{'fraction_non_alphanumeric': 0.017816091954022988, 'fraction_numerical': 0.0022988505747126436, 'mean_word_length': 5.173758865248227, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Explicit remarks on the torsion subgroups of homology groups of Reeb spaces of explicit fold maps

Fold maps are higher dimensional versions of Morse functions and fundamental and important tools in studying algebraic and differential topological properties of manifolds: as the theory established by Morse and the higher dimensional version, started by Thom and Whitney, later actively studied by Eliashberg, Levine etc. and recently studied by Kobayashi, Saeki, Sakuma etc., explicitly show this. 
One of fundamental, important and difficult studies on this field is, constructing explicit fold maps and investigating their source manifolds. As fundamental and strong tools for systematic construction, the author has introduced surgery operations (bubbling operations) to fold maps, motivated by studies of Kobayashi etc. since 1990. The author has explicitly shown that homology groups of Reeb spaces of maps constructed by iterations of these operations are generally flexible and restricted in several specific cases. The Reeb space of a map is defined as the space of all connected components of inverse images of the maps, inheriting fundamental invariants of manifolds such as homology groups etc. in considerable cases and fundamental and important tools in the field. 
In this paper, we explicitly remark on the torsion subgroups of the homology groups. More precisely, under explicit algebraic constraints, we see explicit strong restrictions on the torsion subgroups, where the homology groups seem to be very flexible in general. We note that this work is similar to several works by the author before but a work in a new situation and that new technique such as well-known fundamental theory of abstract algebra will be used.