Is it possible to move 'itemAttachmentsBlock' (this is the download attachment place) more upwards in the ItemLinks table? For example, before the div class 'itemCategory?I searched for the solution, and played in some core .php files, but unfortunately nothing...Thanks!

Read about templating here:

community.getk2.org/notes/Templating_in_K2
and
simon.getk2.org/component/k2/item/91-templating
Set up a k2 template and edit the file item.php and move the block with attachments to the top (for item view)