Whitney disjunction removes algebraically cancelling double points

Choose disjoint source arc neighbourhoods $P,Q\subset M$, with smaller closed arc neighbourhoods inside them. Along either arc the immersion is an embedding because its interior contains no double-point preimage; shrink the source neighbourhoods using local embedding charts and compactness of the arcs. Their images meet each other only at the two prescribed corners, and meet no other branch there: otherwise a sequence of unwanted incidences as the neighbourhoods shrink would converge to an additional double-point preimage on an arc or to an extra branch at a prescribed double point. The genuine-double-point hypothesis says that there are exactly two preimages at each selected corner; all other collision images are avoided by the arcs. Use the fixed Whitney bigon $B=\{(u,v):-1\le u\le 1,\mathrm{\mid }v\mathrm{\mid }\le 1-u^2\}$ with the explicit quadrant charts of [L1]. At the two corners use the complementary-sheet coordinates; opposite local signs allow the standard Whitney collar to be sewn between the arc collars. Its interior leaves the selected sheet in a transverse direction, and at a corner lies in the quadrant between the two sheet axes, so it meets neither axis. Compactness/properness excludes every remaining branch from a sufficiently small collar. As the circle is null-homotopic, the inner boundary of this collar also bounds a continuous disk (it is homotopic through the collar to the original circle). Attach that disk to the collar and smooth relative to a smaller fixed bigon boundary neighbourhood using relative Whitney approximation on the open interior. To see this as an ordinary-manifold application, the attached map is already smooth on the collar interior; apply the relative theorem on $\mathrm{int}B$ with a closed inner collar where it is smooth, and extend by the original boundary model. Thus both corners stay fixed and no arbitrary rounding off the sheets is used. Apply the immersion-image adapter to obtain an embedded disk $W$ with interior disjoint from all of $f(M)$, fixed on the collar and avoiding neighbourhoods of every other double point.

The branch-normal data along the Whitney circle give an admissible splitting into ranks $m-1$ and $m-1$ of the rank-$2m-2$ normal bundle of the collared disk. The orientations involved here are those over the contractible disk and the two source arc neighbourhoods; the original opposite-sign hypothesis is the corner compatibility. No orientation of all of $M$ or $X$ is added to clause 1. The corrected framing supplier asserts existence of an admissible extendible choice, permitting correction of the frame within one summand while fixing its subspace and its values at both corners. The normal-summand surjection proved in that supplier realizes the inverse of the discrepancy loop in $SO(2m-2)$ within $SO(m-1)$, valid also at $m=3$, so this correction kills the discrepancy from a global disk normal frame. The correction is supported in the interior of one boundary arc; it changes no sheet or disk. Obtain a clean framed disk. This uses an admissible extendible framing, never an arbitrary fixed circle-normal framing.

Choose the framing tube and the selected smaller source patch $P$ so that the local Whitney model isotopy $G_t$ is the identity near the image of $\mathrm{\partial }P$, moves only the first sheet as compared with the fixed second sheet, and is supported in a compact tube around $W$. The local construction of [L3], applied to these sheet patches, provides an isotopy of the first sheet, relative to its boundary, whose final image intersects the fixed second sheet with precisely the prescribed cancelling pair removed. The tube is disjoint from all other source-image branches except the chosen sheet collars by step 1.1; shrink its radius using the compact disk core and the compact image of the source outside slightly enlarged $P,Q$, and use the fixed collar charts near the boundary. The local model is embedded on $P$ at every time and unchanged near $\mathrm{\partial }P$.

Define $f_t(x)=G_t(f(x))$ for $x\in P$, and $f_t(x)=f(x)$ for $x\mathrm{\notin }P$. The two formulas agree on an open neighbourhood of $\mathrm{\partial }P$ for every $t$, so they glue to a smooth map on $M\times I$. On $P$, $d{f}_t=d{G}_t\circ df$ is injective because $G_t$ is a local ambient diffeomorphism, and on the complement $d{f}_t=df$ is injective; the agreement region handles patch boundaries. Thus every $f_t$ is an immersion and the family is a regular homotopy. The second source sheet $Q$ remains fixed. Applying $G_t$ to the entire immersion would preserve all coincidences and is explicitly not the construction. At the cancellation parameter the two branches may be tangent, although each source branch remains immersed.

At time one the model removes exactly the intersections at $r_1,r_2$ between the selected sheets and creates none. It creates no self-intersection within the moved patch because that patch stays embedded. A new intersection with any other branch is excluded by the tube cleanliness in step 3.1, and all other double-point neighbourhoods are fixed. Thus $\mathrm{\Sigma }(f_1)=\mathrm{\Sigma }(f)\setminus \{r_1,r_2\}$ and all remaining crossings retain their original self-transverse local models. The endpoint $f_1$ is self-transverse. Together with step 4.1 this proves clause 1 with the corrected endpoint-only transversality condition.

There are finitely many double points: local injectivity excludes a neighbourhood of the diagonal in compact $M\times M$, and the transverse coincidence locus is a discrete closed subset of its compact complement. For a finite admissible pairing apply clause 1 successively. After each move the unmoved double-point germs and signs remain fixed. Transport the endpoints of any planned source arcs by the already constructed regular homotopy; the corresponding ambient Whitney circles remain null-homotopic because their parametrized loops vary by a homotopy. Rechoose the arcs by small source perturbations relative to the endpoint collars to avoid the finitely many remaining preimages and each other, using the finite relative bump/diagonal argument of [L7] with expected dimension $2-m<0$. Small perturbations preserve embeddedness by the local-injectivity and compact separated-pair estimate. This changes no ambient homotopy class. At every stage apply the adapter to the current immersion, rather than assume pairwise disjointness of all prior disks or closeness to an arbitrary null-homotopy preserves cleanliness. After finitely many removals the endpoint is an injective immersion of compact $M$, hence an embedding. Reparametrize each regular homotopy to be constant near its endpoints before concatenating, giving a smooth regular homotopy.

For the final integral criterion, $M$ and $X$ are oriented, $m$ is even, $X$ is simply connected and $I(f)=0$. Apply [L6] first to obtain a self-transverse immersion $g$ with no triple image and with $I(g)=I(f)=0$. Its finitely many unordered branch pairs now correspond bijectively to genuine collision images, so the vanishing finite sum pairs each positive sign with a negative sign. Connectedness and [L5] supply embedded source arcs avoiding the finitely many other collision preimages; To make the arcs disjoint, keep small disjoint endpoint collars fixed and perturb the second arc using finitely many source-chart translations times bumps spanning all target value directions along its compact interior. The evaluation into $M$ is transverse to the first embedded arc; [L7] yields a slice transverse to it. Its intersection preimage has expected dimension $1-(m-1)=2-m<0$, so is empty. Small $C^1$ perturbations keep the arc embedded: local convex-chart projection estimates give uniform local injectivity, and pairs outside their common chart neighbourhood have a positive separation on a compact set. Smallness also preserves avoidance of the finite collision preimages; the whole perturbation is relative to the endpoint collars and gives a homotopy of the selected arc. Ambient simple connectivity makes each resulting Whitney circle null-homotopic. The pairs are admissible, so step 6.1 gives an embedding endpoint. Concatenate with the small regular homotopy from $f$ to $g$, smoothing the time parameter as in step 6.1. This preserves the criterion for the original pairwise self-transverse immersion, including its possible initial triple images.