Гомологические сферы

M. I. Kornev

Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 84 (2023) no. 2, pp. 243-296.

Bibliographie

[1] S. I. Adyan, “Nerazreshimost nekotorykh algoritmicheskikh problem teorii grup”, Tr. MMO, 6, 1957, 231–298 | Zbl

[2] I. A. Volodin, V. E. Kuznetsov, A. T. Fomenko, “O probleme algoritmicheskogo raspoznavaniya standartnoi trekhmernoi sfery”, UMN, 29:5(179) (1974), 71–168 | Zbl

[3] A. Markov, “Nerazreshimost problemy gomeomorfii”, DAN SSSR, 121:2 (1958), 218–220 | Zbl

[4] S. V. Matveev, “Algoritm raspoznavaniya trekhmernoi sfery (po A. Tompson)”, Matem. sb., 186:5 (1995), 69–84 | Zbl

[5] Dzh. Milnor, Vvedenie v algebraicheskuyu $K$-teoriyu, Mir, M., 1974

[6] S. P. Novikov, Rukopis (ne opublikovano), 2020

[7] V. A. Rokhlin, “Novye rezultaty v teorii chetyrekhmernykh mnogoobrazii”, DAN SSSR, 84 (1952), 221–224 | Zbl

[8] A. Adem, I. Hambleton, “Minimal Euler characteristics for even-dimensional manifolds with finite fundamental group”, Forum Math. Sigma, 11 (2023), e24 | DOI | MR | Zbl

[9] G. Baumslag, E. Dyer, A. Heller, “The topology of discrete groups”, J. Pure Appl. Algebra, 16:1 (1980), 1–47 | DOI | MR | Zbl

[10] G. Baumslag, E. Dyer, C. F. Miller, “On the integral homology of finitely presented groups”, Topology, 22:1 (1983), 27–46 | DOI | MR | Zbl

[11] A. J. Berrick, J. A. Hillman, “Perfect and acyclic subgroups of finitely presentable groups”, J. London Math. Soc. (2), 68:3 (2003), 683–698 | DOI | MR | Zbl

[12] A. K. Bousfield, D. M. Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics, 304, Springer-Verlag, Berlin–New York, 1972 | DOI | MR | Zbl

[13] E. Brieskorn, “Beispiele zur Differentialtopologie von Singularitäten”, Invent. Math., 2 (1966), 1–14 | DOI | MR | Zbl

[14] K. S. Brown, Cohomology of groups, Grad. Texts in Math., 87, Springer-Verlag, New York-Berlin, 1982 | DOI | MR | Zbl

[15] M. Chiodo, M. E. Hill, Preserving torsion orders when embedding into groups with `small` finite presentations, 2016, arXiv: 1610.00977 | Zbl

[16] J. Choi, J.-H. Lee, “Binary icosahedral group and 600-cell”, Symmetry, 10:8 (2018), 326 | DOI | Zbl

[17] I. Dai, J. Hom, M. Stoffregen, L. Truong, An infinite-rank summand of the homology cobordism group, 2018, arXiv: 1810.06145

[18] A. Deleanu, “On a theorem of Baumslag, Dyer and Heller linking group theory and topology”, Cah. Topol. Géom. Différ. Catég., 23:3 (1982), 231–242 | MR | Zbl

[19] E. Dror, “Homology spheres”, Israel J. Math., 15 (1973), 115–129 | DOI | MR | Zbl

[20] E. Dyer, A. T. Vasquez, “Some small aspherical spaces”, J. Aust. Math. Soc., 16:3 (1973), 332–352 | DOI | MR | Zbl

[21] D. B. A. Epstein, “Finite presentations of groups and 3-manifolds”, Quart. J. Math. Oxford Ser. (2), 12 (1961), 205–212 | DOI | MR | Zbl

[22] M. H. Freedman, “The topology of four-dimensional manifolds”, J. Differential Geometry, 17:3 (1982), 357–453 | DOI | MR | Zbl

[23] K. A. Frøyshov, “Equivariant aspects of Yang–Mills Floer theory”, Topology, 41:3 (2002), 525–552 | DOI | MR

[24] K. A. Frøyshov, Mod 2 instanton Floer homology, Unpublished, 2016

[25] D. E. Galewski, R. J. Stern, “Classifications of simplicial triangulations of topological manifolds”, Bull. Amer. Math. Soc., 82:6 (1976), 916–918 | DOI | MR | Zbl

[26] P. G. Goerss, J. F. Jardine, Simplicial homotopy theory, Birkhäuser, Basel, 2009 | MR

[27] F. J. González Acuña, On homology spheres, PhD thesis, Princeton University, 1970 | MR

[28] A. Haefliger, “Plongements différentiables de variétés dans variétés”, Comment. Math. Helv., 36 (1961), 47–82 | DOI | MR | Zbl

[29] I. Hambleton, M. Kreck, “On the classification of topological 4-manifolds with finite fundamental group”, Math. Ann., 280:1 (1988), 85–104 | DOI | MR | Zbl

[30] J.-C. Hausmann, Sh. Weinberger, “Caractéristiques d'Euler et groupes fondamentaux des variétés de dimension 4”, Comment. Math. Helv., 60:1 (1985), 139–144 | DOI | MR | Zbl

[31] J. Hempel, “Residual finiteness for 3-manifolds”, Combinatorial group theory and topology (Alta, Utah, 1984), Ann. of Math. Stud., 111, Princeton Univ. Press, Princeton, 1987, 379–396 | MR

[32] J. Hempel, 3-Manifolds, AMS Chelsea Publishing, Providence, RI, 2004 | MR | Zbl

[33] K. Hendricks, J. Hom, M. Stoffregen, I. Zemke, Surgery exact triangles in involutive Heegaard Floer homology, 2020, arXiv: 2011.00113

[34] G. Higman, “A finitely generated infinite simple group”, J. London Math. Soc., 26 (1951), 61–64 | DOI | MR | Zbl

[35] G. Higman, “Subgroups of finitely presented groups”, Proc. Roy. Soc. London. Ser. A, 262 (1961), 455–475 | DOI | MR | Zbl

[36] G. Higman, B. H. Neumann, H. Neumann, “Embedding theorems for groups”, J. London Math. Soc., 24 (1949), 247–254 | DOI | MR

[37] J. A. Hillman, Four-manifolds, geometries and knots, Geom. Topol. Monogr., 5, Univ. Warwick, Coventry, UK, 2002 | MR

[38] H. Hopf, “Fundamentalgruppe und zweite Bettische Gruppe”, Comment. Math. Helv., 14 (1942), 257–309 | DOI | MR | Zbl

[39] D. M. Kan, W. P. Thurston, “Every connected space has the homology of a $K(\pi,1)$”, Topology, 15:3 (1976), 253–258 | DOI | MR | Zbl

[40] M. A. Kervaire, “Smooth homology spheres and their fundamental groups”, Trans. Amer. Math. Soc., 144 (1969), 67–72 | DOI | MR | Zbl

[41] M. A. Kervaire, J. W. Milnor, “Groups of homotopy spheres. I”, Ann. of Math. (2), 77 (1963), 504–537 | DOI | MR | Zbl

[42] W. B. R. Lickorish, “A representation of orientable combinatorial 3-manifolds”, Ann. of Math. (2), 76 (1962), 531–540 | DOI | MR | Zbl

[43] C. Livingston, “Four-manifolds of large negative deficiency”, Math. Proc. Cambridge Philos. Soc., 138:1 (2005), 107–115 | DOI | MR | Zbl

[44] J. Lurie, Diffeomorphisms and PL homeomorphisms (Lecture 6), Topics in Geometric Topology, MIT Course, 2009 https://www.math.ias.edu/l̃urie/937notes/937Lecture6.pdf

[45] C. Manolescu, Pin(2)-equivariant Seiberg–Witten Floer homology and the triangulation conjecture, 2015, arXiv: 1303.2354 | MR

[46] C. Manolescu, Lectures on the triangulation conjecture, 2016, arXiv: 1607.08163 | MR

[47] T. Matumoto, “Triangulation of manifolds”, Algebraic and geometric topology (Stanford, Calif.,), v. 2, Proc. Sympos. Pure Math., XXXII, AMS, Providence, RI, 1976, 3–6 | MR

[48] C. R. F. Maunder, “A short proof of a theorem of Kan and Thurston”, Bull. London Math. Soc., 13:4 (1981), 325–327 | DOI | MR | Zbl

[49] J. P. May, Simplicial objects in algebraic topology, Chicago Lectures in Math., Univ. Chicago Press, Chicago, IL, 1992 | MR | Zbl

[50] J. McCleary, A user's guide to spectral sequences, Cambridge Stud. Adv. Math., 58, Cambridge Univ. Press, Cambridge, 2001 | MR

[51] J. Milnor, “Groups which act on $S_n$ without fixed point”, Amer. J. Math., 79 (1957), 623–630 | DOI | MR | Zbl

[52] J. Milnor, “Microbundles. Part I”, Topology, 3 (1964), 53–80 | DOI | MR | Zbl

[53] J. Milnor, Singular points of complex hypersurfaces, Ann. of Math. Stud., 61, Princeton Univ. Press, Princeton, 1968 | MR | Zbl

[54] J. Milnor, “On the 3-dimensional Brieskorn manifolds $M(p,q,r)$”, Knots, groups, and 3-manifolds, Ann. of Math. Stud., 84, Princeton Univ. Press, Princeton, 1975, 175–225 | MR

[55] J. W. Milnor, J. D. Stasheff, Characteristic classes, Ann. of Math. Stud., 76, Princeton Univ. Press, Princeton, 1974 | MR | Zbl

[56] N. Monod, “Variations on a theme by Higman”, Expo. Math., 35:2 (2017), 226–235 | DOI | MR | Zbl

[57] R. Myers, “Homology cobordisms, link concordances, and hyperbolic 3-manifolds”, Trans. Amer. Math. Soc., 278:1 (1983), 271–288 | DOI | MR | Zbl

[58] M. H. A. Newman, “The engulfing theorem for topological manifolds”, Ann. of Math. (2), 84 (1966), 555–571 | DOI | MR | Zbl

[59] S. Ochanine, Signature modulo $16$, invariants de Kervaire généralisés et nombres caractéristiques dans la $K$-théorie réelle, Mém. Soc. Math. France (N.S.), no. 5, 1981 | MR

[60] G. Perelman, The entropy formula for the Ricci flow and its geometric applications, 2002, arXiv: 0211159

[61] S. P. Plotnick, A. I. Suciu, “$k$-Invariants of knotted $2$-spheres”, Comment. Math. Helv., 60:1 (1985), 54–84 | DOI | MR | Zbl

[62] H. Poincaré, “Analysis situs”, Journal de l'école polytechnique, 1 (1895), 1–121

[63] H. Poincaré, “Cinquième complément à l'analysis situs”, Rendiconti del Circolo Matematico di Palermo, 18 (1904), 45–110 | DOI

[64] J. G. Ratcliffe, S. T. Tschantz, “Some examples of aspherical 4-manifolds that are homology 4-spheres”, Topology, 44:2 (2005), 341–350 | DOI | MR | Zbl

[65] C. Rezk, The MO answer, April 2019 https://mathoverflow.net/questions/328022/classification-of-bundles-postnikov-towers-obstruction-theory-local-coefficie

[66] H. Rubinstein, The solution to the recognition problem for $\mathbb{S}^3$, Unpublished lectures, May 1992

[67] N. Saveliev, “Invariants for homology 3-spheres”, Low-Dimens. Topol., v. I, Encyclopaedia of Mathematical Sciences, 140, Springer-Verlag, Berlin, 2002 | DOI | MR | Zbl

[68] O. Şavk, A survey of the homology cobordism group, 2023, arXiv: 2209.10735 | MR

[69] A. Scorpan, The wild world of $4$-manifolds, AMS, Providence, RI, 2005 | MR | Zbl

[70] G. P. Scott, “Compact submanifolds of 3-manifolds”, J. London Math. Soc. (2), 7 (1973), 246–250 | DOI | MR | Zbl

[71] J.-P. Serre, Trees, Springer-Verlag, Berlin, 1980 | MR | Zbl

[72] J. Singer, “Three-dimensional manifolds and their Heegaard diagrams”, Trans. Amer. Math. Soc., 35:1 (1933), 88–111 | DOI | MR | Zbl

[73] S. Smale, “Generalized Poincaré's conjecture in dimensions greater than four”, Ann. of Math. (2), 74 (1961), 391–406 | DOI | MR | Zbl

[74] S. Smale, “On the structure of manifolds”, Amer. J. Math., 84 (1962), 387–399 | DOI | MR | Zbl

[75] J. R. Stallings, “Polyhedral homotopy-spheres”, Bull. Amer. Math. Soc., 66 (1960), 485–488 | DOI | MR | Zbl

[76] J. Stallings, “A finitely presented group whose 3-dimensional integral homology is not finitely generated”, Amer. J. Math., 85 (1963), 541–543 | DOI | MR | Zbl

[77] J. Stillwell, Classical topology and combinatorial group theory, Grad. Texts in Math., 72, Springer, New York, 2012 | MR

[78] A. Suciu, “Homology $4$-spheres with distinct $k$-invariants”, Topology Appl., 25:1 (1987), 103–110 | DOI | MR | Zbl

[79] A. I. Suciu, “Iterated spinning and homology spheres”, Trans. Amer. Math. Soc., 321:1 (1990), 145–157 | DOI | MR | Zbl

[80] R. G. Swan, “Periodic resolutions for finite groups”, Ann. of Math. (2), 72 (1960), 267–291 | DOI | MR | Zbl

[81] A. Thompson, “Thin position and the recognition problem for $\mathbb{S}^3$”, Math. Res. Lett., 1:5 (1994), 613–630 | DOI | MR | Zbl

[82] user02138, When are Brieskorn manifolds homeomorphic?, April 2013 https://mathoverflow.net/questions/126807/when-are-brieskorn-manifolds-homeomorphic

[83] user98602, Homology 4-balls with boundary $S^3$, June 2016 https://math.stackexchange.com/questions/1835405/homology-4-balls-with-boundary-s3

[84] A. H. Wallace, “Modifications and cobounding manifolds”, Canadian J. Math., 12 (1960), 503–528 | DOI | MR

[85] C. A. Weibel, An introduction to homological algebra, Cambridge Stud. Adv. Math., 38, Cambridge Univ. Press, Cambridge, 1994 | MR | Zbl

[86] J. Whitehead, “On the asphericity of regions in a 3-sphere”, Fundamenta Mathematicae, 32:1 (1939), 149–166 | DOI | Zbl

[87] J. H. C. Whitehead, “On $C^1$-complexes”, Ann. of Math. (2), 41:4 (1940), 809–824 | DOI | MR | Zbl

[88] E. C. Zeeman, “The Poincaré conjecture for $n\geq 5$”, Topology of 3-manifolds and related topics, Proc. The Univ. of Georgia Institute (1961), Prentice-Hall, Inc., Englewood Cliffs, NJ, 1961, 198–204 | MR