Equivariant Thom-Gysin Identities for Fixed Submanifolds

Average any Riemannian metric on $M$ over normalized Haar measure on $T$ to obtain a $T$-invariant metric. The normal exponential map is $T$-equivariant; its derivative is invertible along the zero section, so the inverse function theorem and compactness of $N$ give a $T$-invariant tubular neighborhood identified with $\nu$. Since $T$ fixes $N$ pointwise, $ET{\times }_{T}N\cong BT\times N$, and the Borel normal bundle $ET{\times }_T\nu$ is an oriented rank-$c$ vector bundle over the space $BT\times N$ of CW type.

The Borel principal bundle $ET\times N\to BT\times N$ is the pullback of the selected product Milnor bundle. If $r=0$, this is the trivial principal bundle over a point; otherwise, for each circle factor, [F3] supplies a numerable trivializing cover and subordinate partition. Products of these finitely many covers and partition functions are locally finite and subordinate to product principal-$T$ charts. Pulling the numeration back to $BT\times N$ shows that the associated Borel normal bundle $ET{\times }_T\nu$ is numerable. It is oriented by the given $T$-invariant orientation. By [F2] its base is paracompact Hausdorff of CW type, so [F4] gives the normalized Thom class ${\tau }_{\nu }$ and the cup-product isomorphism in every degree.

Use the tubular-neighborhood identification and excision to transfer ${\tau }_{\nu }$ to $H_T^c(M,M\setminus N;\mathbb{R})$. For each $\beta \in H_T^{\ast }(N;\mathbb{R})$, define $i_{\ast }^T(\beta )$ as the image of $q^{\ast }\beta ⌣{\tau }_{\nu }$, with the pulled-back $\beta$ on the left and the Thom class on the right, where $q$ is the tube projection under the relative-to-absolute map $H_T^{\ast }(M,M\setminus N;\mathbb{R})\to H_T^{\ast }(M;\mathbb{R})$. This canonical composite is the Gysin map and is defined for every $\beta$ by the Thom isomorphism.

Pull the relative class in step 3.1 back along the zero section. By the definition of $e_T(\nu )$ in [F1], the pulled-back Thom class is $e_T(\nu )$, and pullback respects cup products. This gives $i^{\ast }{i}_{\ast }^T(\beta )=\beta ⌣e_T(\nu )$. If $c$ is even, graded commutativity equals the stated $e_T(\nu )⌣\beta$. If $c$ is odd, fiberwise multiplication by $-1$ fixes the zero section and reverses the normal orientation. Its pullback of the normalized Thom class is $-{\tau }_{\nu }$ by [F6], hence $e_T(\nu )=-e_T(\nu )$, and $e_T(\nu )=0$ over $\mathbb{R}$. Thus both orders again agree. This retains the case of arbitrary submanifolds of $M^T$, whose codimension need not be even.

The Gysin class in step 3.1 is represented in a tubular neighborhood of $N$. Choose such neighborhoods disjoint for disjoint compact fixed components $N,N^{\mathrm{\prime }}$. The representative restricts to zero on $N^{\mathrm{\prime }}$, so $i_{N^{\mathrm{\prime }}}^{\ast }{i}_{N\ast }^T(\beta )=0$.

Step 2.1 supplies the Borel Thom class, and step 3.1 uses precisely the right-Thom product of [F5]. Its trace theorem therefore applies: at every finite stage the relative Thom class has an interior-supported real representative integrating to $1$ along each oriented normal disk; normal integration followed by integration over $N$ agrees with integration over $M$ in the tangent-then-normal orientation. The stable Borel maps detect this equality at sufficiently large finite stages. Hence ${\int }_M{i}_{\ast }^T(\beta )={\int }_N\beta$.

The self-intersection identity in step 4.1, the disjoint-support identity in step 4.2, and the trace identity in step 4.3 establish every stated property. All maps are $\mathcal{R}$-linear: the Thom product commutes with coefficients pulled back from $BT$, which have even degree, and [F5] supplies linearity of integration.