Maps to the projective plane

We prove the projective plane $ℝ P^{2}$ is an absolute extensor of a finite-dimensional metrizable space $X$ if and only if the cohomological dimension mod $2$ of $X$ does not exceed $1$ . This solves one of the remaining difficult problems (posed by A N Dranishnikov) in Extension Theory. One of the main tools is the computation of the fundamental group of the function space $Map (ℝ P^{n}, ℝ P^{n + 1})$ (based at the inclusion) as being isomorphic to either $ℤ_{4}$ or $ℤ_{2} \oplus ℤ_{2}$ for $n \geq 1$ . Double surgery and the above fact yield the proof.

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Keywords

absolute extensor, cohomological dimension, covering dimension, extension dimension, extension of maps, projective space

Mathematical Subject Classification 2000

Primary: 54F45

Secondary: 54C65, 55M10

References

Publication

Received: 4 April 2007

Revised: 22 February 2009

Accepted: 23 February 2009

Published: 30 March 2009

Authors

Jerzy Dydak
Department of Mathematics University of Tennessee Knoxville, TN 37996-1300 United States
http://www.math.utk.edu/~dydak
Michael Levin
Department of Mathematics Ben Gurion University of the Negev P.O.B. 653 Be’er Sheva 84105 Israel

Volume 9, issue 1 (2009)

Jerzy Dydak and Michael Levin

Abstract