Real and symplectic clutching groups in low degrees

The last-column maps $SO(r)\to SO(r+1)\to S^r$ and $Sp(r-1)\to Sp(r)\to S^{4r-1}$ are locally trivial bundles. To see local sections near a unit vector $v_0$, first choose an orthonormal basis with last vector $v_0$. For $v$ near $v_0$, project the other basis vectors onto $v^{\perp }$ and Gram–Schmidt them; their Gram determinants stay nonzero nearby. Append $v$ as last column. In the real case keep the orientation positive by the sign chosen at $v_0$; it remains positive by continuity. In the quaternionic case use the right-Hermitian inner product and divide by positive real lengths; the resulting frame change is right-linear. Translate these local sections to all points. Their stabilizers are exactly the indicated smaller groups. A finite partition on each compact sphere makes the bundles numerable, so [F2] applies. Starting with $SO(2)$, the circle of plane rotations, and $Sp(1)=S^3$, the component part of the exact sequence and connectedness of the spheres inductively prove connectedness of all $SO(r)$ for $r\ge 2$ and all $Sp(r)$ for $r\ge 1$. The groups $SO(1)$ and $Sp(0)$ are points.

Cover $S^n$ by two disks. Disk bundles are trivial by [F1]; hence a rank-$r$ real bundle has transition map $S^{n-1}\to G{L}_r(\mathbb{R})$, deformable by polar normalization into $O(r)$. The same proof over $\mathbb{H}$ gives transition maps into $Sp(r)$: positive square roots exist by the convergent real matrix power-series construction, and all matrices commute with right scalar multiplication. An isomorphism between clutched bundles consists of two disk-defined gauge matrices, exactly as in [F1]; a homotopy of transitions gives an isomorphism by homotopy invariance. Thus the clutching and gauge classification holds over $\mathbb{H}$ as well.

The covering in [F4] has simply connected total space by [F3]. By [F5], lifting a loop in $SO(3)$ from $1\in S^3$ gives endpoint $1$ or $-1$. This is unchanged by based homotopy, is multiplicative under loop concatenation because the cover is a group homomorphism, and has kernel the loops whose lifts close. Such a closed lift contracts in $S^3$ by [F3], so the kernel is zero. A path from $1$ to $-1$ in $S^3$ projects to a loop, showing surjectivity. Hence ${\pi }_1(SO(3))=\mathbb{Z}\mathrm{/}2$. The second group is zero by [F4].

For $n\ge 2$, choose a point on the connected equator and multiply a transition map by the inverse of its value there. This normalizes it to a based map into the identity component. Disk gauges restrict to maps homotopic to constants; after this normalization their only residual action is conjugation by a constant group element. This action is trivial for $Sp(r)$ once its path connectedness is proved in step 1.1. For the real case conjugation by a determinant-positive constant is trivial in homotopy by path connectedness of $SO(r)$; conjugation by a reflection $R$ becomes trivial after adding one coordinate, because $\mathrm{diag}(R,-1)\in SO(r+1)$ is path connected to identity and conjugates $\mathrm{diag}(g,1)$ in exactly the same way as $R$ conjugates $g$. Therefore the real orientation-orbit ambiguity disappears on stabilization. For $n=1$, a transition on $S^0$ is a pair of matrices; change gauges to make the first identity. The component of the second is the complete invariant, since each hemisphere is an interval. This yields the component-group formulation.

The determinant map $O(r)\to \{\pm 1\}$ is onto for $r\ge 1$, with fibers $SO(r)$ and a reflection translate of it. Step 1.1 therefore proves ${\pi }_0(O(r))=\mathbb{Z}\mathrm{/}2$. Stabilization preserves determinant, and block sum multiplies determinants. Thus the first colimit is $\mathbb{Z}\mathrm{/}2$. The symplectic component colimit is zero by the same connectedness result.

For $r\ge 3$, the exact sequence of $SO(r)\to SO(r+1)\to S^r$ and ${\pi }_2(S^r)={\pi }_1(S^r)=0$ give an isomorphism on ${\pi }_1$. Therefore the stable first group is the $\mathbb{Z}\mathrm{/}2$ of step 1.3. For $r=3$, the segment ${\pi }_2(SO(3))\to {\pi }_2(SO(4))\to {\pi }_2(S^3)=0$ shows ${\pi }_2(SO(4))=0$, since its source is zero; no false assertion that ${\pi }_3(S^3)=0$ is used. For $r\ge 4$ both ${\pi }_3(S^r)$ and ${\pi }_2(S^r)$ vanish by [F3], so stabilization on ${\pi }_2$ is an isomorphism and its stable value is zero. Based positive homotopy groups of $O(r)$ equal those of its identity component.

Direct sum of bundles corresponds to block sum of transitions. Stable block sum agrees with homotopy-group addition: in any topological group pointwise multiplication agrees with concatenation on based homotopy classes, by cutting a square into horizontal and vertical paths; after moving one block into the other by a path of real block rotations, the two commuting block matrices multiply to the block sum. For ${\pi }_0$ the same block movement proves the component assertion. Every stable class has an additive inverse because $g\oplus g^{-1}$ contracts by $\mathrm{diag}(g,I)R_t\mathrm{diag}(I,g^{-1})R_t^{-1}$ from $t=0$ to $\pi \mathrm{/}2$, with $R_t$ the two-block rotation. Group completion therefore introduces no new stable classes. Conversely, equality of reduced differences is stable isomorphism after a common trivial summand by [F1], and the clutching criterion in step 1.2 and step 2.1 identifies exactly the same stable transitions. This proves both stated colimit identifications; it does not silently identify finite-rank unoriented bundles with a based homotopy group.

For $Sp(1)=S^3$, [F3] gives ${\pi }_1={\pi }_2=0$, and step 1.1 gives ${\pi }_0=0$. For $r\ge 2$ the sphere in $Sp(r-1)\to Sp(r)\to S^{4r-1}$ has dimension at least seven. Its homotopy groups in degrees one, two and three vanish by [F3]. The exact sequence in [F2] therefore makes stabilization an isomorphism on ${\pi }_1$ and ${\pi }_2$; the component groups remain zero. Finally bundles over a point are classified by their nonnegative integer real or quaternionic dimension, whose group completion is $\mathbb{Z}$. The definition $K{O}^{-j}(\ast )={\widetilde{KO}}^0(S^j)$, its quaternionic analogue, and the colimit identifications of step 3.1 now yield every displayed coefficient. No periodicity assertion enters any step.