# Question: If M is a k dimensional manifold

If M is a k-dimensional manifold with boundary, prove that ∂M is a (k - 1) -dimensional manifold and M - ∂M is a k=dimensional manifold.

## Answer to relevant Questions

Find a counter-example to Theorem 5-2 if condition (3) is omitted. Following the hint, consider f: (- 2π, 2π) →R2 defined byLet Kn = {xЄRn: x1 = 0 and x2. . . x n−1 > 0}. If MCKn is a k-dimensional manifold and N is obtained by revolving M around the axis x1 = . = xn-1=0, show that N is a (k + 1) -dimensional manifold. Example: the ...If M C R n is an orientable (n - 1)-dimensional manifold, show that there is an open set A C Rn and a differentiable g: A→ R1 so that M = g-1 (0) and g1 (x) has rank 1 for x ЄM.If M1CRN is an -dimensional manifold-with-boundary and M 2 C M1 - ∂M1 is an -dimensional manifold with boundary, and M1, M2 are compact, prove thatLet f, g: [0, 1] → R3 be nonintersecting closed curves. Define the linking number l (f, g) of and g by (cf. Problem 4-34Post your question