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Special generic maps are smooth maps at each singular point of which we canrepresent as $(x_1, \cdots, x_m) \mapsto(x_1,\cdots,x_{n-1},\sum_{k=n}^{m}{x_k}^2)$ for suitable coordinates. Morsefunctions with exactly two singular points on homotopy spheres and canonicalprojections of unit spheres are special generic. They are known to restrict thetopologies and the differentiable structures of the manifolds in varioussituations. On the other hands, various manifolds admit such maps.  This article first presents a special generic map on a $7$-dimensionalmanifold and the image. This results also seems to present a new example of$7$-dimensional closed and simply-connected manifolds having non-vanishingtriple Massey products and seems to be a new work related to similar works byDranishnikov and Rudyak. We also review results on vanishing of products ofcohomology classes, previously obtained by the author. The images of specialgeneric maps are smoothly immersed manifolds whose dimensions are equal to thedimensions of the targets. The author studied the topologies of these imagespreviously and studies on homology groups, cohomology rings and structures ofthem for special generic maps having simple structures are presented as newresults.

The topologies and the differentiable structures of the images of special generic maps having simple structures