The inclination or λ-lemma is a fundamental tool in finite dimensional hyperbolic dynamics. In contrast to finite dimension, we consider the forward semi-flow on the loop space of a closed Riemannian manifold M provided by the heat flow. The main result is a backward λ-lemma for the heat flow near a hyperbolic fixed point x. There are the following novelties. Firstly, infinite versus finite dimension. Secondly, semi-flow versus flow. Thirdly, suitable adaption provides a new proof in the finite dimensional case. Fourthly and a priori most surprisingly, our λ-lemma moves the given disk transversal to the unstable manifold backward in time, although there is no backward flow. As a first application we propose a new method to calculate the Conley homotopy index of x. © 2014 Springer-Verlag Berlin Heidelberg.35903/04/15929967Abbondandolo, A., Majer, P., Lectures on the Morse complex for infinite dimensional manifolds, in Morse theoretic methods in nonlinear analysis and in symplectic topology (2006) NATO Science Series II: Mathematics, Physics and Chemistry, pp. 1-74. , In: Biran, P., Cornea, O., Lalonde, F. (eds.) Springer, New YorkChow, S.-N., Hale, J.K., Methods of bifurcation theory (1996) Grundlehren Math. Wissensch., 251. , (Springer, 1982, corrected second printing)Chow, S.-N., Lin, X.-B., Lu, K., Smooth invariant foliations in infinite dimensional spaces (1991) J. Differ. Equ., 94, pp. 266-291Conley, C.C., Isolated invariant sets and the Morse index (1978) In: CBMS Regional Conference Series in Mathematics, 38. , American Mathematical Society, ProvidenceGrobman, D., Homeomorphisms of systems of differential equations (1959) Dokl. Akad. Nauk. SSSR, 128, pp. 880-881Hadamard, J., Sur l'iteration et les solutions asymptotiques des équations differentielles (1901) Bull. Soc. Math. Fr., 29, pp. 224-228Hartman, P., A lemma in the theory of structural stability of differential equations (1960) Proc. Am. Math. Soc., 11, pp. 610-620Henry, D., Geometric theory of semilinear parabolic equations (1993) In: Lecture Notes in Mathematics, 840. , Springer-Verlag, Berlin (1981, third printing)Lorenzi, L., Lunardi, A., Metafune, G., Pallara, D., (2005) Analytic semigroups and reaction-diffusion problems, , http://www.math.unipr.it/~lunardi/LectureNotes/I-Sem2005.pdf, Internet sem 2004-2005. Accessed 10 Oct 2011Palis, J., (1967) On Morse-Smale Diffeomorphisms, , http://www.math.sunysb.edu/~joa/PUBLICATIONS/, Ph. D. thesis, UC BerkeleyPalis, J., On Morse-Smale dynamical systems (1968) Topology, 8, pp. 385-404Palis Jr., J., de Melo, W., (1982) Geometric Theory of Dynamical Systems, , New York: Springer-VerlagPerron, O., Über die Stabilität und asymptotisches Verhalten der Integrale von Differential gleichungssystemen (1928) Math. Z., 29, pp. 129-160Reed, M., Simon, B., (1980) Functional Analysis, , Methods of modern mathematical physics I, London: Academic PressSalamon, D.A., Morse theory, the Conley index and Floer homology (1990) Bull. LMS, 22, pp. 113-140Salamon, D.A., Weber, J., Floer homology and the heat flow (2006) Gafa, 16, pp. 1050-1138Weber, J., (2012) The heat flow and the homology of the loop space, , http://www.math.sunysb.edu/~joa/PUBLICATIONS/2010%20habilitation.pdf, HU, Berlin (2010). Accessed 19 AprWeber, J., Morse homology for the heat flow (2013) Math. Z., 275 (1), pp. 1-54Weber, J., The backward λ-lemma and Morse filtrations (2013) In: Proceedings Nonlinear Differential Equations, pp. 1-9. , http://arxiv.org/abs/1211, 17-21 September 2012, João Pessoa, Brazil, 2180 (2012) (to appear in PNLDE). Accessed 16 MarWeber, J., (2014) Stable foliations and the homology of the loop space, , (in preparation)Weber, J., (2014) The Heat Flow and the Homology of the Loop Space, , (in preparation)