Based gauge-group homotopy via hemisphere clutching

By [F1], the gauge group under consideration consists of smooth sections with value $e$ at $p_m$, and [F3] identifies this subgroup with the evaluation fiber and its subspace topology. First calculate its continuous-section model, which is related by [F2]. Choose a finite trivializing cover $(U_i)$ of compact $S^r$. In each equivariant chart $P{\mathrm{\mid }}_{U_i}\cong U_i\times SU(N)$, set ${\omega }_i(v,W)=g^{-1}W$ for a tangent vector $(v,W)$ at $(x,g)$. Direct calculation gives ${\omega }_i(X_P)=X$ and $R_h^{\ast }{\omega }_i={\mathrm{Ad}}_{h^{-1}}{\omega }_i$. By [F8] choose a subordinate partition $({\rho }_i)$ and extend $({\rho }_i\circ \pi ){\omega }_i$ by zero outside $P{\mathrm{\mid }}_{U_i}$. Their finite sum $A_0={\sum }_i({\rho }_i\circ \pi ){\omega }_i$ is smooth, right equivariant, and reproduces vertical vectors because ${\sum }_i{\rho }_i=1$; [F9] makes it a smooth principal connection. Choose hemisphere frames with the upper frame equal to $p_m$ and transport them radially using $A_0$. The ODE statement [F6] gives smooth sections up to the equator, including the disk centers, and hence the smooth transition $\psi$ in [A1]. In that model a based section is a pair of maps $f_{+}:D_{+}^r\to SU(N)$ and $f_{-}:D_{-}^r\to SU(N)$ satisfying [A1] and $f_{+}(m)=e$. Let $J={\mathrm{Map}}_{\ast }((D_{+}^r,m),(SU(N),e))$. For $f\in J$ define a map on the lower disk by ${\sigma }_f(\rho q)=\psi (q{)}^{-1}f(\rho q)\psi (q),0<\rho \le 1,{\sigma }_f(0)=e,$ where $q\in S^{r-1}$ is the radial direction and the upper-disk center $m$ is identified with $0$. At $\rho =1$ this has boundary value ${\psi }^{-1}f(q)\psi$, and as $\rho \to 0$ it tends uniformly to $e$ because $f(0)=e$ and $S^{r-1}$ is compact; hence it is continuous on $D_{-}^r$.

For each compatible pair $(f_{+},f_{-})$, the product $v={\sigma }_{f_{+}}^{-1}{f}_{-}$ is the identity on $\mathrm{\partial }{D}_{-}^r$, so it descends continuously to a based map $D_{-}^r\mathrm{/}\mathrm{\partial }{D}_{-}^r\cong S^r\to SU(N)$. Conversely, given $f\in J$ and such a based map $v$, the pair $(f,{\sigma }_{f}v)$ satisfies the clutching condition. On maps from a compact disk to the metric space $SU(N)$, the compact-open topology is the uniform topology. Unitary conjugation preserves the Hilbert--Schmidt metric, so $\mathrm{\parallel }{\sigma }_f-{\sigma }_g{\mathrm{\parallel }}_{\mathrm{\infty }}\le \mathrm{\parallel }f-g{\mathrm{\parallel }}_{\mathrm{\infty }}$ for $f,g\in J$; at the disk center both extensions equal $e$. Thus $f\mapsto {\sigma }_f$ is continuous. Restrictions, multiplication, inversion, and finite pasting are continuous in the same topologies, so both assignments are continuous and inverse. Hence the continuous based section space is homeomorphic to $J\times {\mathrm{Map}}_{\ast }(S^r,SU(N))$.

The space $J$ is contractible by $f_t(x)=f((1-t)x)$, which keeps the center value $e$. Hence projection of the product in step 2.1 onto ${\mathrm{Map}}_{\ast }(S^r,SU(N))$ induces a bijection on components and isomorphisms on all homotopy groups. Equivalently, the restriction to the upper disk is a trivial fibration with contractible base $J$; [F5] gives the same conclusion from its exact sequence.

By [F4], the homotopy groups of ${\mathrm{Map}}_{\ast }(S^r,SU(N))$ are ${\pi }_{j+r}(SU(N))$ for $j\ge 1$, and its pointed components are based homotopy classes $[S^r,SU(N){]}_{\ast }={\pi }_r(SU(N))$. Finally [F2] transfers these continuous-section identifications to the smooth based gauge group.