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...ned from the chain complex in the totally twisted case CF ◦ (Y, t) by a change of coefficients; thus, the corresponding homology groups are related by a universal coefficients spectral sequence (c.f. =-=[1]-=-) . In particular, when M is the trivial Z[H 1 (Y ; Z)]-module Z, the M-twisted chain complex is the same as the untwisted chain complex stated earlier. Moreover, it is easy to see that Tor i Z[H 1 (Y...

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...F K,s : HF ◦ (Y, s, M) −→ HF ◦ (Y, s, M) Proof. The proof follows from “turning around” the proof of Lemma 4.16. Proof of Theorem 3.8 Consider the Kirby calculus picture for the cobordism W (see [5], =-=[4]-=-). Any two such pictures can be connected by a sequence of pair cancellations and additions, and a sequence of handleslides (Kirby moves). In fact, since W is a connected, manifold-with-boundary, W ha...

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...2001 PETER OZSVÁTH AND ZOLTÁN SZABÓ Abstract. The aim of this article is to introduce invariants of oriented, smooth, closed four-manifolds, built using the Floer homology theories defined in [8] and =-=[12]-=-. This four-dimensional theory also endows the corresponding three-dimensional theories with additional structure: an absolute grading of certain of its Floer homology groups. The cornerstone of these...

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...polynomials and Seiberg-Witten invariants. We return to calculations of Φ in [11], and its non-vanishing properties for symplectic manifolds, after calculating HF + for a number of three-manifolds in =-=[10]-=-. These calculations are based on the surgery long exact sequences from [12], combined with the absolute Q grading defined in the present article. We content ourselves here with some general propertie...

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...f which have natural analogues in Seiberg-Witten theory. 1.4. Basic properties of the closed invariant. The following is an analogue of Donaldson’s connected sum theorem for his polynomial invariants =-=[2]-=-. Unlike its gaugetheoretic counterpart, the result follows rather directly from the definition of the invariants. Theorem 1.3. Let X1 and X2 be a pair of smooth, oriented four-manifolds with b + 2 (X...

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...,s ◦ F K,s : HF ◦ (Y, s, M) −→ HF ◦ (Y, s, M) Proof. The proof follows from “turning around” the proof of Lemma 4.16. Proof of Theorem 3.8 Consider the Kirby calculus picture for the cobordism W (see =-=[5]-=-, [4]). Any two such pictures can be connected by a sequence of pair cancellations and additions, and a sequence of handleslides (Kirby moves). In fact, since W is a connected, manifold-with-boundary,...

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...orm in terms the first Chern classes of Spin c structures for which HF + is non-trivial. These “adjunction inequalities” have a straightforward generalization in the four-dimensional context (compare =-=[6]-=-, [7], [9]). Theorem 1.5. Let Σ ⊂ X be a homologically non-trivial embedded surface with genus g ≥ 1 and with non-negative self-intersection number. Then, for each Spin c structure s ∈ Spin c (X) for ...

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...n terms the first Chern classes of Spin c structures for which HF + is non-trivial. These “adjunction inequalities” have a straightforward generalization in the four-dimensional context (compare [6], =-=[7]-=-, [9]). Theorem 1.5. Let Σ ⊂ X be a homologically non-trivial embedded surface with genus g ≥ 1 and with non-negative self-intersection number. Then, for each Spin c structure s ∈ Spin c (X) for which...

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...r the connected sum X = X1#X2 vanish identically, for all Spin c structures. The blow-up formula for the cobordism invariant translates directly into a corresponding blow-up formula for ΦX,s (compare =-=[3]-=-): Theorem 1.4. Let X be a closed, smooth, four-manifold with b + 2 (X) > 1, and let ̂ X = X#CP 2 be its blowup. Then, for each Spin c structure ̂s ∈ Spin c ( ̂ X), with d( ̂ X,̂s) ≥ 0 we have the rel...

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... 18 Oct 2001 PETER OZSVÁTH AND ZOLTÁN SZABÓ Abstract. The aim of this article is to introduce invariants of oriented, smooth, closed four-manifolds, built using the Floer homology theories defined in =-=[8]-=- and [12]. This four-dimensional theory also endows the corresponding three-dimensional theories with additional structure: an absolute grading of certain of its Floer homology groups. The cornerstone...

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...theory free” proofs of some facts about smooth four-manifolds which have been previously established by means of Donaldson polynomials and Seiberg-Witten invariants. We return to calculations of Φ in =-=[11]-=-, and its non-vanishing properties for symplectic manifolds, after calculating HF + for a number of three-manifolds in [10]. These calculations are based on the surgery long exact sequences from [12],...