Sentence:  <WSD>Cap off</WSD>, with discs just above the annulus, any circuits that bound in the annulus; then use annuli to <WSD>cap off</WSD> adjacent pairs of curves that encircle the annulus in opposite directions. 
Meaning of the word marked with <WSD> tags:  To glue a relatively uncomplicated compact manifold to a manifold of the same dimension along (a component of the latter's boundary).