On the homotopy type of CW-complexes with aspherical fundamental group

Partial Euler characteristic, normal generations and the stable $D(2)$ problem

“…Since H n+1 (X) = 0,X is contractible and X is homotopy equivalent to BG. [6] already prove that a (G, n)-complex realizing µ g n (G) is homotopy equivalent to BG. Note that a (G, n)-complex is a special case of X in Theorem 4.3.…”

Two remarks on Wall's D2 problem

“…If G is a polycyclic-by-finite group, the group ring ZG is again noetherian and d G = h G + 1, where h G denotes the Hirsch length of G (see [27, 6•6•1]). The examples of [9,15,16,17] show that for general infinite fundamental groups (for example, the fundamental group of the trefoil knot), there can be (infinitely) many distinct 2-complexes with the same Euler characteristic.…”

Homology Cylinders and Sutured Manifolds for Homologically Fibered Knots

“…Given any admissible presentation P of π 1 (M) as in (4.2), we construct a 2-complex X(P ) having one 0-cell as a basepoint, (4g+2n−2+l) 1-cells indexed by the generators and (2g+n−1+l) 2-cells indexed by the relations and attached according to the words. Then we can use a theorem of Harlander-Jensen [22,Theorem 3] with the fact that the deficiency of π 1 (M) is 2g + n − 1 (see Epstein [5,Lemmas 1.2,2,2]) to show that X(P ) and M ′ are homotopy equivalent. In fact, there exists a basepoint preserving cellular map f : X(P ) → M ′ which is a homotopy equivalence and maps the union S 0 of the 1-cells of P 0 corresponding to i + (γ 1 ), .…”