($a$) the foreground-subtracted WMAP Q, V, and W band maps with angular resolution of $0\fdg60$, $0\fdg45$, and $0\fdg35$ FWHM, respectively, and from ($b$) ILC ($\simeq 1\degr$), TCM, and Wiener-filtered TCM (WF, $0\fdg35$). Before the genus is measured, all maps are stereographically projected. The Kp0 mask is applied. Curves in each panel show the functional forms expected for a random-phase Gaussian field, $A \nu e^{-\nu^2 /2}$, and have been fitted to the measured genus points by adjusting $A$.](jfig1.eps){width="84mm"}

![Average genus per steradian measured from 500 WMAP Q (triangles), V (open circles), and W band (filled circles) mock observations. The error bars are given by the standard deviations of the genus values estimated from the mock observations, and the fitting curves are the functional form expected for a random-phase Gaussian field (see the caption of Figure 1).](jfig2.eps){width="84mm"}

We present the genus curves as a function of the area fraction threshold level $\nu_A$. The $\nu_A$ is defined to be the temperature threshold level at which the corresponding iso-temperature contours encloses a fraction of the survey area equal to that at the temperature threshold level $\nu_A$ for a Gaussian field $$f_A = {1 \over {(2\pi)^{1/2}}} \int_{\nu_A}^{\infty} e^{-x^2 /2} dx .$$ The $\nu_A=0$ level corresponds 