18. For the closed orientable surface M of genus g >1, show that for each nonzero x e H' (M; Z) there exists ß E H' (M; Z) with «ß + 0. Deduce that M is not homotopy equivalent to a wedge sum X vY of CW complexes with nontrivial reduced homology. Do the same for closed nonorientable surfaces using cohomology with Z, coefficients.

18. For the closed orientable surface M of genus g >1, show that for each nonzero x e H' (M; Z) there exists ß E H' (M; Z) with «ß + 0. Deduce that M is not homotopy equivalent to a wedge sum X vY of CW complexes with nontrivial reduced homology. Do the same for closed nonorientable surfaces using cohomology with Z, coefficients.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

18. For the closed orientable surface M of genus g >1, show that for each nonzero
x e H' (M; Z) there exists B E H' (M;Z) with aß + 0. Deduce that M is not homotopy
equivalent to a wedge sum X V Y of CW complexes with nontrivial reduced homology.
Do the same for closed nonorientable surfaces using cohomology with Z, coefficients.

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