A class of naturally generalized special generic maps

Smooth maps like special generic maps

“…Essentially same problems are also asked in the previous preprint of the author [18]. There the author introduced simply generalized special generic maps.…”

“…Hereafter, as the coefficient ring A, we take a principal ideal domain having the unique identity element different from the zero element. According to our main work of [18], as Remark 1 presents shortly, we can easily construct a natural simply generalized special generic map on some suitable m-dimensional closed and connected manifold M such that in Example 2 (2), we can choose j p as an arbitrary integer 1 ≤ j ≤ l+1 for some point p ∈ ∂ N (if the boundary of the manifold N has sufficiently many connected components). According to our construction, the resulting reference homomorphisms along the curves between the homology groups of degrees at least 1 are always the zero homomorphisms.…”

“…See also [2] for example. Exposition related to this history is also in [18] In our new class, the preimage of each point in the interior of the immersed manifold and Morse(-Bott) functions used around the boundary are generalized.…”

“…This does not depend on which point we choose in the connected component C ∋ p of ∂W f . Note that in [18], simply generalized special generic maps in simplest cases here and the homology groups and the cohomology rings of the manifolds are studied.…”

“…This is important in construction and used in the proof of Main Theorem 2. Related to Example 2 (2), one of our main work of [18] is regarded as a work constructing simply generalized special generic maps with prescribed reference homomorphisms along given curves and studying the homology groups and the cohomology rings of the obtained manifolds of the domains. Of course reference homomorphisms along curves are not introduced there.…”