A Note on Non-Isolated Real Singularities and Links

The topological nature of analytic map germs has grown to this day to becomee an immense field of research. For real analytic maps, and more particularly singular such maps, we can for the purpose of this paper mention the works of H. Hamm (e.g [4]), Z. Szafraniec ([7]) and more recently the paper of Séade, Cisneros-Molina and Snoussi ([1]) and the preprint ([3]) of N. Dutertre. The result presented in following paper is in fact a mere consequence of the work of the latter authors, and should ideally be called a corollary of [6, Theorem 13.5] and [7, Lemma 1].

The result in this paper is a topological relation between the link of a singularity of real analytic function germs $f:(\mathbb{R}^{n},0)\to(\mathbb{R},0)$, its Milnor-Lê fiber, and the Milnor fiber of an isolated singularity of the form $f-c\omega^{k}$ where $\omega$ is the sum of the squares of the coordinate functions on $\mathbb{R}^{n}$, $c>0$ a constant, and $k\in\mathbb{N}$ a sufficiently large natural number.

Suppose that the real analytic function germ $f:(\mathbb{R}^{n},0)\to(\mathbb{R},0)$ has an isolated critical value in the origin, satisfies the transversality property of D.B. Massey ([5, Definition 13.4]) and that $\dim V(f)>0.$ Then the following theorem holds

Let $p>0$ be a positive integer. Then the transversality property is in fact equivalent to the existence of the first fibration for locally surjective map germs

$$g:(\mathbb{R}^{n},0)\to(\mathbb{R}^{p},0),\qquad\dim V(g)>0$$

with an isolated critical value in the origin.

In our case ($p=1$) the transversality property implies the equivalence of the bundles. This is done, following Milnor, by integrating an appropriate vector field to ”inflate” the Milnor tube $N_{f}(\epsilon,\delta)$ and push the fibers of the first fibration outwards towards the sphere. For $p>1$, the equivalence of the bundles is more delicate yet has recently been shown ([1]) by J. L. Cicneros-Molina, J. Séade and J. Snoussi that the notion of $d$-regularity which they introduced, plays an essential rôle.

Let us now recall a result of Szafraniec.

In particular it follows from the proof of the lemma that if $r=r(c,k)$ is chosen sufficiently small and if

$$N_{f}^{-}(r):=\{x\in\mathbb{S}_{r}:f(x)\leq 0\}$$

$$N_{g}^{-}(r):=\{x\in\mathbb{S}_{r}:g(x)\leq 0\}$$

then $N_{f}(r)\subset\text{int}N_{g}(r)$ and $N_{f}(r)$ is a deformation retract of $N_{g}(r)$.

Let us from now on assume that $f:(\mathbb{R}^{n},0)\to(\mathbb{R},0)$ satisfies the hypotheses of Theorem 1 . We then apply the theorem to $-f$ and consequently let $N_{f}^{-}(\epsilon,\delta)$ be the negative Milnor tube of $f$, so that

$$f:N_{f}^{-}(\epsilon,\delta)\to(-\infty,0)$$

is the projection of a trivial fibre bundle. Let

$$\mathcal{F}^{-}(f)=f^{-1}(\epsilon)\cap\mathbb{B}_{\delta}.$$

By inflating the Milnor tube as in the proof of Theorem 1 one obtains a homeomorphism

$$\mathcal{F}^{-}(f)\cong\{x\in\mathbb{S}_{\delta}:\quad f(x)\leq-\epsilon\}.$$

As a consequence,

$$\mathcal{F}^{-}(f)\cup_{\partial\mathcal{F}^{-}(f)}(\overline{N_{f}^{-}(\epsilon,\delta)}\cap\mathbb{S}_{\delta})\cong N_{f}(\delta)$$

where

$$\overline{N_{f}^{-}(\epsilon,\delta)}\cap\mathbb{S}_{\delta}=f^{-1}([-\epsilon,0])\cap\mathbb{S}_{\delta}.$$

On the other hand, by Lemma 1.0.1 , $N_{f}(\delta)\hookrightarrow N_{g}(\delta)$ is a deformation retract so

$$\mathcal{F}^{-}(f)\cup_{\partial\mathcal{F}^{-}(f)}(\overline{N_{f}^{-}(\epsilon,\delta)}\cap\mathbb{S}_{\delta})\hookrightarrow N_{g}^{-}(\delta)$$

is a homotopy equivalence. Note that by Lemma 1.0.1 , $g$ has an isolated critical point in the origin.

Let us now recall a result due to A. Durfee. Suppose given an algebraic set $M$ in the affine space $\mathbb{R}^{n}$ and let $X\subset M$ be a compact algebraic subset. Suppose that $M\setminus X$ is nonsingular. Recall that a subset $T\subset M$ is an algebraic neighborhood of $X$ if:
there exists a nonnegative proper polynomial map $\alpha:M\to\mathbb{R}$ and a positive real number $\gamma$ smaller than any critical value of $\alpha$ such that

$$X=\alpha^{-1}(0),\qquad T=\alpha^{-1}([0,\gamma])$$

In this situation Durfee ([2]) proved

We now apply this with

$$X=K_{f}=g^{-1}(0)\cap\mathbb{S}_{\delta},\quad T=g^{-1}([-\epsilon^{\prime},\epsilon^{\prime}])\cap\mathbb{S}_{\delta}$$

for $\epsilon^{\prime}<<\delta$, and assume that

$$g^{-1}([-\epsilon^{\prime},0))\cap\mathbb{S}_{\delta}\neq\emptyset.$$

Of course this is always the case whenever the boundary of the negative Milnor fiber $\mathcal{F}^{-}(g)$ of $g$ is nonempty. Then Lemma 1.0.1 together with Theorem 1 applied to the isolated singularity $g$ give

$$N_{g}^{-}(\delta)\sim\{g\leq-\epsilon^{\prime}\}\cap\mathbb{S}_{\delta}\cong\mathcal{F}_{\eta^{\prime}}^{-}(g).$$

We have proven