Manifolds with Poly-Surface Fundamental Groups

On minimal Poincaré $4$-complexes

“…(one applies among others: π 3 (D ′ ) = π 3 (D) = 0 and the Hurewicz theorem). The obstruction in Z ⊗ Λ π 3 (P ′ ) is given by the image of [h (3) • ϕ] ∈ π 3 (K ′ ) under the composite map…”

“…In the example F × S 2 there are two models, namely, F × S 2 and the non-trivial S 2 -bundle E → F with second Stiefel-Whitney class = 0 (see, for example, [3], Appendix). Here it is also convenient to consider the map respectively (see [3]). It is shown in [9], Section 5, that F × S 2 and E are the only models up to homotopy equivalence.…”

Homotopy self-equivalences of 4-manifolds with PD2 fundamental group

“…Hence the above differential is surjective onto the H 3 (B; Z/2). Therefore, on the line p + q = 4, the groups which survive to E ∞ are Z in (0, 4 …”

“…We combine the approach of [15], involving bordism techniques and the modified surgery theory of Kreck [18], with the minimal models of Hillman [12] to obtain the following result (we use the notion of w 2 -type, which is recalled in Definition 4.1 below). Theorem B is also stated in [4] (see also [10]), but our proof is primarily based on understanding homotopy selfequivalences, and this technique may have other applications. Note that E • (B) is not path connected, in fact π 0 (E • (B)) = Aut • (B).…”

On fibering and splitting of 5-manifolds over the circle

“…By a result of J. Hillman [Hil91, Lemma 6] for closed, aspherical surface bundles over surfaces, condition (iii) is satisfied. Indeed, the Mayer-Vietoris argument extends to compact, aspherical surfaces with boundary: each circle C j in the connected-decomposition of the aspherical surface Σ b i = F 1 # · · · #F r generates an indivisible element in the free fundamental group of the many-punctured torus or Klein bottle F k , hence each inclusion π 1 (C j ) → π 1 (F k ) of fundamental groups is square-root closed (see [CHS06,Thm. 2.4] for detail).…”