/sci/ - Science & Math

>>12163796
>Is it required that the ring is non-empty in every degree?
Non-empty, yes, but not non-trivial. Take the cohomology ring [math]H^*(\mathbb{Z} /p\mathbb{Z}; \mathbb{Z})[/math] for your favourite prime [math]p[/math]. Here, [math]H^n(\mathbb{Z} /p\mathbb{Z}; \mathbb{Z})=0[/math] for [math]n[/math] odd, [math]H^n(\mathbb{Z} /p\mathbb{Z}; \mathbb{Z}) \cong \mathbb{Z}/p \mathbb{Z}[/math] for [math]n>0[/math], and [math]H^0(\mathbb{Z} /p\mathbb{Z}; \mathbb{Z}) \cong \mathbb{Z}[/math]. It is a graded ring.

>>12150355
Don't do it like that. Notice that your end points are not fixed, contract both paths to a constant path and then connect those points with a path.

>>12150322
<3