# Confusion with exterior algebra

I have a graded vector space $$V$$ such that $$V=V^2\bigoplus V^3$$, i.e $$V^k=0\quad \forall k\neq2,3.$$

Now I can't understand what is $$\bigwedge V^{\leq 2}$$.

Is it $$\bigwedge V^{\leq 2}=\bigwedge^0 V^{\leq 2}\bigoplus \bigwedge^1 V^{\leq 2}\bigoplus...$$ with $$V^{\leq 2}=V^2$$?

Also what is $$(\bigwedge V^{\leq 2})^2$$?

Thanks!

• Whatever it is, it's horrible notation. Where does this come from? – Lord Shark the Unknown Jul 7 at 6:25
• From the book Rational Homotopy Theory – sasho98 Jul 7 at 6:26
• I don't know that book ... does it have authors? – Lord Shark the Unknown Jul 7 at 6:26
• Felix-Halperin-Thomas – sasho98 Jul 7 at 6:28
• Can someone provide a pdf of the book Exterior algebras:elementary tribute to Grassman's ideas – sasho98 Jul 8 at 9:18