Description: I. Foundations of homotopy theory and proper homotopy theory.- 1 Compactifications and compact maps.- 2 Homotopy.- 3 Categories with a cylinder functor.- 4 Cofibration categories and homotopy theory in I-categories.- 5 Tracks and cylindrical homotopy groups.- 6 Homotopy groups.- 7 Cofibres.- Appendices.- 8 Appendix. Compact maps.- 9 Appendix. The Freudenthal compactification.- II. Trees and spherical objects in the category Topp of compact maps.- 1 Locally finite trees and Freudenthal ends.- Appendix. Halin's tree lemma.- 2 Unions in Topp.- Appendix. The proper Hilton--Milnor theorem.- 3 Spherical objects and homotopy groups in Topp.- 4 The homotopy category of n-dimensional spherical objects in Topp.- Appendix. Classification of spherical objects under a tree.- III. Tree-like spaces and spherical objects in the category End of ended spaces.- 1 Tree-like spaces in End.- 2 Unions in End.- 3 Spherical objects and homotopy groups in End.- 4 The homotopy category of n-dimensional spherical objects in End.- Appendix. Classification of spherical objects under a tree-like space.- 5 Z-sets and telescopes.- 6 ARZ-spaces.- IV. CW-complexes.- 1 Relative CW-complexes in Top.- 2 Strongly locally finite CW-complexes.- 3 Relative CW-complexes in Topp.- 4 Relative CW-complexes in End.- 5 Normalization of CW-complexes.- 6 Push outs of CW-complexes.- 7 The Blakers--Massey theorem.- 8 The proper Whitehead theorem.- V. Theories and models of theories.- 1 Theories of cogroups and Van Kampen theorem for proper fundamental groups.- 2 Additive categories and additivization.- 3 Rings associated to tree-like spaces.- 4 Inverse limits of gr(T)-models.- 5 Kernels in ab(T).- VI. T-controlled homology.- 1 R-modules and the reduced projective class group.- 2 Chain complexes in ringoids and homology.- 3 Cellular T-controlled homology.- 4 Coefficients for T-controlled homology and cohomology.- 5 The Hurewicz theorem in End.- 6 The proper homological Whitehead theorem (the 1-connected case).- 7 Proper finiteness obstructions (the 1-connected case).- VII. Proper groupoids.- 1 Filtered discrete objects.- 2 The fundamental groupoid of ended spaces.- 3 The proper homotopy category of 1-dimensional reduced relative CW-complexes.- 4 Free D-groupoids under G.- 5 The proper fundamental groupoid of a 1-dimensional reduced relative CW-complex.- 6 Simplicial objects in proper homotopy theory.- VIII. The enveloping ringoid of a proper grou-poid.- 1 The homotopy category of 1-dimensional spherical objects under T.- 2 The ringoid S (X, T) associated to a pair (X, T) in End.- 3 The enveloping ringoid of the proper fundamental group.- 4 The enveloping ringoid of the proper fundamental groupoid.- IX. T-controlled homology with coefficients.- 1 The T-controlled twisted chain complex of a relative CW-complex (X, T).- 2 The T-controlled twisted chain complex of a CW-complex X.- 3 T-controlled cohomology and homology with local coefficients.- 4 Proper obstruction theory.- 5 The twisted Hurewicz homomorphism and the twisted ?-sequence in ?End.- 6 The proper homological Whitehead theorem (the 0-connected case).- 7 Proper finiteness obstructions (the 0-connected case).- X. Simple homotopy types with ends.- 1 The torsion group Kl.- 2 Simple equivalences and proper equivalences.- 3 The topological Whitehead group.- 4 The algebraic Whitehead group.- 5 The proper algebraic Whitehead group.- List of symbols.

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