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A Weiss–Williams theorem for spaces of embeddings and the homotopy type of spaces of long knots

Samuel Muñoz-Echániz

Geometry & Topology 30 (2026) 701–780
Bibliography
1 A Adem, R J Milgram, Cohomology of finite groups, 309, Springer (2004) MR2035696
2 D R Anderson, E K Pedersen, Semifree topological actions of finite groups on spheres, Math. Ann. 265 (1983) 23 MR719349
3 G Arone, V Turchin, On the rational homology of high-dimensional analogues of spaces of long knots, Geom. Topol. 18 (2014) 1261 MR3228453
4 G Arone, V Turchin, Graph-complexes computing the rational homotopy of high dimensional analogues of spaces of long knots, Ann. Inst. Fourier (Grenoble) 65 (2015) 1 MR3449147
5 A J Blumberg, M A Mandell, The homotopy groups of the algebraic K-theory of the sphere spectrum, Geom. Topol. 23 (2019) 101 MR3921317
6 M Bökstedt, W C Hsiang, I Madsen, The cyclotomic trace and algebraic K-theory of spaces, Invent. Math. 111 (1993) 465 MR1202133
7 M Bökstedt, G Carlsson, R Cohen, T Goodwillie, W C Hsiang, I Madsen, On the algebraic K-theory of simply connected spaces, Duke Math. J. 84 (1996) 541 MR1408537
8 R Budney, Little cubes and long knots, Topology 46 (2007) 1 MR2288724
9 R Budney, A family of embedding spaces, from: "Groups, homotopy and configuration spaces" (editors N Iwase, T Kohno, R Levi, D Tamaki, J Wu), Geom. Topol. Monogr. 13, Geom. Topol. Publ. (2008) 41 MR2508201
10 R Budney, F Cohen, On the homology of the space of knots, Geom. Topol. 13 (2009) 99 MR2469515
11 D Burghelea, Z Fiedorowicz, Cyclic homology and algebraic K-theory of spaces, II, Topology 25 (1986) 303 MR842427
12 D Burghelea, R Lashof, Geometric transfer and the homotopy type of the automorphism groups of a manifold, Trans. Amer. Math. Soc. 269 (1982) 1 MR637027
13 D Burghelea, R Lashof, M Rothenberg, Groups of automorphisms of manifolds, 473, Springer (1975) MR380841
14 G E Carlsson, R L Cohen, The cyclic groups and the free loop space, Comment. Math. Helv. 62 (1987) 423 MR910170
15 G E Carlsson, R L Cohen, T Goodwillie, W C Hsiang, The free loop space and the algebraic K-theory of spaces, K-Theory 1 (1987) 53 MR899917
16 D. Chua, H. Chatham, J. Beauvais-Feisthauer, M. Young, The spectral sequences project, (2022)
17 E Dotto, K Moi, I Patchkoria, On the geometric fixed points of real topological cyclic homology, J. Lond. Math. Soc. 109 (2024) MR4704156
18 B I Dundas, Relative K-theory and topological cyclic homology, Acta Math. 179 (1997) 223 MR1607556
19 B I Dundas, R McCarthy, Stable K-theory and topological Hochschild homology, Ann. of Math. 140 (1994) 685 MR1307900
20 B I Dundas, T G Goodwillie, R McCarthy, The local structure of algebraic K-theory, 18, Springer (2013) MR3013261
21 W Dwyer, K Hess, Long knots and maps between operads, Geom. Topol. 16 (2012) 919 MR2928985
22 B Fresse, V Turchin, T Willwacher, The rational homotopy of mapping spaces of En operads, preprint (2017) arXiv:1703.06123
23 P G Goerss, J F Jardine, Simplicial homotopy theory, 174, Birkhäuser (1999) MR1711612
24 T G Goodwillie, Relative algebraic K-theory and cyclic homology, Ann. of Math. 124 (1986) 347 MR855300
25 T G Goodwillie, J R Klein, M S Weiss, Spaces of smooth embeddings, disjunction and surgery, from: "Surveys on surgery theory, II" (editors S Cappell, A Ranicki, J Rosenberg), Ann. of Math. Stud. 149, Princeton Univ. Press (2001) 221 MR1818775
26 T Goodwillie, M Krannich, A Kupers, Stability of concordance embeddings, Proc. Roy. Soc. Edinburgh Sect. A 154 (2024) 1713 MR4883987
27 D Gromoll, Differenzierbare Strukturen und Metriken positiver Krümmung auf Sphären, Math. Ann. 164 (1966) 353 MR196754
28 A Haefliger, Differential embeddings of Sn in Sn+q for q > 2, Ann. of Math. 83 (1966) 402 MR202151
29 Y. Harpaz, T. Nikolaus, J. Shah, Normal L-theory and topological cyclic homology, talk slides (2021)
30 A E Hatcher, Concordance spaces, higher simple-homotopy theory, and applications, from: "Algebraic and geometric topology, I" (editor R J Milgram), Proc. Sympos. Pure Math. XXXII, Amer. Math. Soc. (1978) 3 MR520490
31 A Hatcher, J Wagoner, Pseudo-isotopies of compact manifolds, 6, Soc. Math. France (1973) MR353337
32 R Haugseng, The higher Morita category of 𝔼n-algebras, Geom. Topol. 21 (2017) 1631 MR3650080
33 L Hesselholt, On the Whitehead spectrum of the circle, from: "Algebraic topology" (editors N A Baas, E M Friedlander, B Jahren, P A Østvær), Abel Symp. 4, Springer (2009) 131 MR2597738
34 L. Hesselholt, I. Madsen, Real algebraic K-theory, preprint (2015)
35 M W Hirsch, On normal microbundles, Topology 5 (1966) 229 MR198490
36 M W Hirsch, Differential topology, 33, Springer (1976) MR448362
37 A. Høgenhaven, Real topological cyclic homology of spherical group rings, PhD thesis, University of Copenhagen (2016)
38 J F P Hudson, Extending piecewise-linear isotopies, Proc. London Math. Soc. 16 (1966) 651 MR202147
39 J F P Hudson, Concordance, isotopy, and diffeotopy, Ann. of Math. 91 (1970) 425 MR259920
40 K Igusa, The stability theorem for smooth pseudoisotopies, K-Theory 2 (1988) 1 MR972368
41 D S Kahn, S B Priddy, The transfer and stable homotopy theory, Math. Proc. Cambridge Philos. Soc. 83 (1978) 103 MR464230
42 A. Kupers, Lectures on diffeomorphism groups of manifolds, lecture notes (2019)
43 A Lamas, A guide for the Adams spectral sequence : a computational approach and applications, Master’s thesis, Pontificia Universidad Católica de Chile (2024)
44 E L Lima, On the local triviality of the restriction map for embeddings, Comment. Math. Helv. 38 (1964) 163 MR161343
45 J Lurie, Diffeomorphisms and PL-homeomorphisms, lecture notes (2009)
46 J P May, J Sigurdsson, Parametrized homotopy theory, 132, Amer. Math. Soc. (2006) MR2271789
47 J Milnor, Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966) 358 MR196736
48 J R Munkres, Elementary differential topology, 54, Princeton Univ. Press (1966) MR198479
49 T Nikolaus, P Scholze, On topological cyclic homology, Acta Math. 221 (2018) 203 MR3904731
50 R S Palais, Local triviality of the restriction map for embeddings, Comment. Math. Helv. 34 (1960) 305 MR123338
51 E K Pedersen, Topological concordances, Invent. Math. 38 (1976) 255 MR436154
52 J Rognes, Two-primary algebraic K-theory of pointed spaces, Topology 41 (2002) 873 MR1923990
53 J Rognes, The smooth Whitehead spectrum of a point at odd regular primes, Geom. Topol. 7 (2003) 155 MR1988283
54 C P Rourke, B J Sanderson, An embedding without a normal microbundle, Invent. Math. 3 (1967) 293 MR222904
55 C P Rourke, B J Sanderson, Block bundles, I, Ann. of Math. 87 (1968) 1 MR226645
56 C P Rourke, B J Sanderson, Block bundles, III : Homotopy theory, Ann. of Math. 87 (1968) 431 MR232404
57 K P Scannell, D P Sinha, A one-dimensional embedding complex, J. Pure Appl. Algebra 170 (2002) 93 MR1896343
58 D P Sinha, The topology of spaces of knots: cosimplicial models, Amer. J. Math. 131 (2009) 945 MR2543919
59 V Turchin, Hodge-type decomposition in the homology of long knots, J. Topol. 3 (2010) 487 MR2684511
60 W Vogell, The involution in the algebraic K-theory of spaces, from: "Algebraic and geometric topology" (editors A Ranicki, N Levitt, F Quinn), Lecture Notes in Math. 1126, Springer (1985) 277 MR802795
61 I Volić, Finite type knot invariants and the calculus of functors, Compos. Math. 142 (2006) 222 MR2197410
62 F Waldhausen, Algebraic K-theory of spaces, a manifold approach, from: "Current trends in algebraic topology, I" (editors R M Kane, S O Kochman, P S Selick, V P Snaith), CMS Conf. Proc. 2, Amer. Math. Soc. (1982) 141 MR686115
63 F Waldhausen, Algebraic K-theory of spaces, from: "Algebraic and geometric topology" (editors A Ranicki, N Levitt, F Quinn), Lecture Notes in Math. 1126, Springer (1985) 318 MR802796
64 F Waldhausen, B Jahren, J Rognes, Spaces of PL manifolds and categories of simple maps, 186, Princeton Univ. Press (2013) MR3202834
65 C T C Wall, Surgery on compact manifolds, 69, Amer. Math. Soc. (1999) MR1687388
66 T Watanabe, Configuration space integral for long n-knots and the Alexander polynomial, Algebr. Geom. Topol. 7 (2007) 47 MR2289804
67 M Weiss, Orthogonal calculus, Trans. Amer. Math. Soc. 347 (1995) 3743 MR1321590
68 M Weiss, B Williams, Automorphisms of manifolds and algebraic K-theory, I, K-Theory 1 (1988) 575 MR953917