software update Big Sur does not show in my software update. My Macbook air 2015 runs on El Capitan 10.11.6. When I try the App store I cannot download Big Sur.

See the following: How to upgrade to macOS Big Sur - Apple Support

You're still running a really old version.

Here's what I advise you to:

Make a clean install of Big Sur.

First backup your data to an external drive. Forget Time Machine. Go for good old Command C Command V. Set away the external drive to preserve your data.

Now create a bootable USB drive How to create a bootable installer for macOS - Apple Support

Start your Mac holding Option and choose the installer to load Recovery Mode

Open Disk Utility and completely erase the disk, volumes and partition.

Close DU and continue to install.

Configure it as new, allow all the updates and then copy back your data from the external drive.

WARNING: Since Catalina 32-bit apps don't run anymore. If you have any, check with the developer if it is updated to 64-bit or find an alternative.