({\mathrm{ℂ\mathbb{P}}}^2\#\overline{{\mathrm{ℂ\mathbb{P}}}^2},{\mathrm{\mathbb{Z}}}_2\right)=\cancel{0}$, the second Stiefel-Whitney class vanishes, and there is a spin structure for r = 0.

The homology group for higher values of r equals

$$H_2\left({\mathrm{ℂ\mathbb{P}}}^2\#\left(16r+1\right)\overline{{\mathrm{ℂ\mathbb{P}}}^2}\right)=H_2\left({\mathrm{ℂ\mathbb{P}}}^2\right)\#\left(16r+1\right)H_2\left(\overline{{\mathrm{ℂ\mathbb{P}}}^2},{\mathrm{\mathbb{Z}}}_2\right).$$E47

Given that $\left[u\right]\in H_2\left({\mathrm{ℂ\mathbb{P}}}^2,{\mathrm{\mathbb{Z}}}_2\right)$ and $-\left[u\right]\in H_2\left(\overline{{\mathrm{ℂ\mathbb{P}}}^2},{\mathrm{\mathbb{Z}}}_2\right)$, the element of ${\mathrm{\mathbb{Z}}}_2$ for the topological sum is

$$\phi \left(\left[u\right]\oplus \left(16r+1\right)\left(-u\right)\right)=-16r\left[u\right]$$E48

Multiplication by a non-zero scalar does not affect the generator of the nontrivial second homology class, which does not vanish. Then, ${\mathrm{ℂ\mathbb{P}}}^2\#\left(16r+1\right)\overline{{\mathrm{ℂ\mathbb{P}}}^2}$ is not a spin manifold for r > 1.

There are no counterexamples given by ${\mathrm{ℂ\mathbb{P}}}^2\#\overline{{\mathrm{ℂ\mathbb{P}}}^2}$ to the lower bound of $\frac{11}{8}$ for $\frac{b_2}{\left|\sigma \right|}$. Topological sums with S⁴, S² × S² and K3 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 $\frac{b_2}{\left|\sigma \right|}$ has been increased to $\frac{11}{8}$.

Theorem 5.1. The coefficients of the intersection form mE₈ + nH, where $H=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$ satisfy the inequality $n\ge \frac{3}{2}m-\rho \left(M\right)+\frac{\rho \left(M\right)}{2^i}$, 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 H₁(M; ${\mathrm{\mathbb{Z}}}_2$).

Proof. Let M be a smooth, oriented, simply connected, compact four-manifold with the even intersection form I = mE₈ + nH, signature σ(M) = 8 m and Euler number e(M) = 2 + 8 m + 2n, since the first Betti number can be set equal to zero for a given intersection form. There exists a 2ⁱ 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

$$\begin{array}{l}\sigma \left(N\right)=2^i\sigma \left(M\right)=8\left(2^{i}m\right)\equiv 8r\\ e\left(N\right)=2^{i}e\left(M\right)=2^{i+1}+8\left(2^{i}m\right)+2\left(2^{i}n\right)=2+8r+2s\end{array}$$E49

then N = rE8 + sH, where

$$r=2^{i}ms=2^{i}n+2^i-1.$$E50

Since it has been proven that $s\ge \left|r\right|+1$ for spin manifolds [4],

$$2^{i}n+2^i-1=2^i\left|m\right|+1.$$E51

and

$$n\ge \left|m\right|-\left(1-\frac{1}{2^{i-1}}\right).$$E52

A term b₁(N) can be added to s to give $s+b_1\left(N\right)\ge r+1$. Since $b_1\left(N\right)\le \left(2^i-1\right)\left(\rho \left(M\right)-1\right)$ when b₁(M) = 0, where ρ(M) is the rank of H₁(M; ${\mathrm{\mathbb{Z}}}_2$) [6]. Then

$$\begin{array}{l}{2}^{i}n+2^i-1+\left(2^i-1\right)\left(\rho \left(M\right)-1\right)\ge 2^i\left|m\right|+1.\\ n\ge \left|m\right|-\rho \left(M\right)+\frac{1+\rho \left(M\right)}{2^i}.\end{array}$$E53

If ρ(M) = 1 and N → M is a double covering with H₁(M; $\mathrm{\mathbb{Z}}$) = ${\mathrm{\mathbb{Z}}}_2$ [6], the inequality $n\ge \left|m\right|$ is valid.

With a tighter bound $n\ge \frac{3}{2}\left|m\right|$ for spin manifolds, a similar inequality will be found for non-spin manifolds. By the proof in §4, $s\ge \frac{3}{2}\left|r\right|$ for the spin covering N, or equivalently,

$$\begin{array}{c}{2}^{i}n+2^i-1\ge \frac{3}{2}{2}^i\left|m\right|\\ n\ge \frac{3}{2}\left|m\right|-\left(1-\frac{1}{2^i}\right).\end{array}$$E54

Since (2ⁱ – 1)(ρ(M) – 1) is an upper bound for b₁(N) – b₁(M), after b₁(M) is set equal to zero,

$$2^{i}n+\left(2^i-1\right)\rho \left(M\right)=\frac{3}{2}{2}^i\left|m\right|$$E55

or

$$n\ge \frac{3}{2}\left|m\right|-\rho \left(M\right)+\frac{\rho \left(M\right)}{2^i}.$$E56

It has been proven that there exist nonsmoothable spin manifolds with $b_2\ge \frac{5}{4}\left|\sigma \right|+2$ [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 $b_2\ge \frac{5}{4}\left|\sigma \right|$. The inequality derived for non-spin manifolds $n\ge \left|m\right|-\left(1-\frac{1}{2^{i-1}}\right)$ may be translated to a bound for the second Betti number.

$$\begin{array}{c}\frac{b_2}{\left|\sigma \right|}=\frac{8\left|m\right|+2n}{8\left|m\right|}\\ \ge 1-\frac{1}{4}\left(1+\frac{1}{\left|m\right|}\left(1-\frac{1}{2^{i-1}}\right)\right)\\ =\frac{5}{4}-\frac{2}{\left|\sigma \right|}\left(1-\frac{1}{2^{i-1}}\right).\end{array}$$E57

This lower bound for b₂ is less than or equal to $\frac{5}{4}\left|\sigma \right|$ for i ≥ 1.

The tighter inequality for the coefficients in the intersection form is equivalent to

$$\begin{array}{c}\frac{b_2}{\left|\sigma \right|}\ge 1+\frac{1}{4}\left(\frac{3}{2}-\frac{1}{\left|m\right|}\left(1-\frac{1}{2^i}\right)\right)\\ =\frac{11}{8}-\frac{2}{\left|\sigma \right|}\left(1-\frac{1}{2^i}\right).\end{array}$$E58

The contradiction is resolved in the inequality if the lower bound for $\frac{b_2}{\left|\sigma \right|}$ can be increased. Then, the nonsmooth manifolds can exist in the region $\frac{5}{4}\left|\sigma \right|+2\le b_2\le \frac{11}{8}\left|\sigma \right|$ when a spin structure exists and $\frac{5}{4}\left|\sigma \right|-2\left(1-\frac{1}{2^{i-1}}\right)\le b_2\le \frac{11}{8}\left|\sigma \right|-2\left(1-\frac{1}{2^i}\right)$ when there is no spin structure.

A lower bound for b₂ also can be derived for smooth, oriented non-spin manifolds by the following set of equations

$$\begin{array}{c}{\mathit{dim}}_{{\mathrm{\mathbb{Z}}}_2}{H}^2\left(M;{\mathrm{\mathbb{Z}}}_2\right)=b_2\left(M\right)+2t\\ t={\mathit{dim}}_{{\mathrm{\mathbb{Z}}}_2}\left(T{\mathit{\text{or}}}_2{H}_1\left(M;{\mathrm{\mathbb{Z}}}_2\times {\mathrm{\mathbb{Z}}}_2\right)\right),\end{array}$$E59

$$b_1\left(N;{\mathrm{\mathbb{Z}}}_2\right)\le 2^i{b}_1\left(M;{\mathrm{\mathbb{Z}}}_2\right)-2^i+1,$$E60

and

$$\begin{array}{c}{b}_2\left(N;{\mathrm{\mathbb{Z}}}_2\right)=2^{i}e\left(M\right)-2+2b_1\left(N;{\mathrm{\mathbb{Z}}}_2\right)\\ \le 2^i\left(2-2b_1\left(M;{\mathrm{\mathbb{Z}}}_2\right)+b_2\left(M;{\mathrm{\mathbb{Z}}}_2\right)\right)+2\left(2^i{b}_1\left(M;{\mathrm{\mathbb{Z}}}_2\right)-2^i+1\right)\\ =2^{i+1}-2^{i+1}{b}_1\left(M;{\mathrm{\mathbb{Z}}}_2\right)+2^i{b}_2\left(M;{\mathrm{\mathbb{Z}}}_2\right)+2^{i+1}{b}_1\left(M;{\mathrm{\mathbb{Z}}}_2\right)-2^{i+1}+2\\ =2^i{b}_2\left(M;{\mathrm{\mathbb{Z}}}_2\right)+2,\end{array}$$E61

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

$$b_2\left(N\right)+2\le 2^i\left(b_2\left(M\right)+2\right)+2=2^i{b}_2\left(M\right)+2^{i+1}+2$$E62

or

$$b_2\left(N\right)\le 2^i{b}_2\left(M\right)+2^{i+1}.$$E63

By the strong $\frac{10}{8}$ inequality for spin manifolds,

$$b_2\left(N\right)\ge \frac{5}{4}\left|\sigma \left(N\right)\right|+2=\frac{5}{4}{2}^i\left|\sigma \left(M\right)\right|+2.$$E64

and

$$\begin{array}{c}{b}_2\left(M\right)\ge \frac{1}{2^i}\left(\frac{5}{4}{2}^i\left|\sigma \left(M\right)\right|+2-2^{i+1}\right)\\ =\frac{5}{4}\left|\sigma \left(M\right)\right|-\left(2-\frac{1}{2^{i-1}}\right).\end{array}$$E65

The tighter inequality derived in §4 for spin manifolds yields

$$b_2\left(N\right)\ge \frac{11}{8}\left|\sigma \left(N\right)\right|\ge \frac{11}{8}{2}^i\left|\sigma \left(M\right)\right|$$E66

and

$$\begin{array}{c}{b}_2\left(M\right)\ge \frac{1}{2^i}\left(\frac{11}{8}{2}^i\left|\sigma \left(M\right)\right|-2^{i+1}\right)\\ =\frac{11}{8}\left|\sigma \left(M\right)\right|-2.\end{array}$$E67

The range for b₂ is narrower for i ≥ 1 and there is a region below $\frac{11}{8}\left|\sigma \right|$ 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 $\frac{b_2}{\left|\sigma \right|}\ge \frac{11}{8}$.

Proof. Consider an oriented, simply connected, compact, four-dimensional manifold M and the 2ⁱ-fold spin covering N → M. From the equation $\left({\stackrel{\sim }{f}}^{\ast }\right){\tau }_F={\alpha }_0{\tau }_E$, where α₀ is the degree of the pull-back map ${\stackrel{\sim }{f}}^{\ast }$ from K_G(BF, SF) to K_G(BE, SE), the trace of the relation ${\rho }_{\mathrm{\ell }}\left({\stackrel{\sim }{f}}^{\ast }\right){\tau }_F=\left({\psi }^{\mathrm{\ell }}{\alpha }_0\right){\rho }^{\mathrm{\ell }}\left(E\right){\tau }_E$ when projected to an S¹ module in the subspace E’ and F′, introduces a factor of 2ⁱ in the pull-back of the kernel of the map from Pin₂ to S¹, ${\left\{\left(c\left(\mathbf{1}-\stackrel{\sim }{\mathbf{1}}\right)\right|c\in \mathrm{\mathbb{Z}}\right\}}^{2^i}$, and an overall factor of $2^{2^i}$. Then,

$$\alpha =2^{2^i}{2}^{\left|k\right|}$$E68

and

$$b_{+}\left(N\right)\ge 2\left|k\right|+k+2^i=3\left|k\right|+2^i.$$E69

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

$$b_2\left(N\right)=\frac{11}{8}\left|\sigma \left(N\right)\right|+2^{i+1}.$$E70

By Eq. (63),

$$\begin{array}{c}{2}^i{b}_2\left(M\right)+2^{i+1}=\frac{11}{8}\left|\sigma \left(N\right)\right|+2^{i+1}=\frac{11}{8}{2}^i\left|\sigma \left(N\right)\right|+2^{i+1}\\ b_2\left(M\right)=\frac{11}{8}\left|\sigma \left(M\right)\right|.\end{array}$$E71

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

$$g\left(\Sigma \right)>\left|\frac{2^r+1}{2^{r+1}},{\sum }^2,-,\frac{2^{r+1}}{2^r-1},\sigma ,\left(M\right)\right|-\frac{2^{r-1}}{2^r-1}{b}_2\left(M\right).$$E72

where Σ² is the intersection product of the second cohomology class Σ [19]. If the $\frac{11}{8}$ conjecture is true, the bound can be increased to

$$g\left(\mathrm{\Sigma }\right)>\frac{11}{8}\left|\frac{2^r+1}{2^{r+1}}{\mathrm{\Sigma }}^2-\frac{2^{r+1}}{2^r-1}\sigma \left(M\right)\right|-\frac{2^{r-1}}{2^r-1}{b}_2\left(M\right).$$E73

For an algebraic surface Σ_d of even degree d embedded in $\mathrm{ℂ}{\mathrm{\mathbb{P}}}^2$, with $2‖d,g\left({\sum }_d\right)\ge \frac{11}{32}{d}^2-\frac{19}{9}$ [25].

The Thom conjecture for curves of algebraic curves of degree d states that $g\left({\sum }_d\right)\ge \frac{1}{2}{d}^2-\frac{3}{2}+1$. The replacement of $\frac{11}{32}$ by $\frac{1}{2}$ in the lower bound for the genus, requires the substitution of 2 for $\frac{11}{8}$ as the lower limit for $\frac{b_2}{\left|\sigma \right|}$. Consider the intersection form ${\mathit{mE}}_8\oplus \mathit{nH}$. Since

$$\begin{array}{l}{b}_2\left({\mathit{mE}}_8\oplus \mathit{nH}\right)=8m+2n\\ \sigma \left({\mathit{mE}}_8\oplus \mathit{nH}\right)=8m,\end{array}$$E74

the inequality

$$\frac{8m+2n}{8m}\ge 2$$E75

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 $\frac{3}{2}$ conjecture for irreducible simply connected four-manifolds is $\chi \ge \frac{3}{2}\left|\sigma \right|$, where $\chi =2+\mathit{rank}\left(Q\right)=2+b_2$ is the Euler characteristic. It follows that

$$b_2\ge \frac{3}{2}\left|\sigma \right|-2.$$E76

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

$$g\left(\sum \right)\ge \left(\frac{3}{2}-\frac{2}{\left|\sigma \right|}\right)\left|\frac{2^r+1}{3\cdot 2^{r+1}}{\sum }^2-\frac{2^{r-1}}{2^r-1}\sigma \left(M\right)\right|-\frac{2^{r-1}}{2^r-1}{b}_2\left(M\right),$$E77

where $2^r‖\left[\sum \right]$. Therefore, the genus of an algebraic curve of degree d embedded in an irreducible, simply connected manifold would satisfy the inequality $g\left({\sum }_d\right)\ge \left(\frac{3}{8}-\frac{1}{2\left|\sigma \right|}\right)d^2+{\gamma }_{1}d+{\gamma }_0$ with γ₀ and γ₁ being constants.

7. Conclusion

The classification of four-manifolds has been reduced to the definite signature with odd intersection forms that are diagonal, n1 + m(−1) or indefinite signature with even intersection forms mE₈ + nH, where E₈ is the exceptional Lie group Cartan matrix and H is the matrix $\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$, which is the form of S² × S². All known smooth, oriented four-manifolds with an even intersection form are known to have coefficients satisfying $n\ge \frac{3}{2}\left|m\right|$ or equivalently, a second Betti number satisfying $b_2\ge \frac{11}{8}\left|\sigma \right|$. The oriented, spin geometries in four dimensions have been demonstrated to admit a smooth structure only if n ≥ |m| + 1 or $b_2\ge \frac{5}{4}\left|\sigma \right|+2$. The proof has been extended to non-spin manifolds with the inequality $n\ge \left|m\right|-\left(1-\frac{1}{2^i}\right)$ and $b_2\ge \frac{5}{4}\left|\sigma \right|-\left(2-\frac{1}{2^{i-1}}\right)$. It is found here that the lower bound for the signed Betti number b₊ is larger than 2|k| + 1, where $\left|k\right|=\frac{3}{16}\left|\sigma \right|$. Considering a cyclic covering of a non-spin manifold and introducing the degree into the proof for the spin manifold, the inequality $b_2\ge \frac{11}{8}\left|\sigma \right|$. This increase in the lower bound for the second Betti number allows the existence of nonsmoothable manifolds with $b_2\ge \frac{5}{4}\left|\sigma \right|$ within a strict demarcation between the regions for smooth and nonsmooth structures.

The existence of smooth, compact simply-connected manifolds with $\frac{5}{4}<\frac{b_2}{\left|\sigma \right|}<\frac{11}{8}$ present potential counterexamples to the $\frac{11}{8}$. The topological sums ${\mathrm{ℂ\mathbb{P}}}^2\#\overline{{\mathrm{ℂ\mathbb{P}}}^2}$ 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 ${\mathrm{ℂ\mathbb{P}}}^2\#\mathrm{\ell }\overline{{\mathrm{ℂ\mathbb{P}}}^2}$, only those values of $\mathrm{\ell }$ congruent to 1 modulo 16 would satisfy the condition on the divisibility of the signature by 16. The absence of spin structures on ${\mathrm{ℂ\mathbb{P}}}^2\#\left(16r+1\right)\overline{{\mathrm{ℂ\mathbb{P}}}^2}$ for $\mathrm{\ell }$ ≥ 1 is established through the computation of the second Stiefel-Whitney class. Therefore, ${\mathrm{ℂ\mathbb{P}}}^2\#\overline{{\mathrm{ℂ\mathbb{P}}}^2}$ is the unique spin manifold in this set, which is necessary for consistency of the $\frac{10}{8}$ 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 S⁴, S² × S² and K3 and complex algebraic surfaces, since the condition of the existence of a spin structure requires generally an increased minimum value of $\frac{11}{8}$ for $\frac{b_2}{\left|\sigma \right|}$.

Classification:

MSC: 57N13; 57R19