Khovanov Homology and Link Cobordisms

Abstract: Khovanov homology associates to any link in three-dimensional space a bigraded chain complex of Z[c]-modules, whose graded Euler characteristic (after normalization) is the Jones polynomial of the link. The complex is invariant up to homotopy equivalence under ambient isotopy of the link. Hence, the isomorphism class of the corresponding homology module is a link invariant. If the indeterminate c is set to zero, the resulting chain complex of abelian groups is finite, and the graded Euler characteristic is still the Jones polynomial. In this thesis, we prove that these latter homology groups are functorial under link cobordism. More precisely, every link cobordism induces a group homomorphism between the homology groups of its boundary links, invariant (up to sign) under any ambient isotopy which leaves the boundary links set-wise fixed. This proves (with a necessary modification) part of a conjecture by Khovanov (Duke Math. J. 101 (1999)). We then introduce some derived invariants of link cobordisms from a link to itself. These are graded Lefschetz numbers of the induced homomorphisms above. They can be computed directly on the chain level. The Jones polynomial appears as the graded Lefschetz number of the identity cobordism. We further prove that the statement of functoriality for homology with coefficients in Z[c] is false. This disproves the rest of the above-mentioned conjecture. Finally, we provide an elementary description of the Khovanov homology of tangles.