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Suppose I have a compact 3-dimensional submanifold N of S X (0,1) which has one boundary component, where S is a closed surface. Must N be a handlebody?

Suppose I have a compact submanifold N of S X (0,1) which has one boundary component, where S is a closed surface. Must N be a handlebody?

Suppose I have a compact 3-dimensional submanifold N of S X (0,1) which has one boundary component, where S is a closed surface. Must N be a handlebody?

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Are compact submanifolds of "S X (0,1)" with one boundary component handlebodies, where S is a closed surface?

Suppose I have a compact submanifold N of S X (0,1) which has one boundary component, where S is a closed surface. Must N be a handlebody?