Abstract
Exotic heat equations that allow to prove the Poincaré conjecture and its generalizations to any dimension are considered. The methodology used is the PDE’s algebraic topology, introduced by A. Prástaro in the geometry of PDE’s, in order to characterize global solutions. In particular it is shown that this theory allows us to identify n-dimensional exotic spheres, i.e., homotopy spheres that are homeomorphic, but not diffeomorphic to the standard \({S}^{n}\).
Panos Pardalos and Themistocles M. Rassias (eds.), Essays in Mathematics and its Applications. Dedicated to Stephen Smale, Springer, New York, 2011
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Notes
- 1.
In this paper we will use the following notation: \(\thickapprox\) homeomorphism; \(\cong\) diffeomorphism; \(\approxeq \) homotopy equivalence; \(\simeq \) homotopy.
- 2.
- 3.
In order to allow a more easy understanding, this paper has been written in a large expository style. (A first version of this paper has been put in arxiv: [43].)
- 4.
We shall denote with the same symbol (RF) the corresponding algebraic manifold. For a geometric algebraic theory of PDE’s see the monograph [29], and references quoted there.
- 5.
We used the fact that the fiber of \(E \rightarrow M\) is contractible.
- 6.
A relative CW complex (Y,X) is a space Y and a closed subspace X such that \(Y ={ \bigcup }_{r=-1}^{\infty }{Y }_{r}\) , such that \(X = {Y }_{-1} \subset {Y }_{0} \subset \cdots \) , and \({Y }_{r}\) is obtained from \({Y }_{r-1}\) by attaching r-cells.
- 7.
Let \({H}_{k}(Y,X; F)\) denote the singular homology group of the pair (Y,X) with coefficients in the field F. \({\beta }_{k} ={ \mathrm{dim}}_{F}{H}_{k}(Y,X; F)\) are called the F-Betti numbers of (Y,X). If these numbers are finite and only finitely are nonzero, then the homological Euler characteristic of \((Y,X)\) is defined by the formula: \({\chi }_{hom}(Y,X) ={ \sum \nolimits }_{0\leq k\leq \infty }{(-1)}^{k}{\beta }_{k}\) . When Y is a compact manifold and X is a compact submanifold, then \({\chi }_{hom}(Y,X)\) is defined. The evaluation of homological Euler characteristic for topological spaces coincides with that of Euler characteristic for CW complexes X given by \(\chi (X) ={ \sum \nolimits }_{i\geq 0}{(-1)}^{i}{k}_{i}\) , where \({k}_{i}\) is the number of cells of dimension i. For closed smooth manifolds M, \(\chi (M)\) coincides with the Euler number, that is the Euler class of the tangent bundle TM, evalued on the fundamental class of M. For closed Riemannian manifolds, \(\chi (M)\) is given as an integral on the curvature, by the generalized Gauss-Bonnet theorem: \(\chi (V ) = \frac{1} {{(2\pi )}^{n}}{\int \nolimits \nolimits }_{V }Pf(\Omega )\) , where \(\partial V = \varnothing \), \(\mathrm{dim}V = 2n\), \(\Omega \) is the curvature of the Levi-Civita connection and \(Pf(\Omega ) = \frac{1} {{2}^{n}n!}{ \sum \nolimits }_{\sigma \in {S}_{2n}}\epsilon (\sigma ){ \prod \nolimits }_{i=1}^{n}{\Omega }_{\sigma (2i-1)\sigma (2i)}\) , where \(({\Omega }_{rs})\) is the skew-symmetric \((2n) \times (2n)\) matrix representing \(\Omega : V \rightarrow \mathfrak{s}\mathfrak{o}(2n) \otimes {\Lambda }_{2}^{0}(V )\) , hence \(Pf(\Omega ) : V \rightarrow {\Lambda }_{2n}^{0}(V )\). In Table 2 are reported some important properties of Euler characteristic, that are utilized in this paper.
- 8.
Let us recall that a compact connected manifold M with boundary \(\partial M\not =\varnothing \) , admits a nonvanishing vector field. Furthermore, a compact, oriented n-dimensional submanifold \(M \subset {\mathbb{R}}^{2n}\) has a nonvanishing normal vector field. Therefore, above statements about smooth solutions of (RF) agree with well known results of differential topology. (See, e.g., [17]).
- 9.
Recall that an odd dimensional oriented compact manifold M, with \(\partial M = \varnothing \) has \(\chi (M) = 0\). In particular \(\chi ({S}^{2k+1}) = 0\), instead \(\chi ({S}^{2k}) = 2\). Furthermore, if M and N are compact oriented manifolds with \(\partial M = \partial N = \varnothing \), then \(\chi (M \times N) = \chi (M)\chi (N)\).
- 10.
If \(f : N \rightarrow M\) is an orientable fiber bundle with compact, orientable fiber F, integration over the fiber provides another definition of the transfer map: \(\tau : {H}_{de-Rham}^{\bullet }(N) \rightarrow {H}_{de-Rham}{(M)}^{\bullet -r}\) , where \(r = \mathrm{dim}F\).
- 11.
Note that an homotopy equivalence is an weak homotopy equivalence, but the vice versa is not true. Recall that two pointed topological spaces \((X,{x}_{0})\) and \((Y,{y}_{0})\) have the same homotopy type if \({\pi }_{1}(X,{x}_{0})\cong{\pi }_{1}(Y,{y}_{0})\) , and \({\pi }_{n}(X,{x}_{0})\) and \({\pi }_{n}(Y,{y}_{0})\) are isomorphic as modules over \(\mathbb{Z}[{\pi }_{1}(X,{x}_{0})]\) for \(n \geq 2\) . A simply homotopy equivalence between m-dimensional manifolds, (or finite CW complexes), is a homotopy equivalence \(f : M \approxeq N\) such that the Whitehead torsion \(\tau (f) \in Wh({\pi }_{1}(M))\) , where \(Wh({\pi }_{1}(M))\) is the Whitehead group of \({\pi }_{1}(M)\) . With this respect, let us recall that if A is an associative ring with unity, such that \({A}^{m}\) is isomorphic to \({A}^{n}\) iff \(m = n\) , put \(GL(A) \equiv {\bigcup \nolimits }_{n-1}G{L}_{n}(A)\) , the infinite general linear group of A and \(E(A) \equiv [GL(A),GL(A)] \vartriangleleft GL(A)\) . E(A) is the normal subgroup generated by the elementary matrices \(\left (\begin{array}{cc} 1 & a\\ 0 & 1\\ \end{array} \right )\) . The torsion group \({K}_{1}(A)\) is the abelian group \({K}_{1}(A) = GL(A)/E(A)\) . Let \({A}^{\bullet }\) denote the multiplicative group of units in the ring A. For a commutative ring A, the inclusion \({A}^{\bullet }\hookrightarrow {K}_{1}(A)\) splits by the determinant map \(\det : {K}_{1}(A) \rightarrow {A}^{\bullet }\), \(\tau (\varphi )\mapsto \det (\varphi )\) and one has the splitting \({K}_{1}(A) = {A}^{\bullet }\oplus S{K}_{1}(A)\) , where \(S{K}_{1}(A) =\ker (\det : {K}_{1}(A) \rightarrow {A}^{\bullet })\) . If A is a field, then \({K}_{1}(A)\cong{A}^{\bullet }\) and \(S{K}_{1}(A) = 0\) . The torsion \(\tau (f)\) of an isomorphism \(f : L\cong{K}\) of finite generated free A-modules of rank n, is the torsion of the corresponding invertible matrix \(({f}_{j}^{i}) \in G{L}_{n}(A)\) , i.e., \(\tau (f) = \tau ({f}_{i}^{j}) \in {K}_{1}(A)\) . The isomorphism is simple if \(\tau (f) = 0 \in {K}_{1}(A)\) . The Whitehead group of a group G is the abelian group \(Wh(G) \equiv {K}_{1}(\mathbb{Z}[G])/\{\tau (\mp g)\vert g \in G\}\). \(Wh(G) = 0\) in the following cases: (a) \(G =\{ 1\}\) ; (b) \(G = {\pi }_{1}(M)\) , with M a surface; (c) \(G = {\mathbb{Z}}^{m}\), \(m \geq 1\) . There is a conjecture, (Novikov) that extends the case (b) also to m-dimensional compact manifolds M with universal cover \(\widetilde{M} = {\mathbb{R}}^{m}\). This conjecture has been verified in many cases [6].
- 12.
Let us emphasize that to state that the inclusions \({M}_{i}\hookrightarrow W\), \(i = 0, 1\) , are homotopy equivalences is equivalent to state the \({M}_{i}\) are deformation retracts of W.
- 13.
In general \(W\bigcup \nolimits {h}^{p}\) is not a manifold but a CW-complex.
- 14.
We say also that a p-surgery removes a framed p-embedding \(g : {S}^{p} \times {D}^{n-p}\hookrightarrow M\). Then it kills the homotopy class \([g] \in {\pi }_{p}(M)\) of the core \(g = g\vert : {S}^{p} \times \{ 0\}\hookrightarrow M\).
- 15.
The connected sum of two connected n-dimensional manifolds X and Y is the n-dimensional manifold \(X\sharp Y\) obtained by excising the interior of embedded discs \({D}^{n} \subset X\), \({D}^{n} \subset Y\) , and joining the boundary components \({S}^{n-1} \subset \overline{X \setminus {D}^{n}}\), \({S}^{n-1} \subset \overline{Y \setminus {D}^{n}}\) , by \({S}^{n-1} \times I\).
- 16.
It is interesting to add that another related notion of cobordism is the H-cobordism of n-dimensional manifold, (V ; M,N), \(\partial V = M \sqcup N\) , with \({H}_{\bullet }(M)\cong{H}_{\bullet }(N)\cong{H}_{\bullet }(V )\) . An n-dimensional manifold \(\Sigma \) is a homology sphere if \({H}_{\bullet }(\Sigma ) = {H}_{\bullet }({S}^{n})\) . Let \({\Theta }_{n}^{H}\) be the abelian group of H-cobordism classes of n-dimensional homology spheres, with addition by connected sum. (Kervaire’s theorem.) For \(n \geq 4\) every n-dimensional homology sphere \(\Sigma \) is H-cobordant to a homology sphere and the forgethful map \({\Theta }_{n} \rightarrow {\Theta }_{n}^{H}\) is an isomorphism.
- 17.
A topological manifold M is piecewise linear, i.e., admits a PL structure, if there exists an atlas \(\{{U}_{\alpha },{\varphi }_{\alpha }\}\) such that the composities \({\varphi }_{\alpha } \circ {\varphi }_{\alpha \prime }^{-1}\), are piecewise linear. Then there is a polyehdron \(P \subset {\mathbb{R}}^{s}\), for some s and a homeomorphism \(\varphi : P\thickapprox{M}\) , ( triangulation), such that each composite \({\varphi }_{\alpha } \circ \varphi \) is piecewise linear.
- 18.
\(\kappa \) is \({\mathbb{Z}}_{2}\) -valued and vanishes iff the product manifold \(M \times \mathbb{R}\) can be given a differentiable structure.
- 19.
It is not known which 4-manifolds with \(\kappa = 0\) actually possess differentiable structure, and it is not known when this structure is essentially unique.
- 20.
There exists also a s-cobordism version of such a theorem for non-simply connected manifolds. More precisely, an (n + 1)-dimensional h-cobordism (V ; N, M) with \(n \geq 5\) , is trivial iff it is an s-cobordism. This means that for \(n \geq 5\) h-cobordant n-dimensional manifolds are diffeomorphic iff they are s-cobordant. Since the Whitehead group of the trivial group is trivial, i.e., \(Wh(\{1\})\,=\,0\), it follows that h-cobordism theorem is the simply-connected special case of the s-cobordism.
- 21.
The proof utilizes Morse theory and the fact that for an h-cobordism \({H}_{\bullet }(W,M)\,\cong\,{H}_{\bullet }(W,N)\cong{0}\) , gives \(W\cong{M} \times [0, 1]\) . The motivation to work with dimensions \(n \geq 5\) is in the fact that it is used the Whitney embedding theorem that states that a map \(f : N \rightarrow M\) , between manifolds of dimension n and m respectively, such that either \(2n + 1 \leq m\) or \(m = 2n \geq 6\) and \({\pi }_{1}(M) =\{ 1\}\), is homotopic to an embedding.
- 22.
\({\Gamma }_{n}\) can be identified with the set of twisted n-spheres up to orientation-preserving diffeomorphisms, for \(n\not =4\) . One has the exact sequences given in (37).
If the used diffeomorphism \({S}^{n-1} \rightarrow {S}^{n-1}\) to obtain a twisted n-sphere by gluing the corresponding boundaries of two disks \({D}^{n}\) , is not smoothly isotopic to the identity, one obtains an exotic n-sphere. For \(n > 4\) every exotic n-sphere is a twisted sphere. For \(n = 4\), instead, twisted spheres are standard ones [4].
- 23.
This result by Kirby and Siebenman does not exclude that every manifold of dimension \(n > 4\) can possess some triangulation, even if it cannot be PL-homeomorphic to Euclidean space.
- 24.
Let R be a commutative ring and H a finite generated free R-module. A \(\epsilon \) -symmetric form over H is a bilinear mapping \(\lambda : H \times H \rightarrow R\) , such that \(\lambda (x,y) = \epsilon \lambda (y,x)\) , with \(\epsilon \in \{ +1,-1\}\) . The form \(\lambda \) is nonsingular if the R-module morphism \(H \rightarrow {H}^{{_\ast}}\equiv Ho{m}_{R}(H; R)\), \(x\mapsto (y\mapsto \lambda (x,y))\) is an isomorphism. A \(\epsilon \) -quadratic form associated to a \(\epsilon \)-symmetric form \(\lambda \) over H, is a function \(\mu : H \rightarrow {Q}_{\epsilon }(R) \equiv coker (1 - \epsilon : R \rightarrow R)\) , such that: (i) \(\lambda (x,y) = \mu (x + y) - \mu (x) - \mu (y)\) ; (ii) \(\lambda (x,x) = (1 + \epsilon )\mu (x) \in im (1 + \epsilon : R \rightarrow R) \subseteq \ker (1 - \epsilon : R \rightarrow R)\), \(\forall x,y \in H\), \(a \in R\) . If \(R = \mathbb{Z}\) and \(\epsilon = 1\) , we say signature of \(\lambda \), \(\sigma (\lambda ) = p - q \in \mathbb{Z}\), where p and q are respectively the number of positive and negative eigenvalues of the extended form on \(\mathbb{R}{ \otimes }_{\mathbb{Z}}H\). Then \(\lambda \) has a 1-quadratic function \(\mu : H \rightarrow {Q}_{+1}(\mathbb{Z})\) iff \(\lambda \) has even diagonal entries, i.e., \(\lambda (x,x) \equiv 0\:(mod \,2)\), with \(\mu (x) = \lambda (x,x)/2\), \(\forall x \in H\). If \(\lambda \) is nonsingular then \(\sigma (\lambda ) \equiv 0\:(mod \,8)\). Examples. (1) \(R = H = \mathbb{Z}\), \(\lambda = 1\), \(\sigma (\lambda ) = 1\).
Fig. 2 
Connected sum representations. \({M}_{1}\sharp {M}_{2} = ({M}_{1} \setminus {D}^{n})\bigcup \nolimits ({S}^{n-1} \times {D}^{1})\bigcup \nolimits ({M}_{2} \setminus {D}^{n})\)
(2) \(R = \mathbb{Z}\), \(H = {\mathbb{Z}}^{8}\), \(\lambda = {E}^{8}\)-form, given by \(\left ({\lambda }_{ij}\right )=\left (\begin{array}{cc} {a}_{rs}& {b}_{rs}\\ {c}_{ rs} &{d}_{rs}\\ \end{array} \right )\) with \(({a}_{rs})=\left (\begin{array}{cccc} 2 & 1 & 0 & 0\\ 1 & 2 & 1 & 0 \\ 0 & 1 & 2 & 1\\ 0 & 0 & 1 & 2\\ \end{array} \right )\),\(({b}_{rs}) = \left (\begin{array}{cccc} 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\\ 1 & 0 & 0 & 0\\ \end{array} \right )\) , \(({c}_{rs}) = \left (\begin{array}{cccc} 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ \end{array} \right )\) , \(({d}_{rs}) = \left (\begin{array}{cccc} 2 & 1 & 0 & 1\\ 1 & 2 & 1 & 0 \\ 0 & 1 & 2 & 1\\ 1 & 0 & 1 & 2\\ \end{array} \right )\).
- 25.
A genus for closed smooth manifolds with some X-structure, is a ring homomorphism \({\Omega }_{\bullet }^{X} \rightarrow R\) , where R is a ring. For example, if the X-structure is that of oriented manifolds, i.e., \(X = SO\) , then the signature of these manifolds just identifies a genus \(\sigma {: }^{+}{\Omega }_{\bullet } = {\Omega }_{\bullet }^{SO} \rightarrow \mathbb{Z}\) , such that \(\sigma (1) = 1\) , and \(\sigma {: }^{+}{\Omega }_{p} \rightarrow 0\) if \(p\not =4q\) . Therefore the genus identifies also a \(\mathbb{Q}\) -algebra homomorphism \({\Omega }_{\bullet }^{SO}{\otimes }_{\mathbb{Q}}\mathbb{Q} \rightarrow \mathbb{Q}\) . More precisely, let \(A \equiv \mathbb{Q}[{t}_{1},{t}_{2},\cdots \,]\) be a graded commutative algebra, where \({t}_{i}\) has degree i. Set \(\mathcal{A}\equiv A[[{a}_{0},{a}_{1},\cdots \,]]\) , where \({a}_{i} \in A\) is homogeneous of degree i, i.e., the elements of \(\mathcal{A}\) are infinite formal sums \(a \equiv {a}_{0} + {a}_{1} + {a}_{2} + \cdots \) . Let \({\mathcal{A}}^{\bullet }\subset \mathcal{A}\) denote the subgroup of the multiplicative group of \(\mathcal{A}\) of elements with leading term 1. Let \({K}_{1}({t}_{1})\), \({K}_{2}({t}_{1},{t}_{2})\), \({K}_{3}({t}_{1},{t}_{2},{t}_{3}),\cdots \in A\) , be a sequence of polynomials of A, where \({K}_{n}\) is homogeneous of degree n. For \(a = {a}_{0} + {a}_{1} + {a}_{2} + \cdots \in {\mathcal{A}}^{\bullet }\) , we define \(K(a) \in {\mathcal{A}}^{\bullet }\) by \(K(a) = 1 + {K}_{1}({a}_{1}) + {K}_{2}({a}_{1},{a}_{2}) + \cdots \) . We say that \({K}_{n}\) form a multiplicative sequence if \(K(ab) = K(a)K(b)\), \(\forall a,b \in {\mathcal{A}}^{\bullet }\) . An example is with \({K}_{n}({t}_{1},\cdots \,,{t}_{n}) = {\lambda }^{n}{t}_{n}\), \(\lambda \in \mathbb{Q}\) . Another example is given by the formal power series given in (40) with \({\lambda }_{k} = {(-1)}^{k-1}\frac{{2}^{2k}{B}_{2k}} {(2k)!}\) . For any partition \(I = ({i}_{1},{i}_{2},\cdots \,,{i}_{k})\) of n, set \({\lambda }_{I} = {\lambda }_{{i}_{1}}{\lambda }_{{i}_{2}}\cdots {\lambda }_{{i}_{k}}\) . Now define polynomials \({\mathcal{L}}_{n}({t}_{1},\cdots \,,{t}_{n}) \in A\) by \({\mathcal{L}}_{n}({t}_{1},\cdots \,,{t}_{n}) ={ \sum \nolimits }_{I}{\lambda }_{I}{s}_{I}({t}_{1},\cdots \,,{t}_{n})\) , where the sum is over all partitions of n and \({s}_{I}\) is the unique polynomial belonging to \(\mathbb{Z}[{t}_{1},\cdots \,,{t}_{n}]\) such that \({s}_{I}({\sigma }_{1},\cdots \,,{\sigma }_{n}) = \sum \nolimits {t}^{I}\) , where \({\sigma }_{1},\cdots \,,{\sigma }_{n}\) are the elementary symmetric functions that form a polynomial basis for the ring \({\mathcal{S}}_{n}\) of symmetric functions in n variables. ( \({\mathcal{S}}_{n}\) is the graded subring of \(\mathbb{Z}[{t}_{1},\cdots \,,{t}_{n}]\) of polynomials that are fixed by every permutation of the variables. Therefore we can write \({\mathcal{S}}_{n} = \mathbb{Z}[{\sigma }_{1},\cdots \,,{\sigma }_{n}]\) , with \({\sigma }_{i}\) of degree i. In Table 4 are reported the polynomials \({s}_{I}({\sigma }_{1},\cdots \,,{\sigma }_{n})\) , for \(0 \leq n \leq 4\) .) \({\mathcal{L}}_{n}\) form a multiplicative sequence. In fact \(\mathcal{L}(ab) ={ \sum \nolimits }_{I}{s}_{I}(ab) ={ \sum \nolimits }_{I}{\lambda }_{I}{\sum \nolimits }_{{I}_{1}{I}_{2}=I}{s}_{{I}_{1}}(a){s}_{{I}_{2}}(b) ={ \sum \nolimits }_{{I}_{1}{I}_{2}=I}{\lambda }_{{I}_{1}}{s}_{{I}_{1}}(a){\lambda }_{{I}_{2}}{s}_{{I}_{2}}(b) = \mathcal{L}(a)\mathcal{L}(b)\) . Then for an n-dimensional manifold M one defines \(\mathcal{L}\) -genus, \(\mathcal{L}[M] = 0\) if \(n\not =4k\) , and \(\mathcal{L}[M] =< {K}_{k}({p}_{1}(TM),\cdots \,,{p}_{k}(TM)),{\mu }_{M} >\) if \(n = 4k\) , where \({\mu }_{M}\) is the rational fundamental class of M and \({K}_{k}({p}_{1}(TM),\cdots \,,{p}_{k}(TM)) \in {H}^{n}(M; \mathbb{Z})\).
- 26.
Formula (39) is a direct consequence of Thom’s computation of \({}^{+}{\Omega }_{\bullet }{\otimes }_{\mathbb{Z}}\mathbb{Q}\cong\mathbb{Q}[{y}_{4k}\vert k \geq 1]\) , with \({y}_{4k} = [\mathbb{C}{P}^{2k}]\) . ( \({}^{+}{\Omega }_{j}{\otimes }_{\mathbb{Z}}\mathbb{Q} = 0\) for \(j\not =4k\) .) In Fact, one has \({p}_{j}(\mathbb{C}{P}^{n}) = {p}_{j}(T\mathbb{C}{P}^{n}) = \binom{n + 1}{j} \in {H}^{4j}(\mathbb{C}{P}^{n}) = \mathbb{Z}\), \(0 \leq j \leq \frac{n} {2}\) . For \(n = 2k\) the evaluation \(< {\mathcal{L}}_{k}, [\mathbb{C}{P}^{2k}] >= 1 \in \mathbb{Z}\) coincides with the signature of \(\mathbb{C}{P}^{2k}\) : \(\sigma (\mathbb{C}{P}^{2k}) = \sigma ({H}^{2k}(\mathbb{C}{P}^{2k}),\lambda ) = \sigma (\mathbb{Z}, 1) = 1 \in \mathbb{Z}\) . Therefore, the signature identifies a \(\mathbb{Q}\) -algebra homomorphism \({\Omega }_{\bullet }^{SO}{\otimes }_{\mathbb{Z}}\mathbb{Q} \rightarrow \mathbb{Q}\) . So the Hirzebruch signature theorem states that this last homomorhism induced by the signature, coincides with the one induced by the genus. In Table 5 are reported some Hirzebruch’s polynomials for \(\mathbb{C}{P}^{2k}\) . In Table 6 are reported also the Bernoulli numbers \({B}_{n}\) , with the Kronecker’s formula, and explicitly calculated for \(0 \leq n \leq 18\).
- 27.
In Table 8 are reported the n-stems for \(0 \leq n \leq 17\).
- 28.
\({V }_{n+k,k}\cong{O}(n + k)/O(n)\) is the Stiefel space of orthonormal k-frames in \({\mathbb{R}}^{n+k}\) , or equivalently of isometries \({\mathbb{R}}^{k} \rightarrow {\mathbb{R}}^{n+k}\). \({V }_{n+k,k}\) is ( n − 1)-connected with \({H}_{n}({V }_{n+k,k}) = \mathbb{Z}\), if \(n \equiv 0\:(mod \,2)\) or if \(k = 1\), and \({H}_{n}({V }_{n+k,k}) = {\mathbb{Z}}_{2}\), if \(n \equiv 1\:(mod \,2)\) and k > 1. One has \({G}_{n+k,k} = {V }_{n+k,k}/O(k)\) , where \({G}_{n+k,k}\) is the Grassmann space of k-dimensional subspaces of \({\mathbb{R}}^{n+k}\). Then the classifying space for n-planes is \(BO(n) ={ \lim }_{\overrightarrow{k}}{G}_{n+k,k}\) , and the corresponding stable classifying space is \(BO ={ \lim \atop \overrightarrow{n} BO(n)}\).
- 29.
For \(k = 1\) one has \({j}_{1} = 24\) ; \(\mathfrak{o}(V ) = {p}_{1}(V )/2 \in 24\mathbb{Z} \subset {\pi }_{3}(O) = \mathbb{Z}\).
- 30.
\(\:b{P}_{n+1}\) is a subgroup of \({\Theta }_{n}\) . If \({\Xi }_{1},{\Xi }_{2} \in b{P}_{n+1}\) , with bounding parallelizable manifolds \({W}_{1}\) and \({W}_{2}\) respectively, then \({\Xi }_{1}\sharp {\Xi }_{2}\) bounds the parallelizable manifold \({W}_{1}\sharp {W}_{2}\) , (commutative sum along the boundary).
- 31.
Note that if X is a compact space with boundary \(\partial X\) , the boundary of \(X \times I\), \(I \equiv [0, 1] \subset \mathbb{R}\) , is \(\partial (X \times I) = (X \times \{ 0\}) \bigcup \nolimits (\partial X \times I) \bigcup \nolimits (X \times \{ 1\}) \equiv {X}_{0} \bigcup \nolimits P\bigcup \nolimits {X}_{1}\) , with \({X}_{0} \equiv X \times \{ 0\}\), \({X}_{1} \equiv X \times \{ 1\}\), \(P \equiv \partial X \times I\) . One has \(\partial P = (\partial X \times \{ 0\}) \bigcup \nolimits (\partial X \times \{ 1\}) = \partial {X}_{0} \bigcup \nolimits \partial {X}_{1}\) . On the other hand, whether X is closed, then \(\partial (X \times I) = {X}_{0} \bigcup \nolimits {X}_{1}\) . Furthermore we shall denote by \({[N]}_{{E}_{k}}\) the equivalence class of the integral admissible bordism of \(N \subset {E}_{k}\) , even if N is not necessarily closed. So, if N is closed one has \({[N]}_{{E}_{k}} {\in }^{B}{\Omega }_{\bullet ,s}^{{E}_{k}}\) , and if N is not closed one has \({[N]}_{{E}_{k}} \in \bar{ {B}}_{\bullet }({E}_{k}; B)\).
- 32.
- 33.
From this theorem we get the conclusion that the Ricci flow equation for n-dimensional Riemannian manifolds, admits that starting from a n-dimensional sphere S n, we can dynamically arrive, into a finite time, to any n-dimensional homotopy sphere M. When this is realized with a smooth solution, i.e., solution with characteristic flow without singular points, then \({S}^{n}\cong{M}\) . The other homotopy spheres \({\Sigma }^{n}\) , that are homeomorphic to \({S}^{n}\) only, are reached by means of singular solutions. So the titles of this paper and its companion [42] are justified now ! Results of this paper agree with previous ones by Cerf [4], Freedman [7], Kervaire and Milnor [18, 20], Moise [21, 22] and Smale [45–47], and with the recent proofs of the Poincaré conjecture by Hamilton [11–15], Perelman [24, 25], and Prástaro [1, 40].
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Acknowledgements
I would like thank Editors for their kind invitation to contribute with a my paper to this book, dedicated to Stephen Smale in occasion of his 80th birthday. Work partially supported by MIUR Italian grants “PDE’s Geometry and Applications”.
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Dedicated to the 80th Anniversary of Professor Stephen Smale
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Prástaro, A. (2012). Exotic Heat PDE’s.II. In: Pardalos, P., Rassias, T. (eds) Essays in Mathematics and its Applications. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-28821-0_15
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