On 2-dimensional<i>CW</i>-complexes with a single 2-cell

Jajodia, Sushil

doi:10.2140/pjm.1979.80.191

Cited by 4 publications

(4 citation statements)

References 9 publications

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“…The finite 2-complexes given in the proof of Theorem 9.6 are irreducible and so show that some further conditions must be imposed. This was shown to be true by Jajodia [27,Corollary 4] when the X i , Y i have a single 2-cell.…”

Section: By [31 Proposition 21]mentioning

confidence: 84%

The homotopy type of a finite 2-complex with non-minimal Euler characteristic

Nicholson¹

2021

Preprint

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We resolve two long-standing and closely related problems concerning stably free ZG-modules and the homotopy type of finite 2-complexes. In particular, for all k ≥ 1, we show that there exists a group G and a non-free stably free ZG-module of rank k. We use this to show that, for all k ≥ 0, there exists homotopically distinct finite 2-complexes with fundamental group G and with Euler characteristic k greater than the minimal value over G. This provides a solution to Problem D5 in the 1979 Problems List of C. T. C. Wall.

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Section: By [31 Proposition 21]mentioning

confidence: 84%

The homotopy type of a finite 2-complex with non-minimal Euler characteristic

Nicholson¹

2021

Preprint

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“…Therefore, the complement of the Trefoil Knot, shown in Figure 23, admits a handle decomposition with one 0-handle, two 1-handles and one 2-handle; see [23], exercise 6.2.2. Note that 2-dimensional CW-complexes with a unique 2-cell are classified by their fundamental group, up to homotopy equivalence; see [27].…”

Section: 31mentioning

confidence: 99%

The fundamental crossed module of the complement of a knotted surface

Martins

2009

Trans. Amer. Math. Soc.

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Abstract. We prove that if M is a CW-complex and M 1 is its 1-skeleton, then the crossed module Π 2 (M, M 1 ) depends only on the homotopy type of M as a space, up to free products, in the category of crossed modules, with Π 2 (D 2 , S 1 ). From this it follows that if G is a finite crossed module and M is finite, then the number of crossed module morphisms Π 2 (M, M 1 ) → G can be re-scaled to a homotopy invariant I G (M ), depending only on the algebraic 2-type of M . We describe an algorithm for calculating π 2 (M, M (1) ) as a crossed module over π 1 (M (1) ), in the case when M is the complement of a knotted surface Σ in S 4 and M (1) is the handlebody of a handle decomposition of M made from its 0-and 1-handles. Here, Σ is presented by a knot with bands. This in particular gives us a geometric method for calculating the algebraic 2-type of the complement of a knotted surface from a hyperbolic splitting of it. We prove in addition that the invariant I G yields a non-trivial invariant of knotted surfaces in S 4 with good properties with regard to explicit calculations.

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“…What is known is that if X is aspherical or if X has only one 2-cell (Jajodia [7]) then G(X) -AuU>,(A)), while Schellenberg [15] and Sieradski [18] have shown that G( X) may be a proper subgroup of Aut(7r,( A)) when ttx( X) is a noncyclic finite abelian group. Question 3.…”

Section: -S'(a)-s(a)-g(a)-mentioning

confidence: 99%