2012/02/18 by Nakamura, Kei
#57M27 (Primary) 57N12 #57M50 (Secondary) #57N10 #FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.1202.4062
Let M be an orientable closed connected 3-manifold. We introduce the notion of amalgamated Heegaard genus of M with respect to a closed separating 2-manifold F, and use it to show that the following two statements are equivalent: (i) a compact connected 3-manifold Y can be embedded in M so that the exterior of the image of Y is a union of handlebodies; and (ii) a compact connected 3-manifold Y can be embedded in M so that every knot in M can be isotoped to lie within the image of Y . Our result can be regarded as a common generalization of the reimbedding theorem by Fox [Fox48] and the characterization of 3-sphere by Bing [Bin58], as well as more recent results of Hass and Thompson [HT89] and Kobayashi and Nishi [KN94].