I have a problem concerning the definition of the first cohomotopy group of P-adic solenoid. In [1], E. Spanier defines group operation on the set of homotopy classes of maps from X to n-dimensional sphere S^n with the restriction that dim X should be strictly less than 2n-1. Now, dimension of a P-adic solenoid S_P is equal to 1, so pi^q(S_P) is defined for q > 1. However, S. Godlewski ([2]) talks about first cohomotopy groups of a P-solenoid in his article "Homomorphisms of cohomotopy groups induced by fundamental classes". In case n > 1 he shows how you can get homomorphisms between cohomotopy groups from fundamental classes. What I want to know is what about the case n=1.

Of course, S^1 (considered as a subset of the set of complex numbers) is a topological group, so one can define group operation on the set of homotopy classes of maps [X,S^1] in the usual way by [f]+[g] := [f * g], for any space X (here f*g is defined by f*g(x) = f(x) * g(x), where the second '*' marks multiplication in the set of complex numbers). This is called the Bruschlinsky group.

Now, there are 3 possibilities (I suppose):

1) The restriction dim X < 2n-1 is unnessesary in the case n=1, and the procedure of defining group operation the set [X,S^1] can be performed in the same way as in [1].

2) In the case n=1 one cannot define group operation on [X,S^1] like in 1) but instead defines group operation using group structure of S^1 (the Bruschlinsky group). It is unclear for me how you can get homomorphisms of Bruschlinsky groups from fundamental classes in such way that fundamental equivalences generate isomorphisms. Is there something so "well known" that author didn't mention and that I am missing?

3) Some other "usual" way of defining the group structure on the set [X,S^1] and get the mentioned homomorphisms in a more or less similar way as in [2].

What is really intriguing is the fact that Godlewski mentions nothing for n=1 in [2] but calculates the first cohomotopy groups of solenoids and in [3] uses the theorem from [2] which states that homomorphisms of cohomotopy groups induced by fundamental equivalences are isomorphisms. This led me to believe that the whole procedure in [2] for n>1 can be done in the case n=1 also. On the other hand, even more intriguing is the fact that in [4] (after theorem 2 on page 94) Godlewski calculalates first cohomotopy group of a solenoids considering it as Bruschlinsky group. One could argue that S.T.Hu shows in [5] (proposition 5.3, page 213 and later in exercise A on p. 226) that two mentioned group operations coincide in the case n=1, but the space X must be 1-coconnected, which is something that solenoid doesn't satisfy (because his one-dimensional Cech cohomology group with integer coefficients is isomorphic to the group of integers). So, understanding what Godlewski means by "first cohomotopy group of a solenoid" and how the procedure in the case n>1 applies in the case n=1 presents a problem for me. I searched through many books and articles in a hope to find some clue, but I didn't find anything that would explain the procedure in [2] for n=1. Either it is very subtle or I missing some simple fact which makes it all work. Any help would be very appreciated.

[1] E. Spanier, "Borsuk's cohomotopy groups", Annals of Math., (50), p.203-245, 1949. [2] S. Godlewski, "Homomorphisms of cohomotopy groups induced by fundamental classes", Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys., (17), p.277-283, 1969. [3] S. Godlewski. "On shapes of solenoids", Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys., (17), p.623-627, 1969. [4] S. Godlewski. "Some remarks concerning the mappings of the inverse limit into an absolute neighborhood retract and its application to cohomotopy groups", Fund. Math., (63) p.89-95, 1968. [5] S.-T. Hu, "Homotopy theory", Academic Press, New York and London, 1959.

  • 1
    $\begingroup$ My guess is option (2). $\endgroup$
    – Jeff Strom
    Oct 8 '10 at 11:55
  • $\begingroup$ Thank you for your reply Jeff Strom. Can you expand on your answer? In other words, can you specify why do you think option 2) might be true. Any references or clues? $\endgroup$
    – user9907
    Oct 9 '10 at 12:24
  • 3
    $\begingroup$ Isn't the first cohomotopy group better-known as the first cohomology group with integral coefficients? $\endgroup$ Jan 30 '11 at 4:25

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