The identity manager needs to do a POST to login to access your webmap. Since you're hitting arcgis.com, this is a cross domain request so you need to do this via a proxy. Have you set up a proxy for your app?
Looking at the docs, this isn't clear. We'll update our documentation to say you need to use a proxy if you want to use a private webmap in an app you host yourself (if you havehaven't guessed, I work on the JS API team).