Note on sequences of Mayer-Vietoris type
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.
- [EH] B. Eckmann and P. J. Hilton, Unions and intersections in homotopy theory, Comment. Math. Helv. 38 (1964), 293–307. MR 0167982
- [HMR] Peter Hilton, Guido 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]. MR 0478146
- 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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