Fixed-Component Normal Euler Classes Become Invertible

Average a Riemannian metric over normalized Haar measure on $T$. At $p\in F$, the exponential map of this invariant metric is $T$-equivariant and is injective on a sufficiently small ball in $T_{p}M$. Thus, for $v$ in that ball, ${\exp }_p(v)$ is fixed by every $t\in T$ exactly when $v$ is fixed by every linear map $d_{p}t$; the fixed set near $p$ is the exponential image of $(T_{p}M{)}^T$. Therefore $F$ is locally a smooth submanifold and $T_{p}F=(T_{p}M{)}^T$. The fixed locus is closed in compact $M$, hence compact; its connected components are open in this locally connected manifold and compactness gives only finitely many.

The orthogonal complement of $(T_{p}M{)}^T$ in $T_{p}M$ is the normal fiber, so it contains no $T$-fixed vector. The complexification of an orthogonal representation of the abelian group $T$ is a commuting family of unitary maps and decomposes into simultaneous character eigenspaces; complex conjugation pairs the $\chi$ and $\bar{\chi }$ spaces. A real one-dimensional character would have image in $\{+1,-1\}$ and is trivial because $T$ is connected. Thus every nontrivial real summand is two-dimensional. Averaging the fiber action against a character gives a smooth idempotent projection on the complexified normal bundle. Its rank is locally constant and hence constant on connected $F_j$. At one point only finitely many characters occur, since the fiber has finite dimension; constant ranks therefore imply exactly those finitely many characters occur everywhere on $F_j$. The real conjugate-pair subbundles are consequently smooth.

If ${\nu }_j$ has nonzero weights, choose $X\in \mathfrak{t}$ outside the finite union of their kernels. For a character $\chi$, define the real linear weight ${\omega }_{\chi }(X)$ by $\chi (\exp (sX))=\exp (2\pi is{\omega }_{\chi }(X))$. On its real weight subbundle the infinitesimal action $A_X$ satisfies $A_X^2=-(2\pi {\omega }_{\chi }(X){)}^{2}I$, so $J_{\chi }=A_X\mathrm{/}(2\pi {\omega }_{\chi }(X))$ is invariant and satisfies $J_{\chi }^2=-I$. Their direct sum gives an invariant complex structure on ${\nu }_j$. Its complex orientation may agree or disagree with the chosen normal orientation, with comparison sign ${ϵ}_j$. The weight projections vary continuously, so their ranks, and therefore the character multiplicities, are constant on connected $F_j$. If there are no weights, then ${\nu }_j=0$ and the complex structure is vacuous.

By step 1.1, $F_j$ is compact smooth, so [F2] applies. In local normal charts and Milnor principal charts the associated Borel normal bundle is an oriented rank-$c_j$ vector bundle over $BT\times F_j$; the pulled-back principal numeration, combined with a finite normal-bundle trivialization over compact $F_j$, supplies a numeration. Thus the general Thom theorem supplies its normalized real Thom class independently of any orientation of $M$. The torus acts trivially on $F_j$, hence $ET{\times }_T{F}_j\cong BT\times F_j$ and [F3] gives $H_T^{\ast }(F_j;\mathbb{R})=H^{\ast }(F_j;\mathbb{R})\otimes \mathcal{R}$. Restrict $e_T({\nu }_j)$ to any point of $F_j$. The normal representation is the direct sum of the complex character lines in step 3.1; the Euler class of a direct sum is the product because the product of the normalized Thom classes restricts to the positive generator on each direct-sum fiber and hence is its normalized Thom class. By [F1] each line has Euler class $u_{{\chi }_{j,a}}$, and the orientation comparison gives $f_j={ϵ}_j{\prod }_a{u}_{{\chi }_{j,a}}$ as the degree-zero-in-$H^{\ast }(F_j)$ component. Every factor is nonzero and belongs to $S$, so $f_j$ is a unit in $S^{-1}\mathcal{R}$.

Write $e_T({\nu }_j)=f_j+n_j$, where $n_j$ has positive degree in $H^{\ast }(F_j;\mathbb{R})$. By [F4], the ideal $\mathcal{R}\otimes H^{>0}(F_j;\mathbb{R})$ has $N$th power zero for some $N\ge 1$, hence $n_j^N=0$; this remains true after localization. In the localized ring the inverse is $f_j^{-1}{\sum }_{q=0}^{N-1}(-f_j^{-1}{n}_j{)}^q$, since multiplying by $f_j+n_j=f_j(1+f_j^{-1}{n}_j)$ telescopes to $1-(-f_j^{-1}{n}_j{)}^N=1$. If $c_j=0$, $n_j=0$ and $f_j=1$, giving the empty-product case as well.