#
Quantum Computation and Measurements from an Exotic Space-Time R^{4}

^{1}

^{2}

^{*}

## Abstract

**:**

^{4}; fundamental group; finite geometry; Cayley–Dickson algebras

## 1. Introduction

## 2. Excerpts on the Theory of 4-Manifolds and Exotic ${R}^{4}$s

#### 2.1. Handlebody of a 4-Manifold

#### 2.2. Akbulut Cork

#### 2.3. Exotic Manifold ${R}^{4}$

**Theorem**

**1.**

**Proof.**

## 3. Finite Geometry of Small Exotic R^{4}s and Quantum Computing

#### 3.1. The Boundary $\partial W$ of Akbulut Cork

#### 3.2. The Manifold $\overline{W}$ Mediating the Akbulut Cobordism between Exotic Manifolds V and W

#### 3.3. The Middle Level Q between the Diffeomorphic Connected Sums

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

## Appendix B

**Figure A1.**The Fano plane as the geometry of the subgroup of index 7 of fundamental group ${\pi}_{1}\left(\overline{W}\right)$ for the manifold $\overline{W}$ mediating the Akbulut cobordism between exotic manifold V and W. The cosets of ${\pi}_{1}$ are organized through the permutation group $P=\left(\right)open="\langle "\; close="\rangle ">7\left|\right(1,2,4,5,6,7,3),(2,5,6\left)\right(3,7,4)$. The cosets are labeled as $\left(\right)open="["\; close="]">1,\dots ,7$. The two-point stabilizer subgroups of P for each line are distinct (acting on different subsets) but isomorphic to each other and to the Klein group ${\mathbb{Z}}_{2}\times {\mathbb{Z}}_{2}$. They are as follows: ${s}_{1}=\left(\right)open="\langle "\; close="\rangle ">(2,7)(4,5),(2,4)(5,7)$, ${s}_{2}=\left(\right)open="\langle "\; close="\rangle ">(1,6)(5,7),(1,7)(5,6)$, ${s}_{3}=\left(\right)open="\langle "\; close="\rangle ">(1,5)(2,3),(1,2)(3,5)$, ${s}_{4}=\left(\right)open="\langle "\; close="\rangle ">(3,5)(4,6),(3,6)(4,5)$, ${s}_{5}=\left(\right)open="\langle "\; close="\rangle ">(2,6)(3,7),(2,7)(3,6)$, ${s}_{6}=\left(\right)open="\langle "\; close="\rangle ">(1,7)(3,4),(1,4)(3,7)$, and ${s}_{7}=\left(\right)open="\langle "\; close="\rangle ">(1,2)(4,6),(1,6)(2,4)$.

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Sample Availability: The codes were written on softwares Snappy and Magma. Part of the calculation is detailed in Section 3.1 and Appendix B. |

**Figure 1.**(

**a**) Handlebody of a 4-manifold with the structure of 1- and 2-handles over the 0-handle ${B}^{4}$; and (

**b**) the structure of a 1-handle as a dotted circle ${S}^{1}\times {B}^{3}$.

**Figure 2.**(

**a**) A 0-framed 2-handle ${S}^{2}\times {B}^{2}$ (

**left**) and a dotted 1-handle ${S}^{1}\times {B}^{3}$ (

**right**) are diffeomorphic at their boundary $\partial ={S}^{2}\times {S}^{1}$; and (

**b**) two equivalent pictures of the Akbulut cork W.

**Figure 3.**(

**a**) The loop $\alpha $ is not slice on the Akbulut cork; (

**b**) the non-trivial h-cobordism between small exotic manifolds V and W; and (

**c**) the mediating 4-manifold $\overline{W}$.

**Figure 4.**Exotic ${R}^{4}$ manifolds ${Q}_{1}$ shown in (

**a**) and ${Q}_{2}$ shown in (

**b**). The connected sums ${Q}_{1}\#{S}^{2}\times {S}^{2}$ and ${Q}_{2}\#{S}^{2}\times {S}^{2}$ are diffeomorphic with middle level Q shown in (

**c**).

**Figure 5.**(

**a**) A picture of the smallest finite projective space $PG(3,2)$. It is found at Frans Marcelis’ website [18]. The coset coordinates shown on the Fano plane with red bullets of $PG(3,2)$ correspond the case in Table 2. (

**b**) A picture of the generalized quadrangle of order two $GQ(2,2)$ embedded in $PG(3,2)$. It may also be found at Frans Marcelis’ website.

**Figure 6.**The Cayley–Salmon configuration built around the Desargues configuration (itself built around the Pasch configuration) as in [20] (Figure 12).

**Figure 7.**Mermin pentagram as the contextual geometry occuring for the index 10 subgroup in the fundamental group ${\pi}_{1}\left(\overline{W}\right)$ of the h-cobordism $\overline{W}$ between exotic manifolds V and W.

**Table 1.**Geometric structure of subgroups of the fundamental group ${\pi}_{1}\left(\overline{W}\right)$ for the h-cobordism $\overline{W}$ between exotic manifolds V and W. Bold characters are for the contextual geometries. ${G}_{1092}$ is the simple group of order 1092. When a MIC state can be found, details are given in Column 4, while the number $pp$ of distinct values of pairwise products in the MIC is in Column 5.

d | P | Geometry | MIC Fiducial | pp |
---|---|---|---|---|

5 | ${A}_{5}$ | ${K}_{5}$ | $(0,1,-1,-1,1)$ | 1 |

6 | ${A}_{6}$ | ${K}_{6}$ | $(1,{\omega}_{6}-1,0,0,-{\omega}_{6},0)$ | 2 |

7 | ${A}_{7}$ | ${K}_{7}$ | ||

10 | ${A}_{5}$ | Mermin pentagram | no | |

12 | ${A}_{12}$ | ${K}_{12}$ | ||

${A}_{5}$ | K(2,2,2,2,2,2) | no | ||

13 | ${A}_{13}$ | ${K}_{13}$ | ||

14 | ${G}_{1092}$, ${A}_{14}$ | ${K}_{14}$ | ||

${2}^{6}\u22ca{A}_{5}$ | K(2,2,2,2,2,2,2) | |||

15 | ${A}_{15}$ | ${K}_{15}$ | ||

${A}_{5}$ | K(3,3,3,3,3) | yes | 3 | |

${A}_{7}$ | PG(3,2) | |||

16 | ${A}_{16}$ | ${K}_{16}$ |

**Table 2.**Geometric structure of subgroups of the fundamental group ${\pi}_{1}\left(Q\right)$ for the middle level Q of Akbulut’s h-cobordism between connected sums ${Q}_{1}\#{S}^{2}\times {S}^{2}$ and ${Q}_{2}\#{S}^{2}\times {S}^{2}$. Bold characters are for the contextual geometries. When a MIC state can be found, details are given in Column 4, while the number $pp$ of distinct values of pairwise products in the MIC is in Column 5. $GQ(2,2)$ is the generalized quadrangle of order two embedded in PG(3,2), as shown in Figure 5b.

d | P | Geometry | MIC Fiducial | pp |
---|---|---|---|---|

6 | ${A}_{6}$ | ${K}_{6}$ | $(1,{\omega}_{6}-1,0,0,-{\omega}_{6},0)$ | 2 |

7 | $PSL(2,7)$ | Fano plane | $(1,1,0,-1,0,-1,0)$ | 2 |

8 | $PSL(2,7)$ | ${K}_{8}$ | no | |

10 | ${A}_{6}$ | ${K}_{10}$ | yes | 5 |

14 | $PSL(2,7)$ | K(2,2,2,2,2,2,2) | no | |

15 | ${A}_{6}$ | PG(3,2), GQ(2,2) | yes | 4 |

16 | ${A}_{16}$ | ${K}_{16}$ | no | |

$SL(2,7)$ | K(2,2,2,2,2,2,2,2) | no |

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**MDPI and ACS Style**

Planat, M.; Aschheim, R.; Amaral, M.M.; Irwin, K.
Quantum Computation and Measurements from an Exotic Space-Time *R*^{4}. *Symmetry* **2020**, *12*, 736.
https://doi.org/10.3390/sym12050736

**AMA Style**

Planat M, Aschheim R, Amaral MM, Irwin K.
Quantum Computation and Measurements from an Exotic Space-Time *R*^{4}. *Symmetry*. 2020; 12(5):736.
https://doi.org/10.3390/sym12050736

**Chicago/Turabian Style**

Planat, Michel, Raymond Aschheim, Marcelo M. Amaral, and Klee Irwin.
2020. "Quantum Computation and Measurements from an Exotic Space-Time *R*^{4}" *Symmetry* 12, no. 5: 736.
https://doi.org/10.3390/sym12050736