How to publish a photo sphere to Google Maps ?

You won't. It is not an automated process, there someone (or something?) reviewing the submissions and enable to be shown the Google Map. My submission took about 2..3 days to appear.

How to upload a Photosphere to Google Map View

Since Google is retiring the panoramio service, is worth to look at the new platform Google Maps View. Of course you are required to create a Google account but that is fairly simple process. I did not manage to upload a photo sphere successfully straight away so I summarize the steps here.

It is not enough to create a equirectangular image and upload as photo under the +.

A panoramic image has to deliver specific Photo Sphere XMP Metadata in order to be recognized by Google as a Photo Sphere. You can either do it manually by tinkering the Exif info of the file with a tool like ExifTool or use the easier way by using this Google App. This is a detour which is not straight forward (unlike other Google tools), I guess they will migrate this feature into the upload process sooner or later.

After this step your photo sphere has not yet been put up for Google Maps. The app only adds the basic meta data to your image. Click on the download to get back the image file. You have to download this image and add the image. Now it will be recognized as photo sphere and properly rendered. Note, it takes a while before it actually pops up on Google Maps (not only in your profile). 

Panoramio shutting down

The image platform showing about 60+ million of images (and spheres), collected since 2005 by thousands of photographers and acquired by Google in 2007 is going to be closed down. 

As reason Google gave is to merge with the View platform, they announced the geo-relevant content would be migrated.

I never uploaded any image to the platform but it was always present in Google Maps, often interesting for places without Street View Coverage. But I have to say the panoramas, they call it spheres, are often of low quality, badly stitched or exposed.

There is a petition going (initiated by the original founders of panoramio), though I doubt Google will listen to it, is a corporate decision, it does not make sense to have competing platforms under the same roof.

This should be a learning lesson about relying on services provided for free, you simply can't assume your work won't be lost or the service available for ever.
In this case Google will certainly try to migrate all content, so you did upload images for many ears in vain.
But there are other cases where services are ceased because they company goes down or the service as such was not successful or profitable enough. Google Orkut is also shutting down. (Discontinued Google Services at Wikipedia)

Wikipedia about panoramio.