Good real images of complex maps

Abstract

We prove several results regarding the homology and homotopy type of images of real maps and their complexification. In particular, we study the local behavior of singular points after deformations. In this context, we prove a restrictive necessary condition for a real perturbation to have the same homology than its complexification, which is known as good real perturbation. We prove the conjecture of Marar and Mond stating that for singularities from $\mathbb{C}^n$ to $\mathbb{C}^{n+1}$, a good real perturbation is homotopy equivalent to its complexification, and show a generalization in other dimensions. Applications to $M$-deformations and other concepts as well as examples are given.

Figures (5)

• Figure 1: Topological types of the deformation $f^\mathbb{R}_{a,b,t}(x)=(x^3-tx,x^4-tax^2-tbx)$ for different values of $a,b$ and $t$.
• Figure 2: Representation of a map $g^\mathbb{R}$ together with its whiskers $\mathcal{W}(g)$ (green, dashed line) and a non-immersive point (red).
• Figure 3: Good real perturbations of the $S_1$ (left) and $H_2$ (right) singularities in Mond's classification of map germs from $\mathbb{C}^2$ to $\mathbb{C}^3$Mond1985, $S_{1,s}^\mathbb{R}(x,y)=(x,y^2, y^3+y(x^2-s))$ and $H^\mathbb{R}_{2,s}(x,y)=(x,y^3-sy,xy+y^2(y^3-sy))$.
• Figure 4: Schematic of the ICSS $E^*_{q,r}(f_s^{(\mathbb{R})})$ where the arrows represent the different boundary operators at several pages that have target or source a given entry, $\ast$ are the possibly non-zero entries in the complex case (also the corresponding dimension of the multiple point space in both cases) and $\bullet$ are the homologies that are to be determined zero in the real case, for dimensions $(9,11)$, $(9,12)$ and $(6,7)$ from left to right.
• Figure 5: Schematic of \ref{['ex:triangles']}.

Theorems & Definitions (89)

• Proposition 2.1: cf. Mond1996
• proof
• Lemma 2.2
• proof
• Theorem 2.3
• proof
• Theorem 2.4
• Definition 2.5
• Remark 2.6
• Definition 2.7