Middle-dimensional surgery has an intersection-form obstruction

The improvement results of this page are stated below the middle: they require a framed embedded representative of the class to be killed and the inequality $p\le q-2$, equivalently $2p+2\le m$ (p-surgery kills the represented pi-p class below the middle dimension). When the middle dimension is reached, the following genuinely new phenomena appear.

(i) A kernel class need not be representable by an embedded sphere with trivial normal bundle. The primary obstruction to representing a middle-dimensional class by a framed embedding is a self-intersection class $\mu$ of the corresponding immersion, taking values in a quotient of the group ring of ${\pi }_1$, and the framing obstruction of the framing lemma of this page can be nonzero (The framing obstruction lives in the normal bundle of the surgery sphere). This is the content of the sources' representability criterion: for a pointed immersion $S^k\to M^{2k}$ into a compact connected manifold, with basepoints, a whisker, and an orientation at the ambient basepoint supplied, $k\ge 3$ gives regular homotopy to an embedding if and only if the self-intersection element vanishes (Lück, Theorem 4.8, printed pp. 84–85). The complementary-dimension condition makes its Whitney disk construction available. The library does not prove the self-intersection criterion here; it is recorded from the cited sources as the exact stopping point.

(ii) Even when a class can be killed, the middle-dimensional intersection form need not be preserved: middle-dimensional surgery can change it, as the B-page counterexample computes for $S^2\times S^2$, using the geometric intersection pairing of The geometric intersection pairing on a closed oriented manifold and the self-intersection/Euler-number identification of The self-intersection number is the Euler number of the normal bundle and The self-intersection number of a complementary-dimensional oriented submanifold.

(iii) In the classical oriented high-dimensional programme ($m\ge 5$) over a finite oriented $m$-dimensional Poincaré complex $X$, first perform surgery below the middle. For $m=2n$, the remaining obstruction is represented by the quadratic kernel form: the middle-dimensional intersection pairing together with its self-intersection refinement. For $m=2n+1$, it is represented by a quadratic kernel formation, a nonsingular quadratic form with an ordered pair of lagrangians obtained from a middle-dimensional splitting; it is not merely a refinement of a pairing on a single middle homology group. In either parity the surgery obstruction lies in $L_m(\mathbb{Z}[{\pi }_1(X)])$ and need not vanish. These algebraic constructions are recorded from Ranicki, Chapters 11–12, and are not developed here (Degree-one normal map for the surgery program, Surgery on a normal map preserves its normal bordism class). The homology-effect proposition of this page (The homology effect of surgery away from the middle dimensions) describes which degrees can change and is not a statement about the middle-dimensional form.

The geometric input for ordinary sphere surgery is a framed embedded representative. In the normal-map setting of (iii), the framing must also be compatible with the normal data: one must supply a null-homotopy $h$ of $f\circ {\phi }_0$ and, for the chosen framing and resulting extension $F:W_{\phi }\to X$, a stable bundle isomorphism $B:{\nu }_{W_{\phi }}\to F^{\ast }\xi$ extending $b$, where $\xi$ is the target normal bundle datum ($\xi ={\nu }_X$ in the manifold-target proposition cited above). The trace framing must be orientation-compatible. This is the $b$-framed surgery datum of that proposition, as in Ranicki, Definition 10.6; a framing and a null-homotopy alone do not supply $B$. The page's homotopy comparison with $M_{\phi }$ retains the below-middle bound $p\le q-2$; no such general comparison in the middle dimension is asserted here.