distribution of genus shift and asymmetry parameters drawn from the 500 WMAP W band Gaussian mock observations. The shift and asymmetry parameters well follow the Gaussian distribution. Comparing the observed genus-related statistics with those for mock observations, we conclude that the WMAP temperature fluctuation is consistent with the Gaussian field. This confirms the recent results of Komatsu et al. (2003) and Colley & Gott (2003).

![Histograms of genus ($a$) shifts $\Delta\nu_{-}$ (solid), $\Delta\nu_{+}$ (dotted lines) and ($b$) asymmetry ($\Delta g$) parameters drawn from the 500 WMAP W band Gaussian mock observations. Gaussian functions with standard deviations given in Table 2 are used to fit the histograms by adjusting the amplitudes. ](jfig3.eps){width="84mm"}

The theoretical genus curve expected from the WMAP observation can be obtained by (Park et al. 1998) $$g(\nu) = { 1 \over 2(2\pi)^{3/2}} 
            {{\sum \ell(\ell+1)(2\ell+1) 
        \left[ C_\ell W_\ell^2 + N_\ell \right] F_\ell^2 } \over 
        {\sum (2\ell+1) 
        \left[ C_\ell W_\ell^2 + N_\ell \right] F_\ell^2 }} 
        \nu e^{-\nu^2 / 2},$$ where $C_\ell$ and $N_\ell$ are WMAP CMB and noise power spectra, $W_\ell = B_\ell P_\ell$ is the window function that describes the combined smoothing effects of the beam ($B_\ell$) and the finite sky map pixel size ($P_\ell$), and $F_\ell$ is the 