Regular Homotopy and Sphere Eversion

Use the supplied smooth tangent-bundle structures of [F12] and the supplied metric; [F10] supplies one under ${\mathrm{A}\mathrm{C}}_{\omega }$ if needed. In any local tangent frame the condition that a matrix have rank $m$ is open, so [F1] and [F4] give the smooth monomorphism bundle $\mathcal{M}$. Orthonormal tangent frames give the associated bundle with fibre $V_m({\mathbb{R}}^n)$ in [F6] and [F7], which is exactly its isometric-injection subbundle $E$. The map $B\mapsto (x\mapsto B_x)$ is a bijection onto $\mathrm{\Gamma }(\mathcal{M})$ by [F3]. It is a homeomorphism: section coordinates are precisely the matrix coefficients on each compact base chart piece. A compact piece of $TM$ has compact base projection and bounded vector coordinates, so controlling coefficient jets there controls every jet of the fibre-linear total-space map. Conversely evaluating that map along the finitely many local basis-vector sections over a compact base piece recovers each coefficient and all its derivatives. These two estimates show that the weak total-space and coefficient topologies agree with the section topology in [F2].

For an injective $B$ put $R=(B^{\ast }B{)}^{1\mathrm{/}2}$ and $Q=B{R}^{-1}$. By [F11] these operations are smooth, and $Q^{\ast }Q=I$. Conversely $(Q,R)$ with $Q$ isometric and $R$ positive gives $B=QR$, uniquely, so this is the asserted local polar diffeomorphism. Under a change of orthonormal tangent frame $U\in O(m)$, $B$ changes to $BU$, $R$ to $U^{\ast }RU$, and $Q$ to $QU$; hence it descends globally without a global frame. The path $B_t=B((1-t)I+t{R}^{-1})$ remains injective because its second factor is positive definite, has $B_0=B$ and $B_1=Q$, and fixes every isometric $B$. The operations and this path are continuous for weak $C^{\mathrm{\infty }}$ section topologies: on finitely many compact chart pieces the positive spectra stay bounded away from zero, and the smooth matrix operations and all chain-rule derivatives vary continuously there. Thus it is an equivariant strong deformation retraction of section spaces. Rank zero gives the unique empty matrix and the same formulas.

The canonical identification $T{\mathbb{R}}^n\cong {\mathbb{R}}^n\times {\mathbb{R}}^n$ sends a formal immersion $(f,F)$ to $(f,B)$ with $B_x:T_{x}M\to {\mathbb{R}}^n$ the same fibre map. This is a bijection onto $C^{\mathrm{\infty }}(M,{\mathbb{R}}^n)\times \mathrm{\Gamma }(\mathcal{M})$ and a homeomorphism for the identical local matrix/derivative neighbourhoods, as in step 1.1. The contraction $(f,B)\mapsto ((1-t)f,B)$ deforms the first factor to zero; combining it with step 2.1 on the second factor gives the homotopy equivalence to $\mathrm{\Gamma }(E)$. Its homotopy inverse sends an isometric section $Q$ to $(0,Q)$. For an immersion the smooth formal pair is $(f,df)$ by [F2] and [F12], so the normalized section is the polar part of $df$, as stated. No compactness of $M$ is needed because every basic neighbourhood controls only finitely many compact chart pieces.

If $TM$ has a smooth global frame by [F9], orthonormalizing it in the supplied metric gives a global orthonormal frame and identifies $E$ with $M\times V_m({\mathbb{R}}^n)$. For $M=S^m$ with its standard metric, [F8] gives $T_x{S}^m=x^{\perp }$; its projection $I-x{x}^{\ast }$ varies smoothly, so the Gauss map to the Grassmannian is smooth in its graph charts. The tautological-plane pullback of [F6] therefore is $T{S}^m$, and its isometric-injection bundle pulls back to $E$. On $S^1$ the standard unit angular field is a global orthonormal frame, giving $E\cong S^1\times V_1({\mathbb{R}}^2)=S^1\times S^1$. When $m=0$ the Stiefel and monomorphism fibres are points, and when $n=m$ the normalized fibre is $O(m)$ while the monomorphism fibre retains its positive-definite polar factor. These are included in the same construction.

Evaluation on smooth sections is locally trivial. The group $O(n)$ acts on every section by target multiplication. For any frame $a\in F$, complete it to an orthonormal basis; Gram-Schmidt applied to a nearby frame $b$ followed by the remaining fixed basis vectors gives a smooth orthogonal matrix $R_a(b)$ with $R_a(a)=I$ and $R_a(b)a=b$. Thus $s\mapsto (\mathrm{ev}(s),R_a(\mathrm{ev}(s){)}^{-1}s)$ identifies ${\mathrm{ev}}^{-1}(U_a)$ with $U_a\times {\mathrm{\Gamma }}_a$, including its inverse $(b,s)\mapsto R_a(b)s$. These maps are continuous for the weak $C^{\mathrm{\infty }}$ topology because target multiplication multiplies each derivative by the same finite matrix. The same argument works for continuous sections. If either section space is nonempty the transitive $O(n)$ action makes evaluation surjective; otherwise its fibres are all empty. If a section space is empty, its evaluation has the homotopy lifting property vacuously, so it is a Hurewicz fibration with empty fibres without invoking a surjective bundle convention. For a nonempty section space, compactness of $F$ gives finitely many such neighbourhoods, with a subordinate continuous partition obtained from finitely many ambient bump functions. Order these finitely many charts once, and well-order their finite words first by length and then lexicographically. Substituting this explicit well-order in the well-order construction of the proof of [F3] supplies its sole use of AC; all remaining constructions there apply to the supplied finite numeration without choice. Thus both evaluations are Hurewicz fibrations; their fibres at $s_0(x_0)$ are the prescribed fixed-value section spaces. This argument applies also when $m=n$ and $F$ is disconnected.

Polar normalization is well defined for any fibrewise injection $B$: in the global matrix presentation put $T=B^{\ast }B+x{x}^{\ast }$, which is positive definite, and normalize by $B{T}^{-1\mathrm{/}2}$. By [F11] this operation is smooth on smooth sections and continuous on continuous sections. The fibrewise path $B((1-t)I+t{T}^{-1\mathrm{/}2})$ remains injective and fixes orthonormal sections, giving the deformation retraction to the Stiefel model used in [F1]. We now compare its smooth and continuous fixed-value section spaces constructively. Represent an orthonormal section by matrices $A(x):{\mathbb{R}}^{m+1}\to {\mathbb{R}}^n$ with $A(x)=A(x)P_x$ and $A(x{)}^{\ast }A(x)=P_x$, where $P_x=I-x{x}^{\ast }$, and put $a=A(x_0)$. Extend $A$ radially to an annulus, multiply by a fixed radial cutoff equal to one near the unit sphere, and convolve with a fixed smooth compactly supported Euclidean kernel of radius $\epsilon <1\mathrm{/}8$. Restriction to the sphere and right multiplication by $P_x$ gives a smooth bundle map $B_{\epsilon }(A)$, converging uniformly to $A$. Pin its value by adding $\chi (x)(a-B_{\epsilon }(A)(x_0))P_x$, for a fixed smooth bump $\chi$ with $\chi (x_0)=1$; denote the result by $C_{\epsilon }(A)$. It equals $a$ at $x_0$, converges uniformly to $A$, depends continuously on $(A,\epsilon )$ into the smooth topology for $\epsilon >0$, and these operators have a common uniform norm bound.

Choose an orthonormal basis $e_1,\dots ,e_m$ of $x_0^{\perp }$. For the closed unit disk $D^m$, write $y=ru$ and define $q(y)=\sin (\pi r){\sum }_i{u}_i{e}_i-\cos (\pi r)x_0$, with $q(0)=-x_0$. The functions $\sin (\pi r)\mathrm{/}r$ and $\cos (\pi r)$ are smooth at $r=0$ by their power series, so $q$ is smooth on the closed disk. Its boundary maps to $x_0$, its interior maps homeomorphically to $S^m\setminus \{x_0\}$, and the induced map $D^m\mathrm{/}\mathrm{\partial }{D}^m\to S^m$ is a homeomorphism, since it is a continuous bijection from a compact space to a Hausdorff space. Put $P(t,y)=I-q(ty)q(ty{)}^{\ast }$ and $K=[{\mathrm{\partial }}_{t}P,P]$. Solve ${\mathrm{\partial }}_{t}O=KO$, $O(0,y)=I$. By [F10] the solution exists throughout $[0,1]$ and is jointly smooth in $(t,y)$: the local smooth solution maps patch along the compact time interval by uniqueness. Since $K^{\ast }=-K$, $O^{\ast }O=I$; differentiating $P^2=P$ gives ${\mathrm{\partial }}_{t}P=[K,P]$, and therefore $PO-OP(0)$ solves the linear equation with zero initial value and is zero. Consequently $O(1,y)e_1,\dots ,O(1,y)e_m$ is a smooth orthonormal frame of $q^{\ast }T{S}^m$ on the whole closed disk, including its boundary. By [F1] it trivialises $q^{\ast }E$.

There is a continuous positive smoothing radius on the entire metric space ${\mathrm{\Gamma }}_{\ast }^0$. For every integer $k\ge 1$, put $f_k(A)={\sup }_{0<\epsilon \le 1\mathrm{/}(8k)}\mathrm{\parallel }{C}_{\epsilon }(A)-A{\mathrm{\parallel }}_{\mathrm{\infty }}$ is continuous, since the uniformly bounded operators make these functions uniformly Lipschitz, and $f_k(A)\to 0$. Put $w_k(A)=2^{-k}\max (0,1-8f_k(A))$ and $\epsilon (A)=({\sum }_{k=1}^{\mathrm{\infty }}{w}_k(A)\mathrm{/}(8k))\mathrm{/}{\sum }_{k=1}^{\mathrm{\infty }}{w}_k(A)$. Both series converge uniformly, the denominator is positive, and if $k_0$ is the first positive weight then $\epsilon (A)\le 1\mathrm{/}(8k_0)$ and $\mathrm{\parallel }{C}_{\epsilon (A)}(A)-A{\mathrm{\parallel }}_{\mathrm{\infty }}<1\mathrm{/}8$. Every convex combination $L_t=(1-t)A+t{C}_{\epsilon (A)}(A)$ is therefore injective on the tangent fibres. Normalize it by $L_t(L_t^{\ast }{L}_t{)}^{-1\mathrm{/}2}$ on those fibres; the inverse square root is given by the convergent binomial series near the identity, since $\mathrm{\parallel }{L}_t^{\ast }{L}_t-I\mathrm{\parallel }<17\mathrm{/}64$. This normalization is continuous, smooth in $x$ when $A$ is smooth, and fixes $a$ at $x_0$. It gives a homotopy from the identity to a continuous smoothing map $S:{\mathrm{\Gamma }}_{\ast }^0\to {\mathrm{\Gamma }}_{\ast }$; restricted to smooth sections it is continuous in the weak $C^{\mathrm{\infty }}$ topology as well. Thus inclusion and $S$ are homotopy inverses. The same construction without the pinning correction compares the unrestricted section spaces. No family of charts or approximation choices is selected: the kernel, cutoffs and series are fixed.

In this frame the prescribed fibre value $s_0(x_0)$ determines a boundary map $g:\mathrm{\partial }{D}^m\to F$, generally nonconstant. A continuous fixed-value section pulls back to a map $\gamma :D^m\to F$ with $\gamma {\mathrm{\mid }}_{\mathrm{\partial }{D}^m}=g$. Conversely such a map gives a section of $q^{\ast }E$ whose images in $E$ all equal $s_0(x_0)$ on the collapsed boundary, so it descends to a unique continuous section of $E$. This bijection is a homeomorphism ${\mathrm{\Gamma }}_{\ast }^0\cong {\mathcal{M}}_g:=\{\gamma :\gamma {\mathrm{\mid }}_{\mathrm{\partial }{D}^m}=g\}$. For compact-open continuity, pullback is continuous, and if $K\subseteq S^m$ is compact then $q^{-1}(K)$ is compact, so the inverse image of the section neighbourhood $s(K)\subseteq U$ is the corresponding pullback neighbourhood over $q^{-1}(K)$; composing with the bundle frame and its inverse is continuous by the exponential correspondence. The same quotient reasoning applies to parametrized homotopies.

If $\mathrm{\Gamma }\mathrm{\ne }\mathrm{\varnothing }$ and $n\ge m+1$, [F2] makes $F$ path connected. A path from any evaluated value to $s_0(x_0)$ lifts under the smooth evaluation fibration of step 1.1, producing a section in ${\mathrm{\Gamma }}_{\ast }$. The same lifting moves a representative of every component of $\mathrm{\Gamma }$ into ${\mathrm{\Gamma }}_{\ast }$, so the map of component sets ${\pi }_0({\mathrm{\Gamma }}_{\ast })\to {\pi }_0(\mathrm{\Gamma })$ is surjective.

If ${\mathrm{\Gamma }}_{\ast }\mathrm{\ne }\mathrm{\varnothing }$, choose its disk model ${\gamma }_0\in {\mathcal{M}}_g$ and one $y_0\in \mathrm{\partial }{D}^m$, and put $e_0=g(y_0)$. The formula $g_t(u)={\gamma }_0((1-t)u+t{y}_0)$ contracts $g$ to the constant map $e_0$ while fixing $y_0$. Restriction $C(D^m,F)\to C(\mathrm{\partial }{D}^m,F)$ is a Hurewicz fibration: $\mathrm{\partial }{D}^m\subset D^m$ is a closed cofibration, as its radial collar gives the usual homotopy extension retraction of $D^m\times I$ onto $D^m\times \{0\}\cup \mathrm{\partial }{D}^m\times I$; for every test space, compose that retraction with the prescribed disk map and boundary homotopy and transpose by [F6]. Lifting the path $g_t$, with all points of its starting fibre as parameters, gives transport ${\mathcal{M}}_g\to {\mathcal{M}}_{e_0}$. Transport along the reversed path is a homotopy inverse: the two concatenations retrace the same path and contract to constant paths by shortening their excursion; relative homotopy lifting [F4] lifts these contractions to fibre homotopies, with prescribed initial maps. Thus ${\mathcal{M}}_g\simeq {\mathcal{M}}_{e_0}$.

The constant-boundary maps descend to the based mapping space $C_{\ast }((D^m\mathrm{/}\mathrm{\partial }{D}^m,\ast ),(F,e_0))$, again homeomorphically for compact-open topologies. Combining steps 2.1, 2.2 and 3.1 gives ${\mathrm{\Gamma }}_{\ast }\simeq C_{\ast }(S^m,F)$ and hence ${\pi }_0({\mathrm{\Gamma }}_{\ast })\cong {\pi }_m(F)$ by [F8]. A homotopy of sections fixed at $x_0$ gives a path in ${\mathcal{M}}_g$ and, under transport, a based homotopy in ${\mathcal{M}}_{e_0}$. Conversely a based homotopy can be transported back, and the fibre homotopies between the composites and the identities provide a fixed-value section homotopy; the smoothing comparison makes it a homotopy of smooth sections when the endpoints are smooth. These are both directions of the difference-class criterion. The identification depends on the disk frame, reference section and contraction; it is not canonical.

The homotopy exact sequence of evaluation, based at a chosen section in ${\mathrm{\Gamma }}_{\ast }$, is the displayed sequence of [F7]. When $F$ is simply connected, two fibre components that become connected in $\mathrm{\Gamma }$ differ by transport around a loop in $F$; a nullhomotopy of that loop and relative lifting show they were already connected in the fibre. Thus the component map is injective, and step 2.3 makes it surjective, giving ${\pi }_0(\mathrm{\Gamma })\cong {\pi }_m(F)$ by step 4.1. If instead ${\pi }_m(F)=0$, $n\ge m+1$ and $\mathrm{\Gamma }\mathrm{\ne }\mathrm{\varnothing }$, steps 4.1 and 2.3 show that ${\pi }_0(\mathrm{\Gamma })$ is a quotient of a singleton and hence itself a singleton. The connected-fibre hypothesis cannot be omitted: for $m=n=1$ the two orientations of an everywhere nonzero circle field give two section components even though ${\pi }_1(O(1))=0$.

For $1\le m\le n$ the first-vector map $\pi :V_m({\mathbb{R}}^n)\to V_1({\mathbb{R}}^n)=S^{n-1}$, $\pi (v_1,\dots ,v_m)=v_1$, is a locally trivial fibre bundle with fibre $V_{m-1}({\mathbb{R}}^{n-1})$. The open sets $U_{+}=\{u\in S^{n-1}:⟨u,e_n⟩>-\frac{2}{3}\}$ and $U_{-}=\{u\in S^{n-1}:⟨u,e_n⟩<\frac{2}{3}\}$ cover $S^{n-1}$, and on $U_{\pm }$ the denominators $\mathrm{\parallel }{e}_n\pm u{\mathrm{\parallel }}^2=2(1\pm ⟨u,e_n⟩)$ are bounded below by $\frac{2}{3}$, so the Householder reflections $H_u^{\pm }v=v-\frac{2⟨v,e_n\pm u⟩}{\mathrm{\parallel }{e}_n\pm u{\mathrm{\parallel }}^2}(e_n\pm u)$ depend continuously on $u$ and are orthogonal; they satisfy $H_u^{+}(e_n)=-u$ and $H_u^{-}(e_n)=u$, hence map the hyperplane $e_n^{\perp }\cong {\mathbb{R}}^{n-1}$ isometrically onto $u^{\perp }$. Therefore ${\mathrm{\Theta }}_{\pm }(u,f_2,\dots ,f_m)=(u,H_u^{\pm }{f}_2,\dots ,H_u^{\pm }{f}_m)$ are homeomorphisms $U_{\pm }\times V_{m-1}({\mathbb{R}}^{n-1})\to {\pi }^{-1}(U_{\pm })$ over $U_{\pm }$, with inverses $(v_1,\dots ,v_m)\mapsto (v_1,(H_{v_1}^{\pm }{)}^{-1}{v}_2,\dots ,(H_{v_1}^{\pm }{)}^{-1}{v}_m)$.

Base case $m=1$: $V_1({\mathbb{R}}^n)=S^{n-1}$ by [F1], which is nonempty and path connected for $n\ge 2=m+1$ by [F8]; and for $n\ge 3=m+2$ every based loop $S^1\to S^{n-1}$ is nullhomotopic by [F7] with $k=1<r=n-1$, so ${\pi }_1(V_1({\mathbb{R}}^n),e_1)=0$.

The explicit functions ${\sigma }_{\pm }(u)=\max (0,\pm ⟨u,e_n⟩+\frac{1}{3})$ satisfy ${\sigma }_{+}+{\sigma }_{-}\ge \frac{1}{3}$ on $S^{n-1}$ and have closed supports $\mathrm{supp}{\sigma }_{\pm }=\{\pm ⟨u,e_n⟩\ge -\frac{1}{3}\}\subseteq U_{\pm }$, so ${\rho }_{\pm }={\sigma }_{\pm }\mathrm{/}({\sigma }_{+}+{\sigma }_{-})$ is a partition of unity subordinate to the two-element cover of step 1.1. With these charts the bundle of step 1.1 is numerable, so by [F3] it is a Hurewicz fibration and in particular a Serre fibration; the only choice-like step of the cited proof is the well-order of finite words over the chart alphabet, and for the two-element alphabet the words are explicitly enumerated by their binary digits, so this application uses no choice. The identification of the fibre over $u$ with $V_{m-1}({\mathbb{R}}^{n-1})$ is the homeomorphism $H_u^{\pm }$ restricted to $e_n^{\perp }$, so ${\pi }_0$ and ${\pi }_1$ of the fibre are those of $V_{m-1}({\mathbb{R}}^{n-1})$ by [F6].

Step (a) for $m\ge 2$: assume $V_{m-1}({\mathbb{R}}^{n-1})$ is path connected, which by the induction hypothesis holds because $n\ge m+1$ gives $n-1\ge (m-1)+1$. The bundle of steps 1.1 and 2.1 is a based Serre fibration with path-connected base $S^{n-1}$ and fibre $V_{m-1}({\mathbb{R}}^{n-1})$ over $e_1$, and its exact sequence in degree zero reads ${\pi }_0(F)\to {\pi }_0(V_m({\mathbb{R}}^n),e_0)\to {\pi }_0(S^{n-1},e_1)$, a sequence of pointed sets whose two outer terms are singletons; exactness makes the middle term a singleton as well, that is, $V_m({\mathbb{R}}^n)$ is path connected.

Step (b) for $m\ge 2$: assume ${\pi }_1(V_{m-1}({\mathbb{R}}^{n-1}),\cdot )=0$, which by the induction hypothesis holds because $n\ge m+2$ gives $n-1\ge (m-1)+2$. The exact sequence of the same based Serre fibration reads ${\pi }_1(F)\to {\pi }_1(V_m({\mathbb{R}}^n),e_0)\to {\pi }_1(S^{n-1},e_1)$, and the target is trivial by [F7] with $k=1<r=n-1$ because $n\ge 3$; exactness makes the first map surjective, so the triviality of ${\pi }_1(F)$ forces ${\pi }_1(V_m({\mathbb{R}}^n),e_0)=0$. Since $V_m({\mathbb{R}}^n)$ is path connected by step 3.1, triviality at one basepoint gives triviality at every basepoint.

For the final identifications: the map $\mathrm{\Phi }:V_m({\mathbb{R}}^{m+1})\to \mathrm{S}\mathrm{O}(m+1)$ that sends $(v_1,\dots ,v_m)$ to the matrix whose first $m$ columns are $v_1,\dots ,v_m$ and whose last column is the unique unit vector $w$ orthogonal to all $v_i$ with $\det (v_1,\dots ,v_m,w)=+1$ is a bijection onto $\mathrm{S}\mathrm{O}(m+1)$: the orthogonal complement of $\mathrm{span}\{v_1,\dots ,v_m\}$ is a line containing exactly two unit vectors, and exactly one of them gives determinant $+1$; the coordinates of $w$ are the $m\times m$ minors of the matrix $(v_1\mid \cdots \mid v_m)$, namely the coefficients of the Hodge dual, which are polynomial in the entries of the $v_i$, and the inverse is the continuous projection to the first $m$ columns, so $\mathrm{\Phi }$ is a homeomorphism. Hence by [F6] the homotopy invariants of $V_m({\mathbb{R}}^{m+1})$ and $\mathrm{S}\mathrm{O}(m+1)$ agree, and $V_m({\mathbb{R}}^{m+1})$ is connected by step 3.1 applied with $n=m+1$. The case $m=1$ gives $V_1({\mathbb{R}}^2)=S^1\cong \mathrm{S}\mathrm{O}(2)$, consistent with $V_1({\mathbb{R}}^k)=S^{k-1}$. Steps 1.2, 3.1 and 4.1 cover $m=1$ and all $m\ge 2$ in the stated ranges, and together with the definition of simple connectivity [F10] they give that $V_m({\mathbb{R}}^n)$ is simply connected whenever $n\ge m+2$.

The map $\varphi :V_2({\mathbb{R}}^3)\to \mathrm{S}\mathrm{O}(3)$, $\varphi (u,v)=(u\mid v\mid u\times v)$, is a homeomorphism. Its image lies in $\mathrm{S}\mathrm{O}(3)$: for orthonormal $u,v$ the vector $u\times v$ is orthogonal to $u$ and $v$ by [F6] and is unit, since expanding its coordinates gives $\parallel u\times v{\parallel }^2=\parallel u{\parallel }^2\parallel v{\parallel }^2-⟨u,v{⟩}^2=1$, so the three columns are orthonormal, and $\det (u\mid v\mid u\times v)=⟨u\times v,u\times v⟩=1$ by the triple product identity [F6]. It is injective because the first two columns determine the argument. It is surjective: for $R\in \mathrm{S}\mathrm{O}(3)$ with columns $c_1,c_2,c_3$, the vector $c_1\times c_2$ is a unit vector orthogonal to $c_1$ and $c_2$ by [F6], hence equals $\pm c_3$, and the sign is $+$ because $⟨c_1\times c_2,c_3⟩=\det (c_1\mid c_2\mid c_3)=\det R=1$; thus $R=\varphi (c_1,c_2)$ with $(c_1,c_2)\in V_2({\mathbb{R}}^3)$. Both $\varphi$ and the projection $R\mapsto (c_1,c_2)$ to the first two columns are continuous, so $\varphi$ is a homeomorphism.