## A rational splitting of a based mapping space.(English)Zbl 1097.55010

Let $$X$$ be a finite, connected CW complex. Let $$X \bigcup_\alpha e^{k+1}$$ denote the adjunction space corresponding to an attaching map $$\alpha \colon S^k \to X$$. The authors give conditions under which the fibration sequence on spaces of based maps $\Omega^{k+1}Y = \mathcal{F}_*(S^{k+1}, Y) \to \mathcal{F}_*(X \bigcup_\alpha e^{k+1}, Y) \to \mathcal{F}_*(X, Y)$ corresponding to the cofibration sequence $$X \to X \bigcup_\alpha e^{k+1} \to S^{k+1}$$ splits after rationalization. Precisely, the authors prove this fibration splits rationally if $$(1)$$ the rational connectivity of $$Y$$ is at least as large as the dimension of the adjuction space and $$(2)$$ the element $$\alpha$$ is decomposable in $$\pi_*(\Omega X) \otimes \mathbb Q$$ of bracket length strictly larger than the longest bracket in $$\pi_*(\Omega Y) \otimes \mathbb Q.$$ As a consequence, they obtain that if $$X$$ is a finite, simply connected complex with dimension no larger than the connectivity of $$Y$$ and with attaching maps all of bracket length greater than the longest bracket in the rational homotopy of $$Y,$$ then $$\mathcal{F}_*(X, Y)$$ is a rational $$H$$-space.
The authors obtain the main result by studying a model for the based mapping space $$\mathcal{F}_*(X, Y)$$ which can be made explicit when the dimension of $$X$$ is no larger than the connectivity of $$Y.$$ In this case, the first author has constructed a Sullivan model of the form $$(\mathbb Q[V \otimes C_*(L, d_L)], \delta)$$ where $$V$$ is a space of generators for the Sullivan minimal model of $$Y$$, $$(L, d_L)$$ is a Quillen minimal model for $$X$$ and $$C_*(L, d_L)$$ is Quillen’s functor from the category of differential graded Lie algebras to differential graded co-algebras. The differential $$\delta$$ is expressed in terms of the product in the Sullivan minimal model of $$Y$$ and the coproduct in the $$C_*(L, d_L)$$. This model is adapted from the work of E. H. Brown jun. and R. H. Szczarba [Trans. Am. Math. Soc. 349, 4931–4951 (1997; Zbl 0927.55012)]. The paper contains several nice applications of the main result and examples showing the necessity of the hypotheses.

### MSC:

 55P62 Rational homotopy theory

### Keywords:

rational $$H$$-space; model

Zbl 0927.55012
Full Text:

### References:

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