Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Simple homotopy type of finite $ 2$-complexes with finite abelian fundamental group


Author: M. Paul Latiolais
Journal: Trans. Amer. Math. Soc. 293 (1986), 655-662
MSC: Primary 57M20; Secondary 57Q10
DOI: https://doi.org/10.1090/S0002-9947-1986-0816317-X
MathSciNet review: 816317
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Theorem 1. Let $ K$ be a finite $ 2$-dimensional $ CW$-complex with $ {\pi _1}(K)$ finite and abelian. Then every element of the Whitehead group of $ K$ is realizable as the torsion of a self-homotopy equivalence on $ K$.

Theorem 2. Homotopy equivalence and simple homotopy equivalence are the same for finite $ 2$-dimensional $ CW$-complexes with finite abelian fundamental groups.


References [Enhancements On Off] (What's this?)

  • [B] H. Bass, Algebraic $ K$-theory, Benjamin, New York, 1968. MR 0249491 (40:2736)
  • [Br] W. Browning, Finite $ CW$-complexes of cohomological dimension $ 2$ with finite abelian $ {\pi _1}$ (unpublished), Forschungsinstitut fur Mathematik, E.T. H., CH-8092, Zurich, Switzerland.
  • [C] M. Cohen, A course in simple-homotopy theory, Springer-Verlag, 1970. MR 0362320 (50:14762)
  • [D2] M. Dyer, Simple homotopy types for $ (G - m)$-complexes, Proc. Amer. Math. Soc. 81 (1981), 111-115. MR 589149 (81k:57014)
  • [D3] -, An application of homological algebra to the homotopy classification of two-dimensional $ CW$-complexes, Trans. Amer. Math. Soc. 259 (1980), 505-514. MR 567093 (81g:55009)
  • [DS] M. Dyer and A. Sieradski, Trees of homotopy types of two-dimensional $ CW$-complexes, Comment. Math. Helv. 48 (1973), 31-44. MR 0377905 (51:14074)
  • [F] R. Fox, Free differential calculus. I, Ann. of Math. (2) 57 (1953), 547-559. MR 0053938 (14:843d)
  • [L] T.-Y. Lam, Induction theorems for Grothendieck groups and Whitehead groups of finite groups, Ann. Sci. Ecole. Norm. Sup. 1 (1968), 91-148. MR 0231890 (38:217)
  • [M1] W. Metzler, Über den Homotopietyp zweidimensionaler $ CW$-Komplexe und Elementarttransformationen bei Darstellungen von Gruppen durch Erzeugende und definierende Relationen, J. Reine Angew. Math. 285 (1976), 7-23. MR 0440527 (55:13402)
  • [M2] -, Two-dimensional complexes with torsion values not realizable by self-equivalences, Homological Group Theory, London Math. Soc. Lecture Notes Series 36, Cambridge Univ. Press, 1977, pp. 327-337. MR 564435 (81h:55003)
  • [P] D. Puppe, Homotopiemengen und Ihre Induzieten Abbildungen. I, Math. Z. 69 (1958), 299-344. MR 0100265 (20:6698)
  • [S] A. Sieradski, A semigroup of simple homotopy types, Math. Z. 153 (1977), 134-148. MR 0438321 (55:11237)
  • [Sw] R. Swan, Minimal resolutions for finite groups, Topology 4 (1965), 193-208. MR 0179234 (31:3482)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 57M20, 57Q10

Retrieve articles in all journals with MSC: 57M20, 57Q10


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1986-0816317-X
Article copyright: © Copyright 1986 American Mathematical Society

American Mathematical Society