Rational homotopy of the space of self-maps of complexes with finitely many homotopy groups

Smith, Samuel B.

doi:10.1090/S0002-9947-1994-1225575-6

Rational homotopy of the space of self-maps of complexes with finitely many homotopy groups
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Trans. Amer. Math. Soc. 342 (1994), 895-915 Request permission

Abstract:

For simply connected CW complexes X with finitely many, finitely generated homotopy groups,$^{1}$ the path components of the function space $M(X,X)$ of free self-maps of X are all of the same rational homotopy type if and only if all the k-invariants of X are of finite order. In case X is rationally a two-stage Postnikov system the space ${M_0}(X,X)$ of inessential self-maps of X has the structure of rational H-space if and only if the k-invariants of X are of finite order.

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