Secondary contributions to the anisotropy of the cosmic microwave background (CMB), such as the integrated Sachs–Wolfe (ISW) effect, the thermal Sunyaev–Zel’dovich (tSZ) effect, and the effect of gravitational lensing, have distinctive non-Gaussian signatures, and full descriptions therefore require information beyond that contained in their power spectra. The Minkowski Functionals (MF) are well-known as tools for quantifying any departure from Gaussianity and are affected by noise and other sources of confusion in a differentway from the usual methods based on higher-order moments or polyspectra, thus providing complementary tools for CMB analysis and cross-validation of results. In this paper we use the recently introduced skew-spectra associated with the MFs to probe the topology of CMB maps to probe the secondary non-Gaussianity as a function of beam smoothing in order to separate various contributions. We devise estimators for these spectra in the presence of realistic observational masks and present expressions for their covariance as a function of instrumental noise. Specific results are derived for the mixed ISW-lensing and tSZ-lensing bispectra as well as contamination due to point sources for noise levels that correspond to the Planck (143 GHz channel) and Experimental Probe of Inflationary Cosmology (EPIC; 150 GHz channel) experiments. The cumulative signal-to-noise ratio (S/N) for one-point generalized skewness parameters can reach an order of O(10) for Planck and two orders of magnitude higher forEPIC, i.e. O(103). We also find that these three skew-spectra are correlated, having correlation coefficients r ∼ 0.5–1.0; higher l modes are more strongly correlated. Although the values of S/N increase with decreasing noise, the triplets of skew-spectra that determine the MFs become more correlated; the S/N of lensing-induced skew-spectra are smaller compared to that of a frequency-cleaned tSZ map
