Compact parameter pairs and relative families

A compact parameter pair is a pair $(P,Q)$ in which $P=P_0\times [0,1{]}^d$, where $P_0$ is a compact smooth manifold without boundary (possibly empty) and $d\ge 0$, and $Q\subseteq P$ is closed (possibly empty); $P$ is the parameter manifold and $Q$ the relative parameter set. Smoothness in the interval coordinates means local smooth extendibility to open Euclidean neighbourhoods, including at corners. This class contains spheres, cubes and intervals and is closed under products with $[0,1]$, so it also contains the homotopy parameters used below. The smoothing constructions extend the interval coordinates by clamping them before convolution; they do not require a tubular neighbourhood theorem for manifolds with corners.

Let $M$ and $N$ be smooth manifolds and let $(P,Q)$ be a compact parameter pair.

For families and homotopies of formal immersions below, assume $\dim M\le \dim N$, as required by the supplied immersion mapping spaces; the map-family clauses have no dimension restriction.

• A smooth $P$-family of maps $M\to N$ is a smooth map $F:P\times M\to N$, with slices $F_p:=F(p,\cdot )$; $F$ is also called the evaluation map of the family (Smooth families of maps and their evaluation maps).
• A continuous $P$-family is a continuous map $\mathrm{\Phi }:P\to C^{\mathrm{\infty }}(M,N)$ for the weak compact-open $C^{\mathrm{\infty }}$ topology (The weak compact-open C-infinity topology on mapping spaces). By the exponential law its adjoint $P\times M\to N$, $(p,x)\mapsto \mathrm{\Phi }(p)(x)$, is continuous (The exponential law: for a locally compact metric $X$ and any spaces $Z$ and $Y$, transposition is a bijection between $C(X\times Z,Y)$ and $C(Z,C(X,Y))$ with the compact-open topology, The compact-open topology on $C(X,Y)$ for a metric domain $X$, with subbasis $S(K,V)=\{f:f[K]\subseteq V\}$). The family is smooth when it is the transpose of a smooth $P$-family, i.e. $\mathrm{\Phi }(p)=F_p$ for a smooth $F$.
• Under ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) for the canonical tangent bundles and their total-space mapping topology, a family of formal immersions over $P$ is a continuous map $\mathrm{\Phi }:P\to \mathrm{FImm}(M,N)$ (Space of immersions and space of formal immersions); it is smooth when it is the transpose of a pair $(f,F)$ of smooth maps $f:P\times M\to N$, $F:P\times TM\to TN$ with $F$ a fibrewise injective smooth bundle map over $f$ (Formal immersion between smooth manifolds). A family is genuine when its slices lie in $\mathrm{Imm}(M,N)$, and holonomic on a subset $A\subseteq P$ when $F_p=d{f}_p$ for every $p\in A$; it is smoothly holonomic on $A$ when some open neighbourhood $W$ of $A$ in $P$ carries a smooth family $g:W\times M\to N$ with $f=g$ and $F=dg$ on $W\times M$.
• Under the same ${\mathrm{A}\mathrm{C}}_{\omega }$ assumption, a homotopy of $P$-families of formal immersions is a continuous map $H:P\times [0,1]\to \mathrm{FImm}(M,N)$, read as a family parametrized by the compact manifold with boundary $P\times [0,1]$; it is relative to $Q$ when $H(q,s)=H(q,0)$ for all $q\in Q$ and $s\in [0,1]$, and relative to $\mathrm{\partial }P$ in the smooth case when it is constant along $\mathrm{\partial }P\times [0,1]$. A homotopy is smooth when it is given by a smooth family over $P\times [0,1]$; a smooth homotopy of genuine families is a homotopy through genuine families, i.e. a regular homotopy when $P$ is a point.