Representing nonnegative homology classes of ℂℙ²#𝕟\overline{ℂℙ}² by minimal genus smooth embeddings

Abstract

Abstract. For any nonnegative class ξ in H 2 (CP 2 #nCP 2 , Z), the minimal genus of smoothly embedded surfaces which represent ξ is given for n ≤ 9, and in some cases with n ≥ 10, the minimal genus is also given. For the finiteness of orbits under diffeomorphisms with minimal genus g, we prove that it is true for n ≤ 8 with g ≥ 1 and for n ≤ 9 with g ≥ 2.

Minimal genus for 4-manifolds withb+ = 1

“…This is straightforward to see using the description of Aut(Γ) T above for Cases (1), ( 3), (5). For Case (4) this was established by [11], Proposition 1 (This is still true for arbitrary n by [11] and [31]). It remains to deal with Case (2).…”

“…The equality for CP 2 ♯nCP 2 , n ≤ 9 was due to Wall [29] Theorem 2 and the following corollary, and for Enriques surface due to Lönne [18] Theorem 8 (extending Friedman-Morgan [2]). The minimal genus function mg was computed in [13], [25] for S 2 × S 2 , in [12] for S 2 -bundles over T 2 , and in [13], [11] for CP 2 ♯nCP 2 , n ≤ 9. For the Enriques surface, the generalized Thom conjecture in [21] implies that the genus bound ( 6) is valid for any class with non-negative square.…”

A note on the complexity of h–cobordisms

“…Many related (and overlapping) families of homology classes of rational surfaces have been shown to have arbitrarily high complexity; Lawson's survey [19] provides a detailed summary. Most results give lower bounds for the minimum number of positive double points (see Fintushel and Stern [6, Theorem 1.2]) or minimal genus (such as [9], [20], [29]). As mentioned in the introduction, Corollary 1.2 is implied by Ruberman [29] when each Y n is diffeomorphic to CP 2 # nCP 2 , and k ≥ 0.…”

A note on the complexity of h-cobordisms

“…Many related (and overlapping) families of homology classes of rational surfaces have been shown to have arbitrarily high complexity; Lawson's survey [17] provides a detailed summary. Most results in fact give lower bounds for the minimum number of positive double points (see Fintushel and Stern [7, Theorem 1.2]) or minimal genus (such as [10], [18], [25]).…”