Concordance maps in knot Floer homology

We show that a decorated knot concordance $C$ from $K$ to $K^{\mathrm{\prime }}$ induces a homomorphism $F_C$ on knot Floer homology that preserves the Alexander and Maslov gradings. Furthermore, it induces a morphism of the spectral sequences to $\widehat{HF}(S^3)\cong {\mathbb{Z}}_2$ that agrees with $F_C$ on the $E^1$ page and is the identity on the $E^{\mathrm{\infty }}$ page. It follows that $F_C$ is non-vanishing on ${\widehat{HFK}}_0(K,\tau (K))$. We also obtain an invariant of slice disks in homology 4-balls bounding $S^3$. If $C$ is invertible, then $F_C$ is injective, hence $\dim \widehat{HFK}_j(K,i) \le \dim \widehat{HFK}_j(K’,i)$forevery$i$,$j \in $\mathbb{Z}$$. This implies an unpublished result of Ruberman that if there is an invertible concordance from the knot $K$ to $K^{\mathrm{\prime }}$, then $g(K)\le g(K^{\mathrm{\prime }})$, where $g$ denotes the Seifert genus. Furthermore, if $g(K)=g(K^{\mathrm{\prime }})$ and $K^{\mathrm{\prime }}$ is fibred, then so is $K$.