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Square root of R^3

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Asking cute questions is great.

 

Answering such questions is a nice challenge.

 

An article by Mark Dominus about this topics is here ( EN ).

 

 

 

Solution in "R3 Has No Root", Robbert Fokkink, American Mathematical Monthly vol 109 March 2002, p. 285, uses the orientation preserving and reversing arguments.

Topological proof is in "Another Proof That R3 Has No Square Root", Sam B. Nadler, Jr., American Mathematical Monthly vol 111 June–July 2004, pp. 527–528.

Any idea for some new proof???

By the way in "The cartesian product of a certain nonmanifold and a line is E4", R.H. Bing, Annals of Mathematics series 2 vol 70 1959 pp. 399–412, is a construction of a strange space B such that B x R = R4.

GREAT RESULT !!!