env is not really a program to use, but one to get infos from.
Actualy, the variables there show either what you have set in the shell, or have been loaded to the shell (by sourcing or exporting -- but thats, for the moment, beyond your needs).

As you dont it for yourself, why do it at all?

As of now, i might have one my moments, but i'm getting sick by people who cant even accomplish a simple test by their own.
Its nothing difficult, just DO IT and SEE what happens- that is called 'a learning process'... learning by doing...

To be mean, dont attend the accelerated class, but go to the basic class if you cant handle this (i'm even stunned this is called accelerated).
Heck, i'm not allowed in advanced classes because i dont have a 'basic' class visisted... reading this makes me sick...

And to be 'teachfull' nontheless... cat prints out all the content of the passed file.
Which in this case is the variable $filename, which you can set to ANYTHNG you like... like filename="$1"
The $( CODE ) around that command, executes the CODE and prints/execute it (depends on the rest of the line).
Do the above exmaple i told you, and you WILL understand... - if not, read again my 'mean' part...

EDIT2:
I wrote the 'mean' part as i didnt feel you were even following vbe's suggestions...

EDIT3:
Sorry if i was too harsch, might sound harder since i have to translate it to a foreign language...
I just wanted to make you do (show) more 'effort' of your own.

EDIT4:
Sorry didnt see you've edited your post...
You tried:

Code:

pwd
var=$ ( pwd )
if var = $HOME
then
echo you are in the home directory
else
echo you are not in the home directory

Lines:
1) pwd is a command and is executed
2) tries to set var to the output of pwd, fix the space issue, and it'll work
3) the condition misses encapsuling [ and ], however i suggest to (get used to) use [[ and ]]
4) good
5) encapsule the STRING by quotes ", recomended, eventhough in this example it'll work this way...
6) good
7) see 5)
8) missing closing fi

Fix these issues and see what happens