Trees of homotopy types of 2-dimensional $\textrm {CW}$ complexes. II
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- by Micheal N. Dyer and Allan J. Sieradski PDF
- Trans. Amer. Math. Soc. 205 (1975), 115-125 Request permission
Abstract:
A $\pi$-complex is a finite, connected $2$-dimensional CW complex with fundamental group $\pi$. The tree $\text {HT} (\pi )$ of homotopy types of $\pi$-complexes has width $\leq N$ if there is a root $Y$ of the tree such that, for any $\pi$-complex $X,X \vee ( \vee _{i = 1}^NS_i^2)$ lies on the stalk generated by $Y$. Let $\pi$ be a finite abelian group with torsion coefficients ${\tau _1}, \cdots ,{\tau _n}$. The main theorem of this paper asserts that width $\text {HT} (\pi ) \leq n(n - 1)/2$. This generalizes the results of [4].References
- Hyman Bass, Algebraic $K$-theory, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0249491
- H. Bass, $K$-theory and stable algebra, Inst. Hautes Études Sci. Publ. Math. 22 (1964), 5–60. MR 174604
- W. H. Cockcroft and R. G. Swan, On the homotopy type of certain two-dimensional complexes, Proc. London Math. Soc. (3) 11 (1961), 194–202. MR 126271, DOI 10.1112/plms/s3-11.1.193
- Micheal N. Dyer and Allan J. Sieradski, Trees of homotopy types of two-dimensional $\textrm {CW}$-complexes. I, Comment. Math. Helv. 48 (1973), 31–44; corrigendum, ibid. 48 (1973), 194. MR 377905, DOI 10.1007/BF02566109
- Ralph H. Fox, Free differential calculus. II. The isomorphism problem of groups, Ann. of Math. (2) 59 (1954), 196–210. MR 62125, DOI 10.2307/1969686
- Graham. Higman, The units of group-rings, Proc. London Math. Soc. (2) 46 (1940), 231–248. MR 2137, DOI 10.1112/plms/s2-46.1.231
- H. Jacobinski, Genera and decompositions of lattices over orders, Acta Math. 121 (1968), 1–29. MR 251063, DOI 10.1007/BF02391907 T. W. Lam, Induction theorems for Grothendieck groups and Whitehead groups of finite groups, Thesis, Columbia University, 1967.
- Wilhelm Magnus, Abraham Karrass, and Donald Solitar, Combinatorial group theory: Presentations of groups in terms of generators and relations, Interscience Publishers [John Wiley & Sons], New York-London-Sydney, 1966. MR 0207802
- Bruno Scarpellini, Comment on: “Against Mac Lane: ‘The health of mathematics’ by S. Mac Lane” [Math. Intelligencer 10 (1988), no. 3, 12–20; MR0948899 (89j:00043)] by C. Smoryński, Math. Intelligencer 11 (1989), no. 1, 3. MR 979015, DOI 10.1007/BF03023767
- Irving Reiner, A survey of integral representation theory, Bull. Amer. Math. Soc. 76 (1970), 159–227. MR 254092, DOI 10.1090/S0002-9904-1970-12441-7
- Richard G. Swan, $K$-theory of finite groups and orders, Lecture Notes in Mathematics, Vol. 149, Springer-Verlag, Berlin-New York, 1970. MR 0308195
- J. H. C. Whitehead, Combinatorial homotopy. II, Bull. Amer. Math. Soc. 55 (1949), 453–496. MR 30760, DOI 10.1090/S0002-9904-1949-09213-3 —, Simplicial spaces, nuclei, and $m$-groups, Proc. London Math. Soc. 45 (1939), 243-327.
- D. B. A. Epstein, Finite presentations of groups and $3$-manifolds, Quart. J. Math. Oxford Ser. (2) 12 (1961), 205–212. MR 144321, DOI 10.1093/qmath/12.1.205
- B. H. Neumann, On some finite groups with trivial multiplicator, Publ. Math. Debrecen 4 (1956), 190–194. MR 78997
Additional Information
- © Copyright 1975 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 205 (1975), 115-125
- MSC: Primary 55D15
- DOI: https://doi.org/10.1090/S0002-9947-1975-0425957-4
- MathSciNet review: 0425957