  Determination and interpretation of Wigner-like photoionisation delays is one
of the most active fields of attosecond science. Previous results have
suggested that large photoionisation delays are associated with structured
continua, but a quantitative relation between photoionisation cross sections
and time delays has been missing. Here, we derive a Kramers-Kronig-like
relation between these quantities and demonstrate its validity for
(anti)resonances. This new concept defines a topological analysis, which
rationalises the sign of photoionisation delays and thereby sheds new light on
a long-standing controversy regarding the sign of the photoionisation delay
near the Ar 3s Cooper minimum. Our work bridges traditional photoionisation
spectroscopy with attosecond chronoscopy and offers new methods for analysing
and interpreting photoionisation delays.
