Whitney immersion theorem

In differential topology, the Whitney immersion theorem (named after Hassler Whitney) states that for m > 1 {\displaystyle m>1} , any smooth m {\displaystyle m} -dimensional manifold (required also to be Hausdorff and second-countable) has a one-to-one immersion in Euclidean 2 m {\displaystyle 2m} -space, and a (not necessarily one-to-one) immersion in ( 2 m − 1 ) {\displaystyle (2m-1)} -space. Similarly, every smooth m {\displaystyle m} -dimensional manifold can be immersed in the 2 m − 1 {\displaystyle 2m-1} -dimensional sphere (this removes the m > 1 {\displaystyle m>1} constraint). The weak version, for 2 m + 1 {\displaystyle 2m+1} , is due to transversality (general position, dimension counting): two m-dimensional manifolds in R 2 m {\displaystyle \mathbf {R} ^{2m}} intersect generically in a 0-dimensional space.

Further results William S. Massey (Massey 1960) went on to prove that every n-dimensional manifold is cobordant to a manifold that immerses in S 2 n − a ( n ) {\displaystyle S^{2n-a(n)}} where a ( n ) {\displaystyle a(n)} is the number of 1's that appear in the binary expansion of n {\displaystyle n} . (The corresponding dimensions are listed in OEIS: A005187). In the same paper, Massey proved that for every n there is manifold (which happens to be a product of real projective spaces) that does not immerse in S 2 n − 1 − a ( n ) {\displaystyle S^{2n-1-a(n)}} . The conjecture that every n-manifold immerses in S 2 n − a ( n ) {\displaystyle S^{2n-a(n)}} became known as the immersion conjecture. This conjecture was eventually solved in the affirmative by Ralph Cohen (1985).

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