Note on sequences of Mayer-Vietoris type
Eldon Dyer and Joseph Roitberg
Proc. Amer. Math. Soc. 80 (1980), 660-662
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Abstract: In this largely expository note, we reexamine the construction of the homotopical Mayer-Vietoris sequence associated to a homotopy pullback. We show that in this situation, the Mayer-Vietoris sequence may be realized simply as the homotopy sequence of a suitable fibration. The usual approaches to constructing the Mayer-Vietoris sequence involve some auxiliary algebraic result, such as the Barratt-Whitehead lemma; the present approach avoids any such considerations. An additional beneficial feature of our approach is the attention paid to the bottom end of the Mayer-Vietoris sequence. Thus we are led to a cleaner proof of Proposition II.7.11 of [HMR]; moreover, we show that the converse of this latter result is also true.
The homological Mayer-Vietoris sequence associated to a homotopy pushout may be established in a very similar manner, as we point out at the end of the paper.
Eckmann and P.
J. Hilton, Unions and intersections in homotopy theory,
Comment. Math. Helv. 38 (1964), 293–307. MR 0167982
Mislin, and Joe
Roitberg, Localization of nilpotent groups and spaces,
North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier
Publishing Co., Inc., New York, 1975. North-Holland Mathematics Studies,
No. 15; Notas de Matemática, No. 55. [Notes on Mathematics, No. 55].
0478146 (57 #17635)
- B. Eckmann and P. Hilton, Unions and intersections in homotopy theory, Comment. Math. Helv. 38 (1964), 293-307. MR 0167982 (29:5247)
- P. Hilton, G. Mislin and J. Roitberg, Localization of nilpotent groups and spaces, North-Holland Math. Studies, No. 15, North-Holland, Amsterdam, 1975. MR 0478146 (57:17635)
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