Geodesic
Configuration homological ${\mathbb Z}_2$-invariants of manifolds
Čelâbinskij fiziko-matematičeskij žurnal, Tome 6 (2021) no. 4, pp. 427-439

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The paper describes the construction of configuration invariants of 3-manifolds. These invariants are based on defining 3-manifolds by their special spines and can be constructed in the following way. Let $P$ be a special polyhedron and $k\in\mathbb{N}$. To each ordered sequence $\xi$, consisting of $k$ elements of the second homology group of the polyhedron $P$ with coefficients in $\mathbb{Z}_2 $, using a configuration map $\omega$ we assign the number $\omega(P, \xi)\in \{0, 1\}$. The value of the invariant is the ratio of the number of sequences $\xi$ for which $\omega(P, \xi) = 1$ to the total number of all such sequences. The axioms that the configuration map must satisfy ensure the invariance of the resulting rational number under $T$-transformations of special polyhedra.
Keywords: special spine, virtual manifold, invariant, chain complex. 
@article{CHFMJ_2021_6_4_a3,
     author = {F. G. Korablev},
     title = {Configuration homological ${\mathbb Z}_2$-invariants of manifolds},
     journal = {\v{C}el\^abinskij fiziko-matemati\v{c}eskij \v{z}urnal},
     pages = {427--439},
     publisher = {mathdoc},
     volume = {6},
     number = {4},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CHFMJ_2021_6_4_a3/}
}
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F. G. Korablev. Configuration homological ${\mathbb Z}_2$-invariants of manifolds. Čelâbinskij fiziko-matematičeskij žurnal, Tome 6 (2021) no. 4, pp. 427-439. http://geodesic.mathdoc.fr/item/CHFMJ_2021_6_4_a3/