Concordance maps in knot Floer homology

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We show that a decorated knot concordance C from K to K' induces a homomorphism FC on knot Floer homology that preserves the Alexander and Maslov gradings. Furthermore, it induces a morphism of the spectral sequences to HF(S³) Z₂ that agrees with FC on the E¹ page and is the identity on the E∞ page. It follows that FC is non-vanishing on HFK₀(K, τ(K)). We also obtain an invariant of slice disks in homology 4-balls bounding S³. If C is invertible, then FC is injective, hence HFKⱼ(K,i) ≤ HFKⱼ(K',i) for every i, j ∈ Z. This implies an unpublished result of Ruberman that if there is an invertible concordance from the knot K to K', then g(K) ≤ g(K'), where g denotes the Seifert genus. Furthermore, if g(K) = g(K') and K' is fibred, then so is K.