##
**Space-time foam dense singularities and de Rham cohomology.**
*(English)*
Zbl 1005.46020

The second author introduced the notion of space-time foam algebra as the algebra of generalized distributions which may have singularities on dense subsets. By means of the sheaf of space-time algebra as the structure sheaf algebra \({\mathcal A}\) of the differential triad \((X,{\mathcal A},\partial)\) and abstract differential geometry introduced by the first author, de Rham cohomology with dense singularities of a space is introduced and it is shown that this cohomology is the same as the usual de Rham cohomology, when the space is nonsingular (Sect. 4).

The outline of the paper is as follows: Mallios’s abstract differential geometry is reviewed in Sect. 1. It based on the differential triad \((X,{\mathcal A},\partial)\), where \(X\) is a topological space (it need not be a smooth manifold), \({\mathcal A}\) is a commutative and unital sheaf of \(\mathbb{R}\)-algebras on \(X\) and \(\partial:{\mathcal A}\to \Omega^1\) is a derivation, where \(\Omega^1\) is a sheaf of \({\mathcal A}\)-modules on \(X\). It is shown that from these data a de Rham complex is obtained [A. Mallios, Geometry of vector sheaves. I, II (1998; Zbl 0904.18001 and Zbl 0904.18002)]. In Section 2, Rosinger’s generalized functions, or space-time foam algebras are explained. The domain of generalized functions is an open set \(X\) of \(\mathbb{R}^n\). \({\mathcal S}\) is a set of subsets \(\Sigma\), thought to be the set of singularities of a certain generalized function, of \(X\). \({\mathcal S}\) is assumed to be a subset of \({\mathcal S}_{{\mathcal D}}(X)= \{\Sigma\mid X\setminus\Sigma\) is dense in \(X\}\). Examples of such \({\mathcal S}\) are \({\mathcal S}_{\text{nd}}(X)\), the set of closed nowhere dense subsets of \(X\), and \({\mathcal S}_{\text{Baire}1}(X)\), the set of the first Baire category set in \(X\).

Let \(L= (\Lambda,\leq)\) be a right directed partial order, \(({\mathcal C}^\infty(X))^\Lambda\) be the set of sequences of smooth functions indexed by \(\lambda\in \Lambda\), \({\mathcal I}_{L,\Sigma}(X)\) the ideal in \(({\mathcal C}^\infty(X))^\Lambda\) consisting of those sequences \(w= (w_\lambda)\) such that for any \(x\in\Sigma\), there exists \(\lambda\) such that \(D^p w_\mu(x)= 0\) for any \(p\in\mathbb{N}^n\) if \(\mu\geq\lambda\). The ideal \({\mathcal I}_{L,{\mathcal S}}(X)\) is defined to be the union of \({\mathcal I}_{L,\Sigma}(X)\), \(\Sigma\in{\mathcal S}\). The foam algebra \(B_{L,\Sigma}(X)\) and multi-foam algebra \(B_{L,{\mathcal S}}(X)\) are defined to be the quotients of \(({\mathcal C}^\infty(X))^\Lambda\) by \({\mathcal I}_{L,\Sigma}(X)\) and \({\mathcal I}_{L,{\mathcal S}}(X)\), respectively. These algebras are called together space-time foam algebras. Since \((u_\lambda)\), \(u_\lambda= u\), is identified with \(u\in({\mathcal C}^\infty(X))\), elements of a space-time foam algebra can be regarded as generalized functions having singularities at \(\Sigma(\in{\mathcal S})\).

Differentials on a space-time foam algebra are induced from the differentials on \(\mathbb{R}^n\). After showing nontriviality of ideals and remarks on the dependence of \(B_{L,{\mathcal S}}(X)\) on \(L\), fineness and flabbyness conditions of the sheaf of space-time foam algebras \({\mathcal B}_{L,{\mathcal S},X}= (B_{L,{\mathcal S}|U}(U))\), are given (Lemma 2, a proof is given in the Appendix). As a result, \({\mathcal B}_{\mathbb{N}\times \mathbb{N},{\mathcal S}_{\text{Baire}1}(X),x}\) is a fine and flabby sheaf. Most of the results in this Section are taken from former papers of E. E. Rosinger [Differential algebras with dense singularities on manifolds, Technical Report UPWT 99/9, Univ. Pretoria (1999)].

In Sect. 4, space-time foam differential triads are explained. The authors state that the sheaves of Schwartz distributions and Colombeau generalized functions are not flabby. So the use of foam algebras is more convenient in abstract differential geometry. Since the elements of a foam algebra are represented by a series of smooth functions, the Poincaré Lemma holds for the de Rham complex obtained from a space-time foam differential triad (Sect. 4, Theorem 1). This is the final result of the paper.

The authors state that space-time foam algebras are closely related to non-standard analysis in the of spirit of [cf. H. A. Biagioni, “A nonlinear theory of generalized functions”, Lect. Notes Math. 1421 (1990; Zbl 0694.46032)], and seems to be outside of the scope of Kuratowski-Bourbarki topological concept. Studies are motivated and aimed by general relativity and its quantization. For this point, see A. Mallios [“Abstract differential geometry, general relativity, and singularities, unsolved problems on mathematics for the 21st century”, 77–100 (2001; Zbl 1002.53047)].

The outline of the paper is as follows: Mallios’s abstract differential geometry is reviewed in Sect. 1. It based on the differential triad \((X,{\mathcal A},\partial)\), where \(X\) is a topological space (it need not be a smooth manifold), \({\mathcal A}\) is a commutative and unital sheaf of \(\mathbb{R}\)-algebras on \(X\) and \(\partial:{\mathcal A}\to \Omega^1\) is a derivation, where \(\Omega^1\) is a sheaf of \({\mathcal A}\)-modules on \(X\). It is shown that from these data a de Rham complex is obtained [A. Mallios, Geometry of vector sheaves. I, II (1998; Zbl 0904.18001 and Zbl 0904.18002)]. In Section 2, Rosinger’s generalized functions, or space-time foam algebras are explained. The domain of generalized functions is an open set \(X\) of \(\mathbb{R}^n\). \({\mathcal S}\) is a set of subsets \(\Sigma\), thought to be the set of singularities of a certain generalized function, of \(X\). \({\mathcal S}\) is assumed to be a subset of \({\mathcal S}_{{\mathcal D}}(X)= \{\Sigma\mid X\setminus\Sigma\) is dense in \(X\}\). Examples of such \({\mathcal S}\) are \({\mathcal S}_{\text{nd}}(X)\), the set of closed nowhere dense subsets of \(X\), and \({\mathcal S}_{\text{Baire}1}(X)\), the set of the first Baire category set in \(X\).

Let \(L= (\Lambda,\leq)\) be a right directed partial order, \(({\mathcal C}^\infty(X))^\Lambda\) be the set of sequences of smooth functions indexed by \(\lambda\in \Lambda\), \({\mathcal I}_{L,\Sigma}(X)\) the ideal in \(({\mathcal C}^\infty(X))^\Lambda\) consisting of those sequences \(w= (w_\lambda)\) such that for any \(x\in\Sigma\), there exists \(\lambda\) such that \(D^p w_\mu(x)= 0\) for any \(p\in\mathbb{N}^n\) if \(\mu\geq\lambda\). The ideal \({\mathcal I}_{L,{\mathcal S}}(X)\) is defined to be the union of \({\mathcal I}_{L,\Sigma}(X)\), \(\Sigma\in{\mathcal S}\). The foam algebra \(B_{L,\Sigma}(X)\) and multi-foam algebra \(B_{L,{\mathcal S}}(X)\) are defined to be the quotients of \(({\mathcal C}^\infty(X))^\Lambda\) by \({\mathcal I}_{L,\Sigma}(X)\) and \({\mathcal I}_{L,{\mathcal S}}(X)\), respectively. These algebras are called together space-time foam algebras. Since \((u_\lambda)\), \(u_\lambda= u\), is identified with \(u\in({\mathcal C}^\infty(X))\), elements of a space-time foam algebra can be regarded as generalized functions having singularities at \(\Sigma(\in{\mathcal S})\).

Differentials on a space-time foam algebra are induced from the differentials on \(\mathbb{R}^n\). After showing nontriviality of ideals and remarks on the dependence of \(B_{L,{\mathcal S}}(X)\) on \(L\), fineness and flabbyness conditions of the sheaf of space-time foam algebras \({\mathcal B}_{L,{\mathcal S},X}= (B_{L,{\mathcal S}|U}(U))\), are given (Lemma 2, a proof is given in the Appendix). As a result, \({\mathcal B}_{\mathbb{N}\times \mathbb{N},{\mathcal S}_{\text{Baire}1}(X),x}\) is a fine and flabby sheaf. Most of the results in this Section are taken from former papers of E. E. Rosinger [Differential algebras with dense singularities on manifolds, Technical Report UPWT 99/9, Univ. Pretoria (1999)].

In Sect. 4, space-time foam differential triads are explained. The authors state that the sheaves of Schwartz distributions and Colombeau generalized functions are not flabby. So the use of foam algebras is more convenient in abstract differential geometry. Since the elements of a foam algebra are represented by a series of smooth functions, the Poincaré Lemma holds for the de Rham complex obtained from a space-time foam differential triad (Sect. 4, Theorem 1). This is the final result of the paper.

The authors state that space-time foam algebras are closely related to non-standard analysis in the of spirit of [cf. H. A. Biagioni, “A nonlinear theory of generalized functions”, Lect. Notes Math. 1421 (1990; Zbl 0694.46032)], and seems to be outside of the scope of Kuratowski-Bourbarki topological concept. Studies are motivated and aimed by general relativity and its quantization. For this point, see A. Mallios [“Abstract differential geometry, general relativity, and singularities, unsolved problems on mathematics for the 21st century”, 77–100 (2001; Zbl 1002.53047)].

Reviewer: Akira Asada (Takarazuka)

### MSC:

46F30 | Generalized functions for nonlinear analysis (Rosinger, Colombeau, nonstandard, etc.) |

58A12 | de Rham theory in global analysis |

55N30 | Sheaf cohomology in algebraic topology |

54A05 | Topological spaces and generalizations (closure spaces, etc.) |

83C75 | Space-time singularities, cosmic censorship, etc. |