Nonsubmersion examples for compact manifolds On page 129 of lectnotes pdf we noted that regular homotopy classes of smooth submersions from a noncompact connected manifold M m to another manifold N n where m n are in 1 1 correspondence with homotopy classes of continuous tangent bundle morphisms F T M T N satisfying the following conditions i There is a continuous map f M N such that N o F f o M ii For each x M the induced map Fx from Tx M to Tf x N is a linear surjection Furthermore we mentioned that this classification does not extend to submersions defined on compact manifolds and we described a class of pairs F f such that properties i and ii hold but there is no smooth submersion g M N such that f is homotopic to g Our purpose here is to describe some similar but simpler examples for which we can give a self contained proof that the underlying map f M N is not homotopic to a smooth submersion in fact in these examples f is not even homotopic to a topological submersion and it is also possible to prove a homotopy theoretic analog of these results involving a concept of compact fiberings however we shall not prove these generalizations Despite the file name for this note our arguments do not require C Ehresmann s theorem that a proper smooth submersion is a smooth fiber bundle projection see the file ehresmann pdf in the course directory In fact our aguments only use material at the level of this course and its two prerequisites The examples For each positive integer k let T k denote the Cartesian product of k copies of the circle Since the tangent bundle T S 1 is isomorphic to the trivial vector bundle S 1 Rn and the tangent bundle of a product satisfies T A B T A T B it follows that for each k we have T T k T k Rk Suppose now that m n 0 and let f T m T n be the constant map If C Rm Rn is an arbitrary linear transformation of maximum rank for example projection onto the first n coordinates then one can form a map F satisfying properties i and ii with respect to f by taking the mapping f C T m Rm T n Rn and composing it with the isomorphisms T T m T m Rm T T n T n Rn described above Having shown the existence of a tangent bundle surjection F such that N o F f o M we shall have our examples if we can prove the following result Proposition If m n 0 then the constant map f from T m to T n is not homotopic to a smooth submersion 1 2 In the course of the proof we shall need the following elementary fact relating basepoint preserving homotopies and homotopies that are not necessarily basepoint preserving Lemma Let X x be a topological space let G be a connected topological group and take the identity for the basepoint of G Then two basepoint preserving maps from X x to G 1 are basepoint preservingly homotopic if and only if they are homotopic by a homotopy which is not necessarily basepoint preserving This applies to our examples because every torus is a connected topological group Proof of the lemma The implication is trivial so it is only necessary to discuss the opposite implication Let H be a homotopy which is not necessarily basepoint preserving and define a closed continuous curve t H x t Then K y t H y t t 1 defines a basepoint preserving homotopy relating the two original mappings Proof of the proposition Suppose that f is homotopic to some smooth submersion say g 0 Our first step will be to show that g0 is homotopic to a basepoint preserving smooth submersion This is fairly easy to do Since T k is arcwise connected there is a continuous curve joining 1 to g0 1 and since the group operation on T k is smooth it follows that translation by a group element is a diffeomorphism therefore the map g 0 z t 1 defines a regular homotopy from g0 to a smooth submersion which is basepoint preserving Let g denote this basepoint preserving submersion By the lemma we know that g is basepoint preservingly homotopic to the constant map and therefore the associated map of fundamental groups g 1 T m 1 T n is trivial Therefore the lifting criterion implies the existence of a continuous mapping h T m Rn such that g p o h where p Rn T n is the usual simply connected covering space projection In fact since the covering space projection p is a smooth submersion it follows that h must also be a smooth submersion Since smooth submersions are open mappings it follows that the image of h must be open in Rn On the other hand since T m is compact it follows that the image of h must be a compact subset of Rn This is impossible because Rn has no nonempty subsets that are both compact and open The source of the contradiction is our assumption that the constant map is homotopic to a smooth submersion and therefore no such mapping can exist which is exactly what we wanted to prove