Virtual domination of 3-manifolds, III

Virtual domination of 3-manifolds, III HONGBIN SUNWe prove that for any oriented cusped hyperbolic 3-manifold M and any compact oriented 3-manifold N with tori boundary, there exists a finite cover M 0 of M that admits a degree-8 map f W M 0 !N , ie M virtually 8-dominates N .57K32, 57M10; 30F40 IntroductionWe assume all manifolds are compact, connected and oriented, unless otherwise indicated.By a cusped hyperbolic 3-manifold, we mean a compact 3-manifold with nonempty tori boundary, such that its interior admits a complete hyperbolic structure with finite volume, unless otherwise indicated.For two closed oriented n-manifolds M and N , and a map f W M !N , a natural quantity associated to f is its mapping degree.The mapping degree of f is d 2 Z if f .OEM / D dOEN for oriented fundamental classes OEM 2 H n .M I Z/ and OEN 2 H n .N I Z/.The notion of mapping degree can be generalized to proper maps between manifolds with boundary.For two compact oriented n-manifolds M and N with boundary, a map f W M !N is proper if f 1 .@N/ D @M .The mapping degree of a proper map f W M !N is d 2 Z if f .OEM; @M / D dOEN; @N for oriented relative fundamental classes OEM; @M 2 H n