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Algebraic K-Theory and Manifold Topology

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Algebraic K-Theory and Manifold Topology Course Syllabus for Math 281: Algebraic K-theory and Manifold Topology Course Description Let M and N be smooth closed manifolds of dimension n. An h-cobordism from M to N is a compact smooth manifold B of dimension (n + 1) with boundary @B ' M q N having the property that the inclusion maps M,! B,! N are homotopy equivalences. If n ≥ 5 and the manifold M is simply connected, then the celebrated h-cobordism theorem of Smale asserts that B is diffeomorphic to a product M × [0; 1] (and, in particular, M is diffeomorphic to N). If M is not simply connected, then it is generally not true that any h-cobordism B from M to N is diffeomorphic to a product M × [0; 1]. In fact, one can introduce an algebraic invariant τ(B) (called the Whitehead torsion of B) belonging to an abelian group W h(M) (called the Whitehead group of M). The invariant τ(B) vanishes whenever B is diffeomorphic to a product M × [0; 1], and the converse holds provided that the dimension of M is greater than 5 (this statement is known as the s-cobordism theorem). The Whitehead group W h(M) depends only on the fundamental group of M and vanishes when M is simply connected, so that the s-cobordism theorem implies the h-cobordism theorem. For many purposes, it is useful to know whether there is an analogue of the s-cobordism theorem for families of manifolds. Fix a compact smooth n-dimensional manifold M, a finite cell complex X, and suppose we given a fiber bundle B ! X where each of the fibers Bx is an h-cobordism from M to some other n-manifold Nx (where the smooth structures vary continuously with x). Under what circumstances can we deduce that B is equivalent to a product M × [0; 1] × X (so that the fiber bundle fNxgx2X is trivial)? In this course, we will study an analogue of the Whitehead torsion which is adapted to the parametrized setting: • In place of the Whitehead group W (M), we will consider an infinite loop space Wh(M) called the (smooth) Whitehead space of M, with π0Wh(M) = W h(M). • To every fiber bundle B ! X as above, one can associate a map τ(B): X ! Wh(M) which is well-defined up to homotopy. When X is a point, the homotopy class of this map determines an element of π0Wh(M) ' W h(M) which agrees with the classical Whitehead torsion of the bordism B. • The map τ(B) is nullhomotopic whenever B is equivalent to a product M ×[0; 1]×X. One of our main objectives in this course will be to prove the following converse: if τ(B) is nullhomotopic, then the fiber bundles B × [0; 1]k ! XM × [0; 1]k+1 × X are equivalent for k 0. Meeting Time and Place MWF at 12, Science Center 310. Office Hours Wednesday 2-3, or by appointment. Texts Course notes will be provided on the course webpage. Some portions of this course will follow the book \Spaces of PL Manifolds and Categories of Simple Maps" by Wald- hausen, Jahren, and Rognes. Pointers to relevant literature will be provided as the course proceeds. Course Website http://www.math.harvard.edu/∼lurie/281.html Prerequisites Familiarity with the machinery of modern algebraic topology (simplicial sets, spectra, ...). Several other topics (such as piecewise linear topology, microbundles, im- mersion theory, the language of quasi-categories) will receive a cursory review as we need them. A high level of mathematical sophistication will be assumed. Possible Topics • Wall's Finiteness Obstruction • Simple homotopy equivalences and Whitehead torsion. • Polyhedra and simple maps; \higher" simple homotopy theory • Waldhausen's algebraic K-theory of spaces • Assembly; the \parametrized index theorem" of Dwyer-Weiss-Williams. • Regular neighborhood theory. • Piecewise linear manifolds, microbundles, and immersion theory • The parametrized (stable) s-cobordism theorem • Concordance spaces and the Hatcher spectral sequence • Hilbert cube manifolds and infinite-dimensional topology Grading Undergraduates or graduate students wishing to take this course for a grade should speak with the instructor. 2 Overview (Lecture 1) August 27, 2014 Let M and N be smooth closed manifolds of dimension n. An h-cobordism from M to N is a compact smooth manifold B of dimension (n + 1) with boundary @B ' M q N having the property that the inclusion maps M,! B -N are homotopy equivalences. If n ≥ 5 and the manifold M is simply connected, then the celebrated h-cobordism theorem of Smale asserts that B is diffeomorphic to a product M × [0; 1] (and, in particular, M is diffeomorphic to N). This theorem has many pleasant consequences. Application 1 (Generalized Poincare Conjecture). Let X be a smooth manifold of dimension d which has the homotopy type of a sphere Sd. Choose a pair of points x; y 2 X with x 6= y, and let X0 be the manifold obtained from X by removing the interiors of nonoverlapping small disks around x and y. Then X0 is a manifold with boundary Sd−1 q Sd−1. If d ≥ 3, then removing the points x and y from X (or small disks around them) does not change the fundamental group of X, so that X0 is simply connected. A simple calculation with homology shows that the inclusions of each boundary component into X0 is a homotopy equivalence: that is, X0 is an h-cobordism from Sd−1 to itself. Applying the h-cobordism theorem, we deduce that M 0 is diffeomorphic to a product Sd−1 × [0; 1]. The original manifold X can be recovered (up to homeomorphism) from X0 by adjoining a cone on each of its boundary components. It follows that X is homeomorphic to the sphere Sd. Remark 2. Beware that the manifold X need not be diffeomorphic to Sd: the identification of X0 with Sd−1 × [0; 1] may not be compatible with the identification of @X0 with Sd−1 q Sd−1. In the non-simply connected case, it is generally not true that an h-cobordism from M to N is diffeomor- phic to a product M × [0; 1]. To guarantee this, one needs a stronger hypothesis on the inclusion M,! B. Before introducing this hypothesis, we need to embark on a slight digression. Definition 3. Let X be a finite simplicial complex. Suppose that there is a simplex σ ⊆ X containing a face σ0 ⊆ σ such that σ is not contained in any larger simplex of X, and σ0 is not contained in any larger simplex other than σ. Let Y ⊆ X be the subcomplex obtained by removing the interiors of σ and σ0. Then the inclusion ι : Y,! X is a homotopy equivalence. In this situation, we will say that ι is an elementary expansion. Note that Y is a retract of X; a retraction of X onto Y will be called an elementary collapse. Definition 4. Let f : Y ! X be a map between finite simplicial complexes. We will say that f is a simple homotopy equivalence if it is homotopic to a finite composition of elementary expansions and elementary collapses. Any compact smooth manifold can be regarded as a finite simplicial complex (by choosing a triangulation), so it makes sense to talk about simple homotopy equivalence between smooth manifolds. One then has the following result: Theorem 5 (s-Cobordism Theorem). Let B be an h-cobordism theorem between smooth manifolds M and N of dimension ≥ 5. Then B is diffeomorphic to a product M × [0; 1] if and only if the inclusion map M,! B is a simple homotopy equivalence. One of our main goals in this course is to formulate and prove a parametrized version of Theorem 5. Let us now outline the path we will take. 1 1 Theme 1: Higher Simple Homotopy Theory The subject of \higher" simple homotopy theory begins with the following: Question 6. Let B be a finite simplicial complex and suppose we are given a fibration q : E ! B. Under what conditions can we find another finite simplicial complex E0 and a fibration q0 : E0 ! B which is fiber-homotopy equivalent to q? There is an obvious necessary condition: if q : E ! B is fiber-homotopy equivalent to a fibration between finite simplicial complexes, then each fiber fEbgb2B must itself have the homotopy type of a finite complex. Let us say that a fibration q : E ! B is homotopy finite if it satisfies this condition, and finite if the total space E is itself a finite complex. For a fixed base B, there is a bijective correspondence between equivalence classes of homotopy finite fibrations q : E ! B and homotopy classes of maps B ! q BAut(X); where the coproduct is taken over all homotopy equivalence classes of finite simplicial complexes X, and Aut(X) denotes the space of homotopy equivalences of X with itself. In other words, the coproduct q BAut(X) is a classifying space for homotopy finite fibrations. Following Hatcher ([2]), we will introduce another space M which enjoys the following analogous property: for any finite simplicial complex B, there is a bijective correspondence between homotopy classes of maps from B into M and equivalence classes of finite fibrations q : E ! B where E is also a finite simplicial complex (here the right notion of equivalence is concordance; we will return to this point later). We can think of M as a sort of \moduli space of finite simplicial complexes." Every finite simplicial complex X will determine a point [X] 2 M, and one can show that two points [X]; [Y ] 2 M belong to the same connected component of M if and only if there is a simple homotopy equivalence from X to Y .
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