Geography and Botany of Irreducible Non-Spin Symplectic 4-Manifolds With Abelian Fundamental Group

Abstract

Abstract. The geography and botany problems of irreducible non-spin symplectic 4-rnanifolds with a choice of fundamental group from {lp . "llp EB "ll.q, l, Z EB Zp, ZEB Z} are studied by building upon the recent progress obtained on the simply con nected realm. R esults on the botany of simply connected 4-manifolds not available in the literature are extended .2010 Mathematics Subject Classification. Primary 57R 17; Secondary 57M05, 54D051. Introduction. Topologists' understanding of smooth 4-manifolds has witnessed a drastic improvement during the last 20 years. Advances in symplectic geometry [8,25,31,35,[65][66][67]71] and inventions of gluing formulas [39,46] for diff eomorphism invariants (73] have paired elegantly with topological constructions [16,18,[19][20][21], offering a noteworthy insight into the smooth four-dimensional story. The most recent series of successes [2-4, 6, 11, 16, 21 ] has answered as many questions as it has raised new ones, and the 4-manifolds theory remains to be an intriguing and active area of research.Two factors responsible for the recent progress on the simply connected realm are the increase in the repertoire of techniques that manufacture symplectic 4-manifolds with sm all topological invariants, and a new perspective on the usage of already existing mechanisms of construction. The idea of using symplectic sums (25) of non-simply connected building blocks along genus 2 surfaces to kill fundamenta l groups in an efficient way was introduced in (1). Its immediate outcome was the

On entropies, $${\mathcal {F}}$$ F -structures, and scalar curvature of certain involutions

“…If ω 2 (X n ) = 0, the homeomorphism type is (2m 1 + 1)CP 2 #(2m 1 + 2)CP 2 (59) for m 1 ∈ N. If ω 2 (X n ) = 0, the homeomorphism type is…”

“…Infinite sets of pairwise non-diffeomorphic symplectic manifolds with fundamental group Z/ p and Z/ p ⊕ Z/q for every Euler characteristic and signature of the homeomorphic but not diffeomorphic manifolds described in the proof of Theorem 1.6 were constructed in [2,3,59]. Using the pullback of the symplectic form, one concludes that their universal covers also admit a symplectic structure.…”

Families of 4-Manifolds with Nontrivial Stable Cohomotopy Seiberg–Witten Invariants, and Normalized Ricci Flow

“…Moreover, the small manifolds constructed by Akhmedov and Park in [4] leads to a slight enlargement of this region spanned by the minimal symplectic 4-manifolds. These manifolds can also be used to enlarge our families obtained in Theorem B-a similar discussion can be found in [52]. Nevertheless, we are content with the vast families we have got for the applications that will follow in the next chapter, and therefore will not discuss these slight extensions here.…”

“…Notice that X α,β is BF-admissible under (50). Since X is also BF-admissible, under (52), there is no quasi-non-singular solution to the normalized Ricci flow on M for any initial metric by Corollary 22. Moreover, we also obtainλ(M) < 0 by Corollary 26.…”

“…Note that these symplectic 4-manifolds are not brand new; they are produced using the families of [1], and were studied in [52]. The new key observation is that, under the mild condition a + b ≡ 0 (mod 8), they are all BF-admissible.…”