Recursion formulas for the homology of Ω(X ∨ Y )

Giora Dula and Elyahu Katz

Vol. 86 (1980), No. 2, 451–461

DOI: 10.2140/pjm.1980.86.451

Abstract

A recursion formula for H(Ω(X ∨Y )), the homology of the loop space of the wedge of the spaces X and Y is established when ΩX and ΩY are connected, and have finite dimensional homology. The recursion formula is expressed in terms of H(ΩX) and H(ΩY ), and applies to dimensions higher than a fixed integer which depends on the dimension of the highest nonvanishing homologies of ΩX and ΩY . A similar but much simpler recursion formula for H(ΩX)∐ H(ΩY ), the co-product of the two algebras H(ΩX) and H(ΩY ) is also formulated. If G₁ and G₂ are topological groups and G₁ ∗ G₂ is their co-product in the category, then our results definitely hold for H(G₁ ∗ G₂) by replacing ΩX by G₁, ΩY by G₂, and Ω(X ∨ Y ) by G₁ ∗ G₂.

Mathematical Subject Classification 2000

Primary: 57T99

Secondary: 55P35

Milestones

Received: 21 January 1978

Published: 1 February 1980