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Mathematics Without Boundaries pp 471–531Cite as

Exotic PDEs

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Abstract

In the framework of the PDE’s algebraic topology, previously introduced by A. Prástaro, are considered exotic differential equations, i.e., differential equations admitting Cauchy manifolds N identifiable with exotic spheres, or such that their boundaries ∂ N are exotic spheres. For such equations are obtained local and global existence theorems and stability theorems. In particular the smooth (four-dimensional) Poincaré conjecture is proved. This allows to complete the previous Theorem 4.59 in Prástaro (Essays in Mathematics and Its Applications (Dedicated to Stephen Smale for his 80th birthday), Springer, Heidelberg/New York/Dordrecht/London, pp. 369–419, 2012) also for the case n = 4.

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Notes

  1. 1.

    See also [70, 71, 73, 76, 77, 79] for related works in the category of quantum (super)manifolds in the sense introduced by A. Prástaro.

  2. 2.

    In this paper we denote by SX the suspension of a topological space X. (For details and general informations on algebraic topology see e.g. [61, 92, 98, 99, 101]. In this paper we will use the following notation: \(\thickapprox \) homeomorphism; \(\mathop{\cong}\) diffeomorphism; ≊ homotopy equivalence; ≃ homotopy.) Therefore SE n is the suspension of the CW-complex E n .

  3. 3.

    E is finite iff there is an integer N such that E n  = S n−N E N for n ≥ N and the complex E N is finite.

  4. 4.

    If n > 1 then G must be abelian.

  5. 5.

    If F is cofinal and \(K_{n} \subset E_{n}\) is a finite subcomplex, then there is an m such that \(S^{m}K_{n} \subset F_{n+m}\). Intersection of two cofinal subspectra is cofinal and if G ⊂ F ⊂ E are subspectra such that F is cofinal in E and G is cofinal in F, then G is cofinal in E. An arbitrary union of cofinal subspectra is cofinal.

  6. 6.

    Let E and F be spectra and f: E → F a function. If F ′ ⊂ F is a cofinal subspectrum then there is a cofinal subspectrum E ′ ⊂ E with \(f(E^{{\prime}}) \subset F^{{\prime}}\). (Composition of maps is now possible!) In the category \(\mathcal{S}\) any spectrum is equivalent to any cofinal subspectrum of its.

  7. 7.

    For any topological space X we set\(X^{+} \equiv X \sqcup \{{\ast}\}\). (If X is not compact X + is the Alexandrov compactification to a point.) In particular if \(X = I \equiv [0, 1] \subset \mathbb{R}\), then one takes {∗} as base point.

  8. 8.

    For definitions of generalized (co)homology theories see, e.g., [61, 92].

  9. 9.

    Here we have used the fact that E(SX) is a cofinal subspectrum of Σ E(X) and hence the inclusion i: E(SX) → Σ E(X) induces an isomorphism i ∗.

  10. 10.

    Let \(\mathcal{U}\) be an abelian category. A differential object in \(\mathcal{U}\) is a pair (A,d) where \(A \in Ob(\mathcal{U})\) and \(d \in \mathrm{Hom}_{\mathcal{U}}(A; A)\) such that d 2 = 0. Let \(\mathcal{D}(\mathcal{U})\) be the category of differential objects in \(\mathcal{U}\) . We call homology the additive functor \(H: \mathcal{D}(\mathcal{U}) \rightarrow \mathcal{U}\) , given by \(H(A,d) =\ker (d)/\mbox{ im}\ \ (d) = Z(A)/B(A)\) , where Z(A) is the set of cycles of A and B is the set of boundaries of A. H(A) is the homology of (A,d). Then, a spectral sequence in the category \(\mathcal{U}\) is a sequence of differential objects of \(\mathcal{U}\) : {E n ,d n }, n = 1, 2,…, such that H(E n ,d n ) = E n+1 , n = 1, 2,…. (See, e.g., [43, 57].)

  11. 11.

    For the homological definition of orientability see next Remark 3.

  12. 12.

    In particular, for k = n, \(F_{k}\mathop{\cong}O(n)\), and for k = 1, \(F_{1}\mathop{\cong}S^{n-1}\).

  13. 13.

    By the Stiefel–Whitney classes of an n-dimensional manifold M, one means the corresponding classes of TM.

  14. 14.

    Universal classifying space for the group SO(n).

  15. 15.

    If M is a closed and connected manifold of dimension k, admitting a finite triangulation, then \(H_{k}(M; \mathbb{Z}_{2})\mathop{\cong}\mathbb{Z}_{2}\). The fundamental class of M is \([M] =\sum _{i}v_{i}^{k}\), i.e., the sum of all k-simplexes.

  16. 16.

    A partition (i1 ,…,i r ) of n is nondyadic if none of the i β are of the form 2 s − 1.

  17. 17.

    Pontrjagin numbers are determined by means of homonymous characteristic classes belonging to \(H^{\bullet }(BG, \mathbb{Z})\) , where BG is the classifying space for G-bundles, with G = S p (n).

  18. 18.

    \(\mathbb{R}\mathbb{P}^{k}\) are orientable manifolds iff \(k \in \mathbb{N}\) is odd. P(2 r − 1,s2 r ) are orientable manifolds. One has \(\dim P(m,n) = m + 2n\).

  19. 19.

    In Table 4 are resumed relations between spectra, generalized (co)homologies, and some distinguished examples. Let us emphasize the relation with Brown’s representable theorem. A functor \(F: (\mathcal{W}^{\bullet {\prime}})^{op} \rightarrow \mathcal{S}_{et}\) is representable, i.e., F is equivalent to \(\mathrm{Hom}_{\mathcal{W}^{\bullet {\prime}}}(-; C)\) for some CW-complex C, iff the following conditions are satisfied. (i)(Wedge axiom). \(F(\vee _{\alpha }X_{\alpha })\mathop{\cong}\prod _{\alpha }F(X_{\alpha })\); (ii)(Mayer–Vietoris axiom). For any CW complex W covered by two subcomplexes U and V, and any elements u ∈ F(U), v ∈ F(V ), such that u and v restrict to the same element of \(F(U\bigcap V )\), there is an element w ∈ F(W) restricting to u and v, respectively. In the particular case of singular cohomology, one has \(H^{n}(X; A)\mathop{\cong}\mathrm{Hom}(X; K(A,n))\), i.e., the singular cohomology functor is representable. Thanks to extended versions of Brown’s representable theorem one can prove that all homology theories come from spectra, or by considering multiplicative operations, all homology theories come from ring spectra with multiplication μ: E ∧ E → E and the unity η: E(S 0) → E.

  20. 20.

    These module structures \(\mathcal{A}^{\bullet }\times H^{\bullet }(X; \mathbb{Z}_{p}) \rightarrow H^{\bullet }(X; \mathbb{Z}_{p})\) , allow us to understand that there are strong constraints just on the space X in order to obtain cohomology spaces \(H^{\bullet }(X; \mathbb{Z}_{p})\) with a prefixed structure. For example, do not exist spaces X with \(H^{\bullet }(X; \mathbb{Z}_{p}) = \mathbb{Z}[\alpha ]\) , unless α has dimension 2 or 4, where there examples \(\mathbb{C}P^{\infty }\) and \(\mathbb{H}P^{\infty }\).

  21. 21.

    A cohomology operations is a natural transformation between cohomology functors. One says that a cohomology operation is stable if it commutes with the suspension functor S. For example the cup product squaring operator H n (X; R) → H 2n (X; R), x↦x ∪ x, where R is a ring and X a topological space, is an instable cohomology operation. Instead, are stable the following Steenrod operations: \(Sq^{i}: H^{n}(X; \mathbb{Z}_{2}) \rightarrow H^{n+i}(X; \mathbb{Z}_{2})\) and \(P^{i}: H^{n}(X; \mathbb{Z}_{2}) \rightarrow H^{n+2i(p-1)}(X; \mathbb{Z}_{2})\) . (In Table  5 are resumed some fundamental properties of Sq i.)

  22. 22.

    A Galois field (or finite field) is a field that contains only finitely many elements. These are classified by q = p k if they contains q elements. Each Galois field with q elements is the splitting field of the polynomial x q − x. Recall that the splitting field of a polynomial p(x) over a field K is a field extension L of K over which p(x) factorizes into linear factors x − a i , and such that a i generates L over K, i.e, L = K(a i ). (In Table 6 are reported some properties of field extensions useful in the paper.) The extension L of minimal degree over K in which p splits exists and is unique up to isomorphism, identified by the Galois group of p. (In Table 7 are reported some fundamental properties and examples of Galois groups.)

  23. 23.

    \(\mathcal{A}^{\bullet }\) has a natural structure of Hopf algebra with commutative, associative comultiplication \(\psi: \mathcal{A}^{\bullet }\rightarrow \mathcal{A}^{\bullet }\otimes \mathcal{A}^{\bullet }\), given by \(\psi (\mathcal{A}^{k}) =\sum _{i+j=k}\mathcal{A}^{i} \otimes \mathcal{A}^{j}\). Let us denote by \(\mathcal{A}_{\bullet }\equiv \mathrm{Hom}_{\vert \vert }(\mathcal{A}^{\bullet }; \mathbb{F}_{q}) = \oplus _{n}\mathcal{A}_{n} = \oplus _{n}\mathrm{Hom}_{\mathbb{F}_{q}}(\mathcal{A}^{n}; \mathbb{F}_{q})\) the dual vector space to \(\mathcal{A}^{\bullet }\). One has the canonical evaluation pairing \(<,>: \mathcal{A}^{\bullet }\times \mathcal{A}_{\bullet }\rightarrow \mathbb{F}_{q}\), < f, α > = α(f). One has the following isomorphism of \(\mathbb{F}_{q}\)-Hopf algebras \(\mathcal{A}_{\bullet }\mathop{\cong}\mathbb{F}_{q}[\xi _{1},\xi _{2},\xi _{3},\cdots \,,\xi _{k},\cdots \,]\), where \(\deg (\xi _{k}) = q^{k-1}\), \(k \in \mathbb{N}\), and comultiplication \(\phi: \mathcal{A}_{\bullet }\rightarrow \mathcal{A}_{\bullet }\otimes \mathcal{A}_{\bullet }\), given by \(\phi (\xi _{k}) =\sum _{i+j=k}\xi _{i}^{q^{j} } \otimes \xi _{j}\).

  24. 24.

    The (co)homological interpretation of the Bockstein operator is associated with a short exact sequence of chain complexes in an abelian category. In fact to such a sequence there corresponds a long exact sequence . The boundary maps \(\delta _{n+1}: H_{n+1}(C_{\bullet }) \rightarrow H_{n}(A_{\bullet })\) are just the Bockstein homomorphisms. In particular, if is a short exact sequence of abelian groups and \(A_{\bullet } = E_{\bullet } \otimes A\), \(B_{\bullet } = E_{\bullet } \otimes B\), \(C_{\bullet } = E_{\bullet } \otimes C\), with E • a chain complex of free, or at least torsion free, abelian groups, then the Bockstein homomorphisms are induced by the corresponding short exact sequence . Similar considerations hold for cochain complexes. in such cases the Bockstein homomorphism increases the degree, i.e., \(\beta: H^{n}(C^{\bullet }) \rightarrow H^{n+i}(A^{\bullet })\).

  25. 25.

    The third integral Stiefel–Whitney class is the obstruction to a spin c-structure on X. (In Table 8 are also summarized, for any convenience, fundamental properties of the Whitney-Stiefel classes.)

  26. 26.

    Let us also emphasize that we can recognize webs on PDEs, by looking inside the geometric structure of PDEs. By means of such webs, we can solve (lower dimensional) Cauchy problems. This is important in order to decide about the “admissibility” of integral manifolds in integral bordism problems. However these aspects are not explicitly considered in this paper. They are studied in some details in other previous works about the PDE’s algebraic topology by A. Prástaro [3, 4]. For complementary informations on geometry of PDEs, see, e.g., Refs. [4, 8–10, 25, 26, 28, 40, 41, 56, 58, 59, 61–69, 80].

  27. 27.

    In other words the quantum bordism group of E k is the integral bordism group of \(J_{n}^{k}(W)\) relative to E k . (This language reproduces one in algebraic topology for couples (X, Y ) of differentiable manifolds, where Y ⊂ X.)

  28. 28.

    This means that \(N_{1} \in [N_{2}] \in \varOmega _{n-1}^{E_{k}}\), iff \(N_{1}^{(\infty )} \in [N_{2}^{(\infty )}] \in \varOmega _{n-1}^{E_{\infty }}\). (See Refs.[62, 78] for notations.)

  29. 29.

    This is, in general, an extended Hopf algebra. (See Refs. [59, 60].)

  30. 30.

    \(\mathfrak{C}ons(E_{k})\) can be identified with the spectral term \(E_{1}^{0,n-1}\) of the spectral sequence associated with the filtration induced in the graded algebra \(\varOmega ^{\bullet }(E_{\infty }) \equiv \oplus _{q\geq 0}\varOmega ^{q}(E_{\infty })\), by the subspaces \(C\varOmega ^{q}(E_{\infty }) \subset \varOmega ^{q}(E_{\infty })\). (For abuse of language we shall call “conservation laws of k-order”, characteristic integral (n − 1)-forms too. Note that \(C\varOmega ^{0}(E_{k}) = 0\). See also Refs. [57–60, 62].)

  31. 31.

    It is important to note that can be \(\mathcal{I}(E_{k})^{p}\not =0\) even if E k is p-cohomologic trivial, i.e., \(H^{p}(E_{k}; \mathbb{R}) = 0\) . This, for example, can happen if E k is contractible to a point.

  32. 32.

    For example for the d’Alembert equation one can see that for any conservation law ω one has < ω, N > = 0, where N is any admissible one-dimensional compact integral manifold of d ′ A), but \(\omega \not\in [0] \in E_{1}^{0,n-1}\).

  33. 33.

    The following Refs. [7, 11–13, 21, 34–39, 42, 44–46, 46, 48–52, 82, 83, 86–88, 91, 96, 97, 100, 102] are important background for differential structures and exotic spheres.

  34. 34.

    For the definition of the groups Θn , see [74].

  35. 35.

    It is well known that the Schönflies problem is related to extensions of the Jordan–Brouwer theorem. (See, e.g., [42].) Let us emphasize that the lemma does not necessitate to work in the category of topological spaces. In fact, it is well known that topological embeddings f: S2 → S 3 do not necessarily have simply connected the two separate components of S 3 ∖f(S 2 ). In fact this is just the case of the Alexander horned sphere Σ 2 ⊂ S 3 [5].

  36. 36.

    The reduced homology groups \(\tilde{H}_{p}(X)\), of non-empty space X, are the homology groups of the augmented chain complex:

    , where ε can be considered generated by the chain \([\varnothing ]\mapsto X\), sending the simplex with no-vertices (empty simplex) to X, i.e., \(\epsilon (\sum _{i}n_{i}\sigma ) =\sum _{i}n_{i}\). Since ε ∂ i  = 0, ε induces a map \(H_{0}(X) \rightarrow \mathbb{Z}\) with kernel \(\tilde{H}_{0}(X)\), so one has \(H_{0}(X)\mathop{\cong}\tilde{H}_{0}(X)\bigoplus \mathbb{Z}\), and \(H_{p}(X)\mathop{\cong}\tilde{H}_{p}(X)\), \(\forall p > 0\). Therefore, we get \(\tilde{H}_{0}(pt)\mathop{\cong}0\). Furthermore, one has \(\tilde{H}_{p}(X,A)\mathop{\cong}H_{p}(X,A)\), for any couple (X, A), \(X \supset A\not =\varnothing \), and \(\tilde{H}_{p}(X) = H_{p}(X,x_{0})\).

  37. 37.

    The Rokhlin invariant of a spin 3-manifold X is the signature of any spin 4-manifold V, such that ∂ V = X, is well defined mod 16. A spin structure exists on a manifold M, if its second Stiefel–Whitney class is trivial: w 2(M) = 0. These structures are classified by \(H^{1}(M; \mathbb{Z}_{2})\mathop{\cong}H_{1}(M; \mathbb{Z}_{2})\). Therefore, homology 3-spheres have an unique spin structure, hence for them the Rokhlin invariant is well defined. In particular the Poincaré homology sphere bounds a spin 4-manifold with intersection form E 8, so its Rokhlin invariant is 1.

  38. 38.

    These are three-dimensional manifolds endowed with a S 1-bundle structure over a two-dimensional orbifold. (See, e.g., [84].)

  39. 39.

    The existence of such a manifold X can be proved following a strategy similar to the one to prove Lemma 11.

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Acknowledgements

I would like thank my colleague Themistocles Rassias for inviting me to write my contribution for this Springer book. A similar version of this work was early put on arXiv.1101.0283 [math.AT].

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  1. Department SBAI-Mathematics, University of Roma La Sapienza, Via A. Scarpa, 16-10-00166, Roma, Italy

    Agostino Prástaro

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  1. Dept. Industrial & Systems Engineering, University of Florida, Gainesville, Florida, USA

    Panos M. Pardalos

  2. Department of Mathematics, National Technical University of Athens, Athens, Greece

    Themistocles M. Rassias

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Prástaro, A. (2014). Exotic PDEs. In: Pardalos, P., Rassias, T. (eds) Mathematics Without Boundaries. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1124-0_16

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