This book grew out of courses which I taught at Cornell University and
the University of Warwick during 1969 and 1970. I wrote it because of a
strong belief that there should be readily available a semi-historical and geometrically
motivated exposition of J. H. C. Whitehead's beautiful theory of
simple-homotopy types; that the best way to understand this theory is to
know how and why it was built.
The purpose of this paper is to prove that the stable homotopy category of algebraic topology is ‘rigid’ in the sense that it admits essentially only one model: Rigidity Theorem. Let C be a stable model category. If the homotopy category of C and the homotopy category of spectra are equivalent as triangulated categories, then there exists a Quillen equivalence between C and the model category of spectra. Our reference model is the category of spectra in the sense of Bousﬁeld and Friedlander [BF, §2] with the stable model structure. The point of the rigidity theorem is that its...
We deﬁne and study sl2 -categoriﬁcations on abelian categories. We show in particular that there is a self-derived (even homotopy) equivalence categorifying the adjoint action of the simple reﬂection. We construct categoriﬁcations for blocks of symmetric groups and deduce that two blocks are splendidly Rickard equivalent whenever they have isomorphic defect groups and we show that this implies Brou´’s abelian defect group conjecture for symmetric groups. e We give similar results for general linear groups over ﬁnite ﬁelds. ...
For each k ∈ Z, we construct a uniformly contractible metric on Euclidean space which is not mod k hypereuclidean. We also construct a pair of uniformly contractible Riemannian metrics on Rn , n ≥ 11, so that the resulting manifolds Z and Z are bounded homotopy equivalent by a homotopy equivalence which is not boundedly close to a homeomorphism. We show that for these lf spaces the C ∗ -algebra assembly map K∗ (Z) → K∗ (C ∗ (Z)) from locally ﬁnite K-homology to the K-theory of the bounded propagation algebra is not a monomorphism ...
Copyright c 2000 by Allen Hatcher
Paper or electronic copies for noncommercial use may be made freely without explicit permission from the author. All other rights reserved.
.Table of Contents
Chapter 0. Some Underlying Geometric Notions
Homotopy and Homotopy Type 1. Cell Complexes 5.
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Operations on Spaces 8. Two Criteria for Homotopy Equivalence 11. The Homotopy Extension Property 14.
Chapter 1. The Fundamental Group
1. Basic Constructions
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We prove that the classical Oka property of a complex manifold Y, concerning the existence and homotopy classiﬁcation of holomorphic mappings from Stein manifolds to Y, is equivalent to a Runge approximation property for holomorphic maps from compact convex sets in Euclidean spaces to Y . Introduction Motivated by the seminal works of Oka  and Grauert (, , ) we say that a complex manifold Y enjoys the Oka property if for every Stein manifold X, every compact O(X)-convex subset K of X and every continuous map f0 : X → Y which is holomorphic in an...
We exhibit a counterexample to Elliott’s classiﬁcation conjecture for simple, separable, and nuclear C∗ -algebras whose construction is elementary, and demonstrate the necessity of extremely ﬁne invariants in distinguishing both approximate unitary equivalence classes of automorphisms of such algebras and isomorphism classes of the algebras themselves.
There are very few examples of Riemannian manifolds with positive sectional curvature known. In fact in dimensions above 24 all known examples are diﬀeomorphic to locally rank one symmetric spaces. We give a partial explanation of this phenomenon by showing that a positively curved, simply connected, compact manifold (M, g) is up to homotopy given by a rank one symmetric space, provided that its isometry group Iso(M, g) is large. More precisely we prove ﬁrst that if dim(Iso(M, g)) ≥ 2 dim(M ) − 6, then M is tangentially homotopically equivalent to a rank one symmetric space or M...