Abstracts, comments, errata (Publications of Michael Weiss)

[1] Surgery and Bordism Invariants, A.M.S. Bull.9 (1983), 223–226 ; [2] and [3] Surgery and the generalized Kervaire Invariant, Parts I and II, Proc. Lond. Math. Soc.51 (1985), 146–230.

These papers are outgrowths of my thesis. They can be regarded as contributions to Andrew Ranicki’s algebraic surgery program. Ranicki gave in The algebraic theory of surgery, I,II (Proc. of Lond. Math. Soc. 40, 1980, pp. 87–283) a unified definition of the L–groups of a ring with involution R as bordism groups of certain chain complexes over R equipped with “self–duality”. In somewhat more detail, R has two kinds of L–groups, the symmetric L–groups and the quadratic L–groups. They correspond to different levels of self–duality: quadratic (the higher level) and symmetric. Before Ranicki, Mishchenko had already introduced bordism groups of chain complexes over R with symmetric self–duality. But the bordism groups of chain complexes over R with quadratic self–duality which Ranicki invented turned out to be the ones that had long been known and used in surgery theory, with a different and more cumbersome definition. It was clear from the Mishchenko–Ranicki definition that the L–groups of R (symmetric or quadratic) were the homotopy groups of an infinite loop space. This fact, combined with standard ideas from homotopy theory, such as long exact sequence of homotopy groups associated with a fibration, led to many useful long exact sequences in L-theory.

Chain complexes over R with symmetric self–duality were always intended to be algebraic analogs of closed manifolds or more generally Poincaré duality spaces. My contribution at this point was to introduce algebraic analogs of stable vector bundles, called chain bundles. It turned out that a chain complex with symmetric self–duality has a normal chain bundle, analogous to the (Spivak) normal bundle of a closed manifold or Poincaré duality space. Trivializing the normal chain bundle amounts to raising the quality of the self–duality from symmetric to quadratic. This observation led to two results:

(1) A long exact sequence showing that the difference between the symmetric and quadratic L–groups of R was of a homological nature and therefore in principle calculable. (See abstract of [5] for more.)

(2) A formula for the change in the surgery obstruction of a surgery problem f : M → N caused by a change in the normal data. In particular, the formula says there is no change if the normal data are changed by a stable automorphism of ν(M) for which the looped Stiefel–Whitney classes Ωw_{i} satisfy < Ωw_{i}, z >= 0 for all z ∈ ker[H_{i−1}(M; ℤ/2) → H_{i−1}(N; ℤ/2].

[4] (with A.Ranicki) Chain complexes and assembly, Math.Z. 204 (1990), 157–185.

Let f : X → Y be a map between simplicial complexes, and suppose that Y is connected. Let p : Y^{∼} → Y be a universal covering, with translation group π, and let X^{∼} → X be the pulled–back covering of X. Ranicki had observed that the cellular chain complex C(X^{∼}) can be fragmented over Y. That is, it is homotopy equivalent to a chain complex over ℤπ of the form

hocolim_{y} F (p(y))

where F is a contravariant functor from the poset of simplices of Y to the category of chain complexes of abelian groups, and the hocolim is a homotopy colimit in the category of chain complexes, taken over the poset of simplices of Y^{∼}.

Ranicki wanted to know which chain complexes over ℤπ can be fragmented over Y. We worked it out together and found that the homotopy theory of these fragmented chain complexes is equivalent to the homotopy theory of chain complexes over the simplicial ring ℤ[ΩY]. This was found independently by Dwyer and Kan, [Homology with simplicial coefficients, Proceedings, Arcata 1986, SLN 1370, pp. 143–149].

Another formulation: the homotopy theory of the fragmented chain complexes is equivalent to the homotopy theory of ℤπ–module chain complexes equipped with the structure of a comodule over C(Y^{∼}). This last formulation is often more convenient in deciding whether a given chain complex over ℤπ can be fragmented over Y. When Y is a K(π, 1) there are no obstructions to fragmenting.

Errata: In the statement of lemma 4.7, the word “finite” should be deleted.

[5] Visible L-theory, Forum.Math. 4 (1992), 465–498.

This introduces the visible symmetric L–groups, a user–friendly variation on the symmetric L–groups of Mishchenko and Ranicki. The reasons for using symmetric L–groups at all in geometric surgery theory are twofold.

(1) A surgery problem f : M → N can be multiplied with any closed manifold Q, giving a new surgery problem. The manifold Q determines an element α in a symmetric L–group L^{i}, whereas the two surgery problems (old and new) determine elements β, γ in quadratic L–groups L_{j}, L_{i+j}. One wants a formula expressing γ in terms of α and β. Such a formula exists: it is γ = α · β, and it is meaningful because there are product maps from L^{i}(R_{1}) ⊗ L_{j}(R_{2}) to L_{i+j}(R_{1} ⊗ R_{2}), defined by Ranicki.

(2) The image in symmetric L–theory of the surgery obstruction of a surgery problem f : M → N is independent of f and the normal data (except for the identification ∂M → ∂N given by f). It is an invariant of the Poincaré duality space obtained by glueing M and N along their boundaries. So, if the forgetful map from quadratic to symmetric L–theory does not lose too much information, it can be used to calculate or partly calculate surgery obstructions.

The visible symmetric L–groups are substitutes for the symmetric L–groups, available when the ring in question is a group ring, say ℤπ. They have exactly the same uses and general properties (appear in product formulae, and so on). Main result of the paper: There is a long exact sequence involving the quadratic L–theory L_{∗}(ℤπ), the visible symmetric L–theory V L^{∗}(ℤπ), and homology of the group π with coefficients in the graded group L^{∗}(ℤ). This makes the visible symmetric L–groups extremely calculable provided the quadratic L–groups of ℤπ are known—which is the case in practice when π is finite.

My first proof of this main result used the earlier papers [2], [3], but eventually I decided in favor of another proof which Ranicki found, because it was much easier to communicate.

Although this has been applied in only a few cases, these few cases are very pretty. There is the paper by Cappell and Shaneson [On four–dimensional s– cobordisms, II, Comm. Math. Helv. vol. 64 (1989), pp. 338–346] on the classification of 4–dimensional topological s–cobordisms. Kwasik and Schultz [Visible surgery, 4–dimensional s–cobordisms and related questions in geometric topology, K-theory 9 (1995), pp. 323–352] carried this further, in fact completed the program as I understand it, also using the visible L–groups. Frank Connolly and Andrew Ranicki use simple cases of the result to calulate Cappell’s Unil groups in a special case ... (work in progress).

[6] Sphères exotiques et l‘espace de Whitehead, C.R. Acad.Sci.Paris, Série 1, tome 303 no.17, (1986), 885–888. (Serious misprints—see the errata list below.)

Let ϕ : D → D be a diffeomorphism fixing all boundary points, where D = D(ℝ^{n}) is the unit disk in ℝ^{n}. For each line ℓ though the origin in ℝ^{n} we can make an identification D(ℝ^{n}) ≅ D(ℓ) × D(ℓ^{⊥}) and thereby view ϕ as a smooth concordance (alias pseudo–isotopy) of the (n−1)–dimensional disk D(ℓ^{⊥}). Actually we can do so in two ways, depending on the orientation of ℓ chosen, but the two ways are related via the canonical involution on the concordance space of D(ℓ^{⊥}). At any rate, ϕ gives rise to a parametrized family of concordances of (n − 1)–disks, parametrized by the space of lines through the origin in ℝ^{n}, which is ℝP^{n−1}. What does it look like ?

Of course, it depends on ϕ and it depends on our understanding of concordance spaces of disks. Isotopy classes of diffeomorphisms of D(ℝ^{n}) (relative to the boundary) are in one–one correspondence with diffeomorphism classes of oriented (n+1)– dimensional homotopy spheres, via a glueing construction (if n > 4, say). Although the group of oriented homotopy (n + 1)–spheres as such is not understood for all n, large cyclic summands discovered by Kervaire–Milnor are known for n ≡ 2 mod 4. From there we can take our ϕ.

From the work of Waldhausen, there is an intimate relationship between concordance theory and algebraic K–theory ; in particular there is a useful map from the concordance space of any disk to the space Ω^{2}K(ℤ).

The question can now be rephrased. Suppose n ≡ 2 mod 4. Given a diffeomorphism rel boundary ϕ : D(ℝ^{n}) → D(ℝ^{n}), of Kervaire–Milnor type, extract a family of concordances of disks, parametrized by ℝP^{n−1}, as above. Convert into algebraic K–theory ; the result is an equivariant map from 𝕊^{n−1} with the antipodal involution to Ω^{2}K(ℤ) with an appropriate involution (which I prefer not to describe here). What is this map ?

This question is answered in [6] (without indication of proof). Only the 2–primary part of the Kervaire–Milnor cyclic group (for n ≡ 2 mod 4) is detected, but this is fully detected. At the time my proof used generalized Morse functions, but it cannot be said to have been rigorous. Today the proof is part of a longish theory developed by me and Bruce Williams during the last 12 years, and not yet fully written up. The theory consist so far of [7],[8],[13],[14],[15],[16],[17] and unfinished work outlined in [20]. The paper [21] (joint with Dwyer and Williams) is also very closely related.

Errata: The diagrams got mixed up. Repair this as follows: in the diagram p. 888, replace q by 2 and n − 1 by 4k − 2 and f′ by f ; in the diagram p. 887, replace q by k and f′ by f. Also, in the formula for pr_{x}, middle of p. 886, replace + by −.

[7] (with B.Williams) Automorphisms of manifolds and algebraic K-theory, Part 1, K-theory 1 (1988), 575–626 ; [10] Orthogonal Calculus, Trans. Amer.Math.Soc. 347 (1995), 3743–3796.

The paper [10] develops a theory of (abstract, intrinsic) characteristic classes for spaces X equipped with a coordinate free filtration. Roughly, this is a filtration indexed by the finite dimensional linear subspaces of ℝ^{∞} = ∪_{i}ℝ^{i}, as opposed to a filtration indexed by the positive integers. For example, X = BO has such a filtration.

The organization is very much inspired by Goodwillie calculus, so there are certain characteristic classes of degree one (intended to be analogs of the total Stiefel–Whitney class) ; where these vanish, characteristic classes of degree 2 are defined (intended to be analogs of the total Pontryagin class) ; where these also vanish, characteristic classes of degree 3 are defined, and so on, although we run out of analogs.

Although [7] appeared long before [10], it appears in retrospect as a detailed application of parts of [10] to certain spaces with coordinate free filtration which are of interest in surgery theory. The spaces are block diffeomorphism spaces DIFF^{∼}(M) and block homeomorphism spaces TOP^{∼}(M) (defined as simplicial sets, where a k–simplex is a diffeomorphism/homeomorphism of M × Δ^{k} mapping each face M × d_{i}Δ^{k} to itself). It is not obvious why these have an interesting coordinate free filtration, but that’s explained in [7]. Only the degree 1 part of the theory of [10] is applied to these examples. The main result is an approximate description of the homotopy type of the quotients DIFF^{∼}(M)/ DIFF(M) and TOP^{∼}(M)/ TOP(M) in terms of the stable concordance theory of M and the canonical involution on it. It is approximate in the sense that there is a comparison map which is about (dim(M)/3)–connected. This explains, and was to some extent inspired by, a spectral sequence due to Hatcher [Concordance spaces, higher simple homotopy theory, and applications, Proc. Symp. Pure Math. 32, part I, AMS (1978), pp. 3–21].

Errata: Proposition 2.2 of [7] is wrong. (The mistake is in the first lines of the proof where certain equations are supposed to define certain quantities. They don’t.) This has no serious consequences for the paper because the conclusion of 2.2 nevertheless holds in all the relevant cases. This is proved in [10]. See 2.1 and the comments following proposition 2.2 in [10]. The proof of theorem 6.3 of [10] contains a serious gap, which is filled in the published erratum. See publication list.

[8] (with B.Williams) Automorphisms of manifolds and algebraic K-theory, Part 2, J.of Pure and Appl. Algebra 62 (1989), 47–107.

More a construction than a theorem: what is constructed is a map Ξ from the L–theory spectrum (quadratic or symmetric) of a ring with involution R to the Tate cohomology spectrum of ℤ/2 acting on the K–theory spectrum of R. (The Tate cohomology of a finite group G acting on a spectrum E is the mapping cone of a certain norm map from the homotopy orbit spectrum E_{hG} to the homotopy fixed point spectrum E^{hG}.) The map Ξ explains first of all where the long exact Rothenberg sequences relating L–theory and K_{0}, K_{1} really come from and how analogs involving higher K–groups can be manufactured. But it is important for another reason, explained in [20].

Large parts of this paper are subsumed in [16] and, in the opinion of the person who refereed [16], more readable there.

Errata: In the proof of 4.6, third line from beginning, a certain full subcategory is claimed to be a full subcategory with cofibrations and weak equivalences. It is not. A morphism in the subcategory which is a cofibration in the larger category can have a cofiber which is not in the subcategory. The obvious remedy is to define: A morphism in the subcategory is a cofibration if it is a cofibration in the larger category, and its cofiber belongs to the subcategory.

[9] Pinching and Concordance Theory, J. Diff. Geom. 38 (1993), 387–416.

This is a comment on the Rauch–Berger–Klingenberg sphere theorem, which states that if M^{n} is simply–connected, compact, with a Riemannian metric whose sectional curvature is everywhere > 1/4 and ≤ 1, then M admits a Morse function with only two critical points (absolute max and absolute min, necessarily). Elementary Morse theory then shows that M is homeomorphic to 𝕊^{n}.

Stronger result proved in [9]: It is possible to associate with any ordered pair of distinct points x, y in M a Morse function f_{xy} having a min at x and a max at y and no other critical points ; this can be done so that f_{xy} depends continuously on x and y, and so that f_{yx} = −f_{xy}.

J.H.–Eschenburg (Augsburg) showed me subsequently how to prove this on one or two pages, so it is not a really difficult theorem. (There are some harder theorems in [9] though, giving the same conclusion under slightly weaker hypotheses.)

At any rate, I wanted to stress that the stronger result is a more useful result because it shows that many homotopy spheres cannot have a Riemannian metric whose sectional curvature is everywhere > 1/4 and ≤ 1. In fact, finding a parametrized family of Morse functions f_{xy} as above on a homotopy sphere M^{n} amounts to trivializing a family of h–cobordisms: the family {N_{xy}}, where N_{xy} is obtained from M by deleting small open disks about x and y. (This is parametrized by the space of 2–element subsets of M.) The obstruction to trivializing this family is again something that can be looked at and calculated from the point of view of algebraic K-theory, much as in [6]. The calculations (done for the Kervaire–Milnor exotic spheres) are only sketched in [9].

Errata: In the first sentence of section 2, replace the inequality sec(M) < 1 by sec(M) > 1.

[10] Orthogonal Calculus … see the abstract of [7].

[11] Curvature and finite domination, Proc.Amer.Math.Soc. 124 (1996), 615–622.

Gromov showed in [Curvature, diameter and Betti numbers, Comm. Math. Helv. 56 (1981), pp. 179–195] that, for any n ≥ 0, there exists a constant C(n) with the following property. If M is a closed connected Riemannian n–manifold with non–negative sectional curvature, then the sum of the Betti numbers of M is less than C(n).

By improving on Gromov’s use of the Leray spectral sequence I show that sum of the Betti numbers can be replaced by minimal number of cells in a CW–space X dominating M. (X dominates M if there exist maps f : M → X and g : X → M such that gf is homotopic to the identity.)

[12] Calculus of Embedddings, Bull. Amer. Math. Soc. 33 (1996), 177-187 ; [18] Embeddings from the point of view of immersion theory, Part I, DVI file 31 pp. (revised 1996) ; [19] (with T.Goodwillie) Embeddings from the point of view of immersion theory, Part II, DVI file 15 pp. (revised 1996).

The goal is to calculate the homotopy types of spaces of smooth embeddings emb(M^{m}, N^{n}) when n − m ≥ 3. The plan is to use ideas from Goodwillie Calculus to reduce this to the calculation of imm(M, N), the immersion space, and embedding spaces emb(S, N) for finite subsets of M. The “hard fact” needed to do this is a higher excision estimate for spaces of diffeomorphisms provided by Goodwillie. The three papers above explain how the hard fact leads to the calculation. Strictly speaking we are not so good at π_{0} calculations, so this is really about loop spaces of embedding spaces.

I like to express some of the results in sheaf–theoretic terms. Immersion theory teaches us that it is a good idea to introduce the category or poset 𝒪 of open subsets of M, and the cofunctor V ↦ emb(V, N) on 𝒪. It teaches us to take seriously the standard Grothendieck topology 𝒥_{1} on 𝒪, in which a family of morphisms (inclusions) {V_{i} → W } with fixed codomain W qualifies as a cover if every point of W is contained in some V_{i}. For example, the cofunctor V ↦ imm(V, N) is a homotopy sheaf on 𝒪 with respect to 𝒥_{1}. (Which means, essentially, that you can express imm(V_{1} ∪ V_{2}, N) in terms of imm(V_{1}, N) and imm(V_{2}, N) and imm(V_{1} ∩ V_{2}, N) and the restriction maps.)

The cofunctor V ↦ emb(V, N) is not a homotopy sheaf on 𝒪 with respect to 𝒥, but Goodwillie calculus teaches us not to give up. For k > 0 let 𝒥_{k} be the Grothendieck topology on 𝒪 in which a family of morphisms (inclusions) {V_{i} → W } with fixed codomain W qualifies as a cover if every subset of W of cardinality ≤ k is contained in some V_{i}. Create the homotopy sheafification T_{k}emb(?, N) of the cofunctor emb(?, N) with respect to 𝒥_{k}. (This is done in very general terms in [18]. Example: T_{1}emb(?, N) ≃ imm(?, N).) Try to understand the homotopy fiber of the forgetful map T_{k}emb(?, N) → T_{k−1}emb(?, N) for any k ≥ 0. (This is also done in [18].) Show that the T_{k}emb(?, N) converge to emb(?, N) as k tends to infinity. (We deduce this from Goodwillie’s excision estimates in [19].) Some applications can be found in [19].

Errata: The proposed alternative definition of spaces of embeddings with compact support in section 5 of [12] is badly wrong (it gives the wrong homotopy types). Fortunately it was never used in the calculations, so the partial calculations in section 5 (of certain spaces of embeddings with compact support) are o.k. See section 10 of Embeddings from the point of view, ... Part I for a correct approach to spaces of embeddings with compact support. See also Embeddings from the point of view, ... Part II for more efficient calculations.

[13] (with B.Williams) Assembly, in Proceedings of 1993 Oberwolfach conf. on Novikov Conjectures, Index Theorems and Rigidity, vol.2, eds. Ferry-Ranicki-Rosenberg, Cambridge University Press, pp. 333-352.

Tells you how, given a homotopy invariant functor from spaces to spectra, you can best approximate it from the left by a homotopy invariant and excisive functor. No great claims for originality ; greater claims for readability.

[14] (with B.Williams) Pro–excisive Functors, in Proceedings of 1993 Oberwolfach conf. on Novikov Conjectures, Index Theorems and Rigidity, vol.2, eds. Ferry-Ranicki-Rosenberg, Cambridge University Press, pp. 353-364.

Closely related to [13]. Describes and characterizes those functors from euclidean neighborhood retracts (not necessarily compact) to spectra which give rise, on applying π_{∗}, to locally finite homology theories.

Errata: Locally finite homology is defined incorrectly on the first page. For a correct definition, replace the inverse limit of homology groups by the homology group of the appropriate homotopy inverse limit of singular chain complexes. The flawed definition is not used in the rest of the paper.

[15] Hammock localization in Waldhausen categories, to appear in J. Pure and Appl. Algebra.

Waldhausen category is short for category with cofibrations and weak equivalences as introduced in his [Algebraic K–theory of spaces, Proceedings of 1983 Rutgers conf. on Algebraic and Geometric Topology, SLN 1126, pp. 318–419]. Suppose that 𝒞 is such a category, equipped with a cylinder functor. For objects C, D in 𝒞 a quasi–morphism from C to D is a diagram C ⟶ D′ ⟵ D where the second arrw (from D to D′) is a weak equivalence and a cofibration. The quasi–morphisms from C to D are the objects of a category, with classifying space |ℳ(C, D)|. (This may be regarded as a morphism space in a Dwyer–Kan localization alias hammock localization of 𝒞.) The main theorem is about the behaviour of |ℳ(C, D)| as a cofunctor in the first variable C. It states that this cofunctor takes cofiber squares to homotopy pullback squares.

[16] (with B.Williams) Duality in Waldhausen categories, Forum Math. 10 (1998), 533-603.

Develops a theory of Spanier–Whitehead duality in Waldhausen categories. It has a lot of similarity with Ranicki’s theory of chain duality, but it applies to more examples, especially examples where the Waldhausen categories in question are not additive categories. We explain how a Spanier–Whitehead duality structure on a Waldhausen category 𝒞 gives rise to an action of ℤ/2 on the algebraic K–theory spectrum of 𝒞. We also redo much of [8] in a more abstract setting.

[17] (with B.Williams) Products and Duality in Waldhausen categories, DVI file 19 pp. (revised 1996).

Relies on [16]. Shows that the Ξ map introduced in [8] is sometimes a map of ring spectra.

[18] , [19] Embeddings from the point of view of immersion theory, Parts I and II … see the abstract of [12].

[20] Automorphisms of manifolds, preprint (1997), 38 pp., to appear in C.T.C. Wall Birthday volume.

A survey paper on spaces of automorphisms of manifolds and spaces of manifolds in a fixed homotopy type. Includes the main theorems of traditional surgery theory, but also the main theorems of pseudoisotopy theory (=concordance theory) Waldhausen style. Also summarizes the main results of [21] below, and the main results of some unfinished joint work with Bruce Williams: namely, the continuation of [7] and [8].

[21] (with W.Dwyer and B.Williams) A parametrized index theorem for the algebraic K–theory Euler class, prepr. 94 pp. (8–2000 version)

Suppose that p : E → B is a smooth fiber bundle with compact fiber F. That is, F is a smooth manifold, possibly with boundary, and the structure group of p is the diffeomorphism group of F, a topological group. The base B can be any compact CW–space.

Let V be a flat vector bundle on E (flat means: discrete structure group). Let V_{i} be the vector bundle on B whose fiber over b ∈ B is the homology group with twisted coefficients H_{i}(p^{−1}(b); V). The Atiyah–Singer theorem for families implies

tr^{∗}[V] = ∑_{i}(−1)^{i}[V_{i}] ∈ [B, K(ℝ_{t})]

where K(ℝ_{t}) = ℤ × BO is the topological K–theory of ℝ and tr : B_{+} → E_{+} is the Becker–Gottlieb–Dold transfer, a stable map determined by p.

Bismut and Lott [Flat vector bundles, direct images and higher real analytic torsion, Jour. AMS 8 (1995), pp. 291–364] observed that both sides of the equation can be viewed as elements in [B, K(ℝ)], where K(ℝ) is the algebraic K–theory space of the discrete ring ℝ (much more appropriate here, as a kind of classifying space for flat vector bundles). Using analytic index theory, e. g. the Bismut index theorem for families, they gave strong evidence for the conjecture that the equation holds in [B, K(ℝ)]. We prove this conjecture in [21].

More generally: Let R be any ring and let V be a bundle of finitely generated projective left R–modules on E (discrete structure group). From the definition of the algebraic K-theory space K(R), the bundle V determines a homotopy class [V] of maps from E to K(R). Let V_{i} be the bundle of left R–modules on B whose fiber over b ∈ B is the homology group H_{i}(p^{−1}(b); V). Assume that all these homology groups are finitely generated projective ; then each [V_{i}] determines a homotopy class of maps [V_{i}] from B to K(R). Theorem:

tr^{∗}[V] = ∑_{i}(−1)^{i}[V_{i}] ∈ [B, K(R)] .

Both sides of the equation are homotopy theoretic in nature, i.e. they are defined already when p : E → B is a fibration whose fibers are homotopy equivalent to compact CW–spaces. But for equality to hold, we need the smoothness assumption. We know cases where p : E → B is a fiber bundle with closed topological manifolds as fibers, and the equation is violated.

This raises the question: Given p : E → B, a fibration with fibers homotopy equivalent to compact CW–spaces, does there exist a smooth fiber bundle on B (with smooth compact manifold fibers) fiber homotopy equivalent to p ? We obtain an index–theoretic characterization of the fibrations p for which this is the case.

[22] (with J.Klein and T.Goodwillie) Spaces of smooth embeddings, disjunction, and surgery, prepr. 62 pp., 2000.

A survey paper on homotopy and homology of spaces of smooth embeddings, mostly in codimension ≥ 3. First chapter: Whitney embedding theorem, Haefliger-Dax theory of double point elimination and scanning, disjunction approach of Hatcher and Quinn. Second chapter: Surgery methods, specifically the Browder–Casson– Sullivan–Wall theorem and embedded surgery methods of Levine and Rigdon– Williams, finally a study of Poincaré embeddings following Williams, Richter, and Klein. Third chapter: Disjunction theory as seen by Goodwillie, in the tradition of Morlet and Hatcher–Quinn. Also mentions very recent results on disjunction in the Poincaré category (Goodwillie and Goodwillie–Klein). Fourth chapter: Goodwillie– Weiss calculus of embeddings (homotopy aspect). Takes it up where Haefliger left it. Fifth chapter: Calculus of embeddings (homology aspect). Very new. Makes the connection with the Vassiliev theory of knot invariants and some related work by Kontsevich on the rational homology of spaces of smooth embeddings of 𝕊^{1} in ℝ^{n}, where n ≥ 4. The conclusion appears to be that much of this generalizes to domains of arbitrary dimension.

[23] (with J.Klein and T.Goodwillie) A Haefliger style description of the embedding calculus tower, preprint 14 pp., September 2000 version.

Essentially this gives approximations to the homotopy type of a space of smooth embeddings emb(M, N) in terms of the homotopy types of spaces of Σ_{i}–isovariant maps M^{i} → N^{i}, for 1 ≤ i ≤ k. If the codimension is at least 3, the approximations “converge” to the homotopy type of emb(M, N) as k goes to infinity.

[24] Homology of spaces of smooth embeddings, preprint 7 pp., 1999 (draft).

Supplies the proof of lemma 5.2.1 in [22], which is the key to setting up a very generalized Eilenberg–Moore spectral sequence converging, more or less, to the homology of a space of smooth embeddings emb(M, N).

[25] Excision and restriction in controlled K–theory, preprint 29 pp., 2000.

The main result is a description in terms of controlled algebra/topology of the locally finite homology theory associated with the algebraic K–theory spectrum of a ring or the algebraic K–theory spectrum of a topological space. While such a description has been given previously by Carlsson, Pedersen and Vogell [Continuously controlled algebraic K–theory of spaces and the Novikov conjecture, Math. Ann. 310 (1998), pp. 169–182], the one here is better suited as a receptacle for characteristic invariants.