# zbMATH — the first resource for mathematics

Generalized group presentation and formal deformations of CW complexes. (English) Zbl 0764.57012
If $$f: K^ n\to L^ n$$ is a simple-homotopy equivalence of connected, finite CW-complexes, then by C. T. C. Wall [Proc. Lond. Math. Soc., III. Ser. 16, 342-354 (1966; Zbl 0151.313)] there is a formal deformation $$K^ n\overset{n+1}\curvearrowright L^ n$$, provided $$n\geq 3$$. For $$n=2$$, the concepts of equivalence by 3-deformations and simple-homotopy might disagree. (Andrews-Curtis problem). The present paper presents an (algebraic) treatment of the whole picture which combines Whitehead’s approach of homotopy systems with R. Peiffer’s $$\Sigma$$-systems; the latter associate to a 3-complex, generators, relators and identities [Math. Ann. 121, 67-99 (1949; Zbl 0040.150)]. A geometric result of the author’s work is that, if a 3-cell $$e^ 3$$ has several free faces, then there is a characteristic map for $$e^ 3$$ which realizes the freeness of all these faces simultaneously. The analogous question for faces of $$n$$-cells seems to be open.
Initially, the reviewer stumbled over the use of topological embeddings $$D^{n-1}\hookrightarrow\partial D^ n$$ to define free faces, as (for $$n\geq 4)$$ the result might be a solid wild ball in $$\partial D^ n$$, and one has to worry whether there exists a deformation retraction for a collapse. But he confirmed himself that this can nevertheless be proved (use a subdisc of $$D^{n-1}$$ in standard position and apply M. Brown’s Schoenflies theorem). If this point wasn’t just overlooked by the author, a corresponding remark would have been appropriate.

##### MSC:
 57Q10 Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc. 57M05 Fundamental group, presentations, free differential calculus 20F05 Generators, relations, and presentations of groups 57Q05 General topology of complexes
Full Text:
##### References:
 [1] J. J. Andrews and M. L. Curtis, Free groups and handlebodies, Proc. Amer. Math. Soc. 16 (1965), 192 – 195. · Zbl 0131.38301 [2] Ronald Brown, On the second relative homotopy group of an adjunction space: an exposition of a theorem of J. H. C. Whitehead, J. London Math. Soc. (2) 22 (1980), no. 1, 146 – 152. · Zbl 0427.55012 [3] Ronald Brown and Philip J. Higgins, On the algebra of cubes, J. Pure Appl. Algebra 21 (1981), no. 3, 233 – 260. , https://doi.org/10.1016/0022-4049(81)90018-9 Ronald Brown and Philip J. Higgins, Colimit theorems for relative homotopy groups, J. Pure Appl. Algebra 22 (1981), no. 1, 11 – 41. · Zbl 0475.55009 [4] Robert Craggs, Free Heegaard diagrams and extended Nielsen transformations. I, Michigan Math. J. 26 (1979), no. 2, 161 – 186. Robert Craggs, Free Heegaard diagrams and extended Nielsen transformations. II, Illinois J. Math. 23 (1979), no. 1, 101 – 127. · Zbl 0441.57011 [5] James Dugundji, Topology, Allyn and Bacon, Inc., Boston, Mass., 1966. · Zbl 0144.21501 [6] E. Dyer and M.-E. Hamstrom, Completely regular mappings, Fund. Math. 45 (1958), 103 – 118. · Zbl 0083.38704 [7] Wolfgang Metzler, Äquivalenzklassen von Gruppenbeschreibungen, Identitäten und einfacher Homotopietyp in niederen Dimensionen, Homological group theory (Proc. Sympos., Durham, 1977) London Math. Soc. Lecture Note Ser., vol. 36, Cambridge Univ. Press, Cambridge-New York, 1979, pp. 291 – 326 (German). · Zbl 0433.57003 [8] Reneé Peiffer, Über Identitäten zwischen Relationen, J. Reine Angew. Math. 285 (1976), 17-23. [9] Elvira Strasser Rapaport, Groups of order 1: Some properties of presentations, Acta Math. 121 (1968), 127 – 150. · Zbl 0159.30501 [10] John G. Ratcliffe, Free and projective crossed modules, J. London Math. Soc. (2) 22 (1980), no. 1, 66 – 74. · Zbl 0427.20044 [11] Kurt Reidemeister, Einführung in die kombinatorische Topologie, Chelsea Publishing Co., New York, N. Y., 1950 (German). · Zbl 0040.10302 [12] C. T. C. Wall, Formal deformations, Proc. London Math. Soc. (3) 16 (1966), 342 – 352. · Zbl 0151.31302 [13] George W. Whitehead, Elements of homotopy theory, Graduate Texts in Mathematics, vol. 61, Springer-Verlag, New York-Berlin, 1978. · Zbl 0406.55001 [14] J. H. C. Whitehead, Combinatorial homotopy. II, Bull. Amer. Math. Soc. 55 (1949), 453 – 496. · Zbl 0040.38801 [15] Perrin Wright, Group presentations and formal deformations, Trans. Amer. Math. Soc. 208 (1975), 161 – 169. · Zbl 0318.57010 [16] Simon F. Young, Contractible $$2$$-complexes, Masters Thesis, Univ. of Cambridge, 1976.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.