Self-transverse immersions and the double point locus

Let $f:M^m\to X$ be a smooth immersion. Write ${\mathrm{\Delta }}_M$ and ${\mathrm{\Delta }}_X$ for the diagonals, embedded by The diagonal of a smooth manifold is a closed embedded submanifold. Its ordered coincidence locus is ${\mathrm{\Delta }}_2(f)=\{(x,y)\in M\times M\setminus {\mathrm{\Delta }}_M:f(x)=f(y)\}\mathrm{.}$ Its unordered branch-pair set is $D(f)=\{\{x,y\}:x\mathrm{\ne }y,f(x)=f(y)\}={\mathrm{\Delta }}_2(f)\mathrm{/}(x,y)\sim (y,x),$ and its collision image is $\mathrm{\Sigma }(f)=f({\mathrm{pr}}_1{\mathrm{\Delta }}_2(f))\subseteq X$. The common-image map $D(f)\to \mathrm{\Sigma }(f)$ is surjective. It is bijective exactly when no image point has three or more preimages. At an image point with $k$ preimages, $D(f)$ has one element for each of the $\left(\genfrac{}{}{0px}{}{k}{2}\right)$ unordered branch pairs, rather than one element for the image point. A genuine double point is a point of $X$ with exactly two preimages. The term double-point locus refers to the branch-pair locus; it does not exclude higher-multiplicity collision images.

The immersion is self-transverse when $f\times f:M\times M\setminus {\mathrm{\Delta }}_M\to X\times X$ is transverse to ${\mathrm{\Delta }}_X$ (A smooth map transverse to an embedded submanifold). Equivalently, for every distinct $x,y$ with common image $r$, one has $d{f}_x(T_{x}M)+d{f}_y(T_{y}M)=T_{r}X$. This is pairwise transversality and does not assert absence of triple points. Each selected preimage gives a local embedded sheet by Every immersion is locally an embedding; for a self-transverse immersion in ambient dimension $2m$, a selected coincident pair has complementary tangent planes, and its two local sheet disks are supplied by A double point has two disjoint embedded sheet disks meeting transversely ↗. This statement about a selected pair does not say that these are all sheets over $r$.

If $f$ is self-transverse and $M$ and $X^{2m}$ are oriented, every ordered coincident branch pair has the sign $\epsilon (x,y)\in \{\pm 1\}$ of The local oriented intersection sign, computed with the $x$ branch first. Swapping the two oriented $m$-blocks multiplies this sign by $(-1{)}^{m^2}=(-1{)}^m$. Consequently for even $m$ it defines an ordering-independent sign on each element of $D(f)$; for odd $m$ an ordering is needed. At a multiple collision image, different branch pairs need not have the same sign, so no single sign is assigned to that image. None of these definitions asserts finiteness, compactness, orientability, absence of triples or any choice principle.