Algebraic and Homological Aspects of Hermitian $K$-Theory

Th. Yu. Popelensky

Informatics and Automation, Geometry, Topology, and Mathematical Physics, Tome 325 (2024), pp. 244-276

Résumé

In 1970, S. P. Novikov proposed a systematization of algebraic results of the surgery theory based on the Hamiltonian formalism over rings with involution. His results have had a significant impact on the development of Hermitian analogs of algebraic $K$-theory. This article was written at S. P. Novikov's suggestion and aims to present the current state of research at the interface between the problems of manifold theory and Hermitian $K$-theory of rings with involution.

Détail

Keywords: Hermitian $K$-theory, $L$-groups, ring with involution, quadratic form.

Bibliographie

[1] Anderson D.W., “Relationship among $K$-theories”, Algebraic $K$-theory I: Higher $K$-theories: Proc. Conf. Seattle Res. Cent. Battelle Mem. Inst., 1972, Lect. Notes Math., 341, Springer, Berlin, 1973, 57–72 | DOI | MR

[2] Anderson D., Karoubi M., Wagoner J., “Relations between higher algebraic $K$-theories”, Algebraic $K$-theory I: Higher $K$-theories: Proc. Conf. Seattle Res. Cent. Battelle Mem. Inst., 1972, Lect. Notes Math., 341, Springer, Berlin, 1973, 73–81 | DOI | MR

[3] Atiyah M.F., Hirzebruch F., “Riemann–Roch theorems for differentiable manifolds”, Bull. Am. Math. Soc., 65:4 (1959), 276–281 | DOI | MR | Zbl

[4] Atiyah M.F., Hirzebruch F., “Vector bundles and homogeneous spaces”, Differential geometry, Proc. Symp. Pure Math., 3, Am. Math. Soc., Providence, RI, 1961, 7–38 | DOI | MR | Zbl

[5] Bak A., $K$-theory of forms, Ann. Math. Stud., 98, Princeton Univ. Press, Princeton, NJ, 1981 | DOI | MR | Zbl

[6] Bak A., Kolster M., “The computation of odd dimensional projective surgery groups of finite groups”, Topology, 21:1 (1982), 35–63 | DOI | MR

[7] Balmer P., “Triangular Witt groups. I: The 12-term localization exact sequence”, K-Theory, 19:4 (2000), 311–363 | DOI | MR | Zbl

[8] Balmer P., “Triangular Witt groups. II: From usual to derived”, Math. Z., 236:2 (2001), 351–382 | DOI | MR | Zbl

[9] Balmer P., “An introduction to triangular Witt groups and a survey of applications”, Algebraic and arithmetic theory of quadratic forms: Proc. Int. Conf., Univ. Talca, Chile, 2002, Contemp. Math., 344, Am. Math. Soc., Providence, RI, 2004, 31–58 | DOI | MR | Zbl

[10] Balmer P., “Witt groups”, Handbook of $K$-theory, v. 2, Springer, Berlin, 2005, 539–576 | DOI | MR

[11] Barwick C., “On the algebraic $K$-theory of higher categories”, J. Topol., 9:1 (2016), 245–347 ; Bass Kh., Algebraicheskaya $K$-teoriya, Mir, M., 1973 | DOI | MR | Zbl

[12] H. Bass, Algebraic $K$-Theory, W.A. Benjamin, New York, 1968 | MR | Zbl

[13] Bass H., “Unitary algebraic $K$-theory”, Algebraic $K$-theory III: Hermitian $K$-theory and geometric applications: Proc. Conf. Seattle Res. Cent. Battelle Mem. Inst., 1972, Lect. Notes Math., 343, Springer, Berlin, 1973, 57–265 | DOI | MR

[14] Blumberg A.J., Gepner D., Tabuada G., “A universal characterization of higher algebraic $K$-theory”, Geom. Topol., 17:2 (2013), 733–838 | DOI | MR | Zbl

[15] Blumberg A.J., Gepner D., Tabuada G., “Uniqueness of the multiplicative cyclotomic trace”, Adv. Math., 260 (2014), 191–232 | DOI | MR | Zbl

[16] Bökstedt M., Topological Hochschild homology, Preprint, Univ. Bielefeld, Bielefeld, 1985 https://people.math.rochester.edu/faculty/doug/otherpapers/bokstedt1.pdf

[17] Bökstedt M., Hsiang W.C., Madsen I., “The cyclotomic trace and algebraic $K$-theory of spaces”, Invent. math., 111 (1993), 465–539 | DOI | MR | Zbl

[18] Browder W., “Homotopy type of differentiable manifolds”, Colloquium on algebraic topology, Aarhus, 1962, Aarhus Univ., Aarhus, 1962, 42–46 | Zbl

[19] W. Browder, Surgery on Simply-Connected Manifolds, Ergeb. Math. Grenzgeb., 65, Springer, Berlin, 1972 | MR | Zbl

[20] Calmès B., Dotto E., Harpaz Y., et al., “Hermitian $K$-theory for stable $\infty $-categories. I: Foundations”, Sel. math., 29:1 (2023), 10 ; arXiv: 2009.07223 [math.KT] | DOI | MR | Zbl | DOI

[21] Calmès B., Dotto E., Harpaz Y., et al., Hermitian $K$-theory for stable $\infty $-categories. II: Cobordism categories and additivity, E-print, 2020, arXiv: 2009.07224 [math.KT] | DOI | Zbl

[22] Calmès B., Dotto E., Harpaz Y., et al., Hermitian $K$-theory for stable $\infty $-categories. III: Grothendieck–Witt groups of rings, E-print, 2020, arXiv: 2009.07225 [math.KT] | DOI | Zbl

[23] Carlsson G., “Deloopings in algebraic $K$-theory”, Handbook of $K$-theory, v. 1, Springer, Berlin, 2005, 3–37 | DOI | MR | Zbl

[24] Carter D.W., “Localization in lower algebraic $K$-theory”, Commun. Algebra, 8:7 (1980), 603–622 | DOI | MR | Zbl

[25] I. M. Gel'fand and A. S. Mishchenko, “Quadratic forms over commutative group rings and the $K$-theory”, Funct. Anal. Appl., 3:4 (1969), 277–281 | DOI | MR | Zbl

[26] Gersten S.M., “On the functor $K_2$. I”, J. Algebra, 17:2 (1971), 212–237 | DOI | MR | Zbl

[27] Gersten S.M., “On Mayer–Vietoris functors and algebraic $K$-theory”, J. Algebra, 18:1 (1971), 51–88 | DOI | MR | Zbl

[28] Gersten S.M., “On the spectrum of algebraic $K$-theory”, Bull. Am. Math. Soc., 78:2 (1972), 216–219 | DOI | MR | Zbl

[29] Grayson D., “Higher algebraic $K$-theory. II (after Daniel Quillen)”, Algebraic $K$-theory: Proc. Conf. Northwestern Univ., Evanston, 1976, Lect. Notes Math., 551, Springer, Berlin, 1976, 217–240 | DOI | MR

[30] Grothendieck A., “Sur quelques points d'algèbre homologique”, Tohoku Math. J. Ser. 2, 9:2 (1957), 119–183 | MR

[31] Hambleton I., Taylor L.R., “A guide to the calculation of the surgery obstruction groups for finite groups”, Surveys on surgery theory, v. 1, Ann. Math. Stud., 145, Princeton Univ. Press, Princeton, NJ, 2000, 225–274 | DOI | MR | Zbl

[32] Hirzebruch F., “On Steenrod's reduced powers, the index of inertia, and the Todd genus”, Proc. Natl. Acad. Sci. USA, 39:9 (1953), 951–956 | DOI | MR | Zbl

[33] Hirzebruch F., “The signature theorem: Reminiscences and recreation”, Prospects in mathematics, Ann. Math. Stud., 70, Princeton Univ. Press, Princeton, NJ, 1971, 3–31 | DOI | MR

[34] Hornbostel J., “Constructions and dévissage in Hermitian $K$-theory”, K-Theory, 26:2 (2002), 139–170 | DOI | MR | Zbl

[35] Hornbostel J., Schlichting M., “Localization in Hermitian $K$-theory of rings”, J. London Math. Soc., 70:1 (2004), 77–124 | DOI | MR | Zbl

[36] Jensen K.K., Thomsen K., Elements of KK-theory, Birkhäuser, Boston, 2012 | DOI | MR

[37] Karoubi M., “Périodicité de la $K$-théorie hermitienne”, Algebraic $K$-theory III: Hermitian $K$-theory and geometric applications: Proc. Conf. Seattle Res. Cent. Battelle Mem. Inst., 1972, Lect. Notes Math., 343, Springer, Berlin, 1973, 301–411 | DOI | MR

[38] Karoubi M., “Localisation de formes quadratiques. I”, Ann. sci. Éc. norm. supér. Sér. 4, 7 (1974), 359–403 | DOI | MR | Zbl

[39] Karoubi M., “Localisation de formes quadratiques. II”, Ann. sci. Éc. norm. supér. Sér. 4, 8 (1975), 99–155 | DOI | MR | Zbl

[40] Karoubi M., “Théorie de Quillen et homologie du groupe orthogonal”, Ann. Math. Ser. 2, 112:2 (1980), 207–257 | DOI | MR | Zbl

[41] Karoubi M., “Le théorème fondamental de la $K$-théorie hermitienne”, Ann. Math. Ser. 2, 112:2 (1980), 259–282 | DOI | MR | Zbl

[42] Karoubi M., “Bott periodicity in topological, algebraic and Hermitian $K$-theory”, Handbook of $K$-theory, v. 1, Springer, Berlin, 2005, 111–137 | DOI | MR | Zbl

[43] Karoubi M., Villamayor O., “Foncteurs $K^n$ en algèbre et en topologie”, C. r. Acad. sci. Paris. Sér. A, 269 (1969), 416–419 | MR | Zbl

[44] Kervaire M.A., Milnor J.W., “Groups of homotopy spheres. I”, Ann. Math. Ser. 2, 77:3 (1963), 504–537 | DOI | MR | Zbl

[45] A. F. Kharshiladze, “The construction of discriminants in Hermitian $K$-theory”, Russ. Math. Surv., 34:6 (1979), 179–183 | DOI | MR | Zbl | Zbl

[46] A. F. Kharshiladze, “Hermitian $K$-theory and quadratic extensions of rings”, Trans. Moscow Math. Soc., 1982:1 (1982), 1–37 | MR | Zbl

[47] I. S. Klein and A. V. Mikhalev, “Steinberg orthogonal group over a ring with involution”, Algebra Logic, 9 (1971), 88–103 | DOI | MR | Zbl

[48] I. S. Klein and A. V. Mikhalev, “Unitary Steinberg group over a ring with involution”, Algebra Logic, 9 (1971), 305–312 | DOI | MR | Zbl

[49] Klingenberg W., Witt E., “Über die Arfsche Invariante quadratischer Formen mod $2$”, J. reine angew. Math., 193 (1954), 121–122 | DOI | MR | Zbl

[50] Kreck M., Lück W., The Novikov conjecture: Geometry and algebra, Oberwolfach Semin., 33, Birkhüser, Basel, 2005 | MR | Zbl

[51] Loday J.-L., “$K$-théorie algébrique et représentations de groupes”, Ann. sci. Éc. norm. supér. Sér. 4, 9:3 (1976), 309–377 | DOI | MR | Zbl

[52] Lück W., “A basic introduction to surgery theory”, Topology of high-dimensional manifolds: Proc. Sch., Trieste, 2001, ICTP Lect. Notes, 9, Abdus Salam Int. Cent. Theor. Phys., Trieste, 2002, 1–224 | MR | Zbl

[53] Maumary S., “Proper surgery groups and Wall–Novikov groups”, Algebraic $K$-theory III: Hermitian $K$-theory and geometric applications: Proc. Conf. Seattle Res. Cent. Battelle Mem. Inst., 1972, Lect. Notes Math., 343, Springer, Berlin, 1973, 526–539 | DOI | MR

[54] May J.P., Thomason R., “The uniqueness of infinite loop space machines”, Topology, 17:3 (1978), 205–224 | DOI | MR | Zbl

[55] Milnor J., Differentiable manifolds which are homotopy spheres, Preprint, Princeton Univ., Princeton, NJ, 1959 ; Collected papers of John Milnor. III: Differential topology, Am. Math. Soc., Providence, RI, 2007, 65–88 | Zbl | MR

[56] Milnor J., “A procedure for killing homotopy groups of differentiable manifolds”, Differential geometry, Proc. Symp. Pure Math., 3, Am. Math. Soc., Providence, RI, 1961, 39–55 ; Milnor Dzh., Vvedenie v algebraicheskuyu $K$-teoriyu, Mir, M., 1974 | DOI | MR | Zbl

[57] J. Milnor, Introduction to Algebraic $K$-Theory, Ann. Math. Stud., 72, Princeton Univ. Press, Princeton, NJ, 1971 | DOI | MR | Zbl

[58] A. S. Miščenko, “Homotopy invariants of nonsimply connected manifolds. I: Rational invariants”, Math. USSR, Izv., 4:3 (1970), 506–519 | DOI | MR

[59] A. S. Miščenko, “Homotopy invariants of nonsimply connected manifolds. II: Simple homotopy type”, Math. USSR, Izv., 5:3 (1971), 668–679 | DOI | MR

[60] A. S. Miščenko, “Homotopy invariants of nonsimply connected manifolds. III: Higher signatures”, Math. USSR, Izv., 5:6 (1971), 1325–1364 | DOI | MR

[61] A. S. Mishchenko, “Hermitian $K$-theory. The theory of characteristic classes and methods of functional analysis”, Russ. Math. Surv., 31:2 (1976), 71–138 | DOI | MR | Zbl

[62] Nikolaus T., Scholze P., “On topological cyclic homology”, Acta math., 221:2 (2018), 203–409 | DOI | MR | Zbl

[63] S. P. Novikov, “Diffeomorphisms of simply connected manifolds”, Sov. Math., Dokl., 3 (1962), 540–543 | MR | Zbl

[64] Novikov S.P., “Gomotopicheski ekvivalentnye gladkie mnogoobraziya. I”, Izv. AN SSSR. Ser. mat., 28:2 (1964), 365–474

[65] S. P. Novikov, “Manifolds with free Abelian fundamental groups and their applications. (Pontryagin classes, smoothnesses, multidimensional knots.)”, Am. Math. Soc. Transl., Ser. 2, 71 (1968), 1–42 | MR | Zbl

[66] S. P. Novikov, “Algebraic construction and properties of Hermitian analogs of $K$-theory over rings with involution from the viewpoint of Hamiltonian formalism. Applications to differential topology and the theory of characteristic classes. I”, Math. USSR, Izv., 4:2 (1970), 257–292 | DOI | MR | Zbl

[67] S. P. Novikov, “Algebraic construction and properties of Hermitian analogs of $K$-theory over rings with involution from the viewpoint of Hamiltonian formalism. Applications to differential topology and the theory of characteristic classes. II”, Math. USSR, Izv., 4:3 (1970), 479–505 | DOI | MR | Zbl

[68] Novikov S.P., “Pontrjagin classes, the fundamental group and some problems of stable algebra”, Essays on topology and related topics: Mémoires dédiés à Georges de Rham, Springer, Berlin, 1970, 147–155 | DOI | MR

[69] Novikov conjectures, index theorems and rigidity: Oberwolfach 1993, v. 1, 2, LMS Lect. Note Ser., 226, 227, ed. by S.C. Ferry, A. Ranicki, J.M. Rosenberg, Cambridge Univ. Press, Cambridge, 1995 | MR | Zbl

[70] I. A. Panin and C. Walter, “On the relation of symplectic algebraic cobordism to hermitian K-theory”, Proc. Steklov Inst. Math., 307 (2019), 162–173 | DOI | DOI | MR | Zbl

[71] I. Panin and C. Walter, “On the algebraic cobordism spectra $\mathbf {MSL}$ and $\mathbf {MSp}$”, St. Petersburg Math. J., 34:1 (2023), 109–141 | DOI | MR | Zbl

[72] Pardon W.L., The exact sequence of a localization in $L$-theory, PhD thesis, Princeton Univ., Princeton, NJ, 1974 | MR

[73] Pardon W., “The exact sequence of a localization for Witt groups”, Algebraic $K$-theory: Proc. Conf. Northwestern Univ., Evanston, 1976, Lect. Notes Math., 551, Springer, Berlin, 1976, 336–379 | DOI | MR

[74] Pedersen E.K., Ranicki A., “Projective surgery theory”, Topology, 19:3 (1980), 239–254 | DOI | MR | Zbl

[75] Pfister A., “Some remarks on the historical development of the algebraic theory of quadratic forms”, Quadratic and Hermitian forms: Proc. Conf. McMaster Univ., Hamilton, 1983, CMS Conf. Proc., 4, Am. Math. Soc., Providence, RI, 1984, 1–16 | MR

[76] Pfister A., “On the Milnor conjectures: History, influence, applications”, Jahresber. Dtsch. Math.-Ver., 102:1 (2000), 15–41 | MR | Zbl

[77] Quillen D., “The Adams conjecture”, Topology, 10:1 (1971), 67–80 | DOI | MR | Zbl

[78] Quillen D., “Higher $K$-theory for categories with exact sequences”, New developments in topology: Proc. Symp., Oxford, 1972, LMS Lect. Note Ser., 11, Cambridge Univ. Press, Cambridge, 1974, 95–103 | MR

[79] Quillen D., “Higher algebraic $K$-theory. I”, Algebraic $K$-theory I: Higher $K$-theories: Proc. Conf. Seattle Res. Cent. Battelle Mem. Inst., 1972, Lect. Notes Math., 341, Springer, Berlin, 1973, 85–147 | DOI | MR

[80] Ranicki A.A., “Algebraic $L$-theory. III: Twisted Laurent extensions”, Algebraic $K$-theory III: Hermitian $K$-theory and geometric applications: Proc. Conf. Seattle Res. Cent. Battelle Mem. Inst., 1972, Lect. Notes Math., 343, Springer, Berlin, 1973, 412–463 | DOI | MR

[81] Ranicki A., “The algebraic theory of surgery. I: Foundations”, Proc. London Math. Soc. Ser. 3, 40:1 (1980), 87–192 | DOI | MR | Zbl

[82] Ranicki A., “The algebraic theory of surgery. II: Applications to topology”, Proc. London Math. Soc. Ser. 3, 40:2 (1980), 193–283 | DOI | MR | Zbl

[83] Ranicki A., Exact sequences in the algebraic theory of surgery, Math. Notes, 26, Princeton Univ. Press, Princeton, NJ, 1981 | MR | Zbl

[84] Scharlau W., “On the history of the algebraic theory of quadratic forms”, Quadratic forms and their applications: Proc. Conf. Univ. College Dublin, 1999, Contemp. Math., 272, Am. Math. Soc., Providence, RI, 2000, 229–259 | DOI | MR | Zbl

[85] Schlichting M., “Hermitian $K$-theory of exact categories”, J. K-Theory, 5:1 (2010), 105–165 | DOI | MR | Zbl

[86] Schlichting M., “Higher algebraic $K$-theory (after Quillen, Thomason and others)”, Topics in algebraic and topological $K$-theory, Lect. Notes Math., 2008, Springer, Berlin, 2011, 167–241 | DOI | MR | Zbl

[87] Schlichting M., “Hermitian $K$-theory, derived equivalences and Karoubi's fundamental theorem”, J. Pure Appl. Algebra, 221:7 (2017), 1729–1844 | DOI | MR | Zbl

[88] Segal G., “Categories and cohomology theories”, Topology, 13:3 (1974), 293–312 | DOI | MR | Zbl

[89] Shaneson J.L., “Wall's surgery obstruction groups for $Z\times G$, for suitable groups $G$”, Bull. Am. Math. Soc., 74:3 (1968), 467–471 | DOI | MR | Zbl

[90] Sharpe R.W., “On the structure of the unitary Steinberg group”, Ann. Math. Ser. 2, 96:3 (1972), 444–479 | DOI | MR | Zbl

[91] Yu. P. Solov'ev, “Algebraic $K$-theory of quadratic forms”, J. Sov. Math., 44:3 (1989), 319–371 | DOI | Zbl

[92] Springer T.A., “Note on quadratic forms in characteristic $2$”, Nieuw Arch. Wiskd. Ser. 3, 10 (1962), 1–10 | MR | Zbl

[93] Suslin A.A., “On the equivalence of $K$-theories”, Commun. Algebra, 9:15 (1981), 1559–1566 | DOI | MR | Zbl

[94] A. A. Suslin, “Algebraic $K$-theory”, J. Sov. Math., 28:6 (1985), 870–923 | DOI | Zbl

[95] A. A. Suslin, “Algebraic $K$-theory (in the Steklov Mathematical Institute of the Academy of Sciences)”, Proc. Steklov Inst. Math., 168 (1986), 161–177 | MR | Zbl | Zbl

[96] Swan R.G., Algebraic $K$-theory, Lect. Notes Math., 76, Springer, Berlin, 1968 | DOI | MR | Zbl

[97] Swan R.G., “Nonabelian homological algebra and $K$-theory”, Applications of categorical algebra: Proc. Symp., New York, 1968, Proc. Symp. Pure Math., 17, Am. Math. Soc., Providence, RI, 1970, 88–123 | DOI | MR | Zbl

[98] Swan R.G., “Excision in algebraic $K$-theory”, J. Pure Appl. Algebra, 1:3 (1971), 221–252 | DOI | MR | Zbl

[99] Swan R.G., “Some relations between higher $K$-functors”, J. Algebra, 21:1 (1972), 113–136 | DOI | MR | Zbl

[100] Taylor L.R., Surgery on paracompact manifolds, PhD thesis, Univ. Calif., Berkeley, 1971 ; arXiv: 1608.05683 [math.GT] | MR | DOI

[101] Thomason R.W., Trobaugh T., “Higher algebraic $K$-theory of schemes and of derived categories”, The Grothendieck Festschrift, v. III, Prog. Math., 88, Birkhäuser, Boston, 1990, 247–435 | DOI | MR | Zbl

[102] Tits J., “Formes quadratiques, groupes orthogonaux et algèbres de Clifford”, Invent. math., 5 (1968), 19–41 | DOI | MR | Zbl

[103] V. A. Vardanyan, “On periodicity in Hermitian theories”, Russ. Math. Surv., 33:2 (1978), 253–254 | DOI | MR | Zbl

[104] L. N. Vasershtein, “Foundations of algebraic $K$-theory”, Russ. Math. Surv., 31:4 (1976), 89–156 | DOI | MR

[105] I. A. Volodin, “Algebraic $K$-theory as extraordinary homology theory on the category of associative rings with unity”, Math. USSR, Izv., 5:4 (1971), 859–887 | DOI | MR | Zbl

[106] Volodin I.A., “Algebraicheskaya $K$-teoriya”, UMN, 27:4 (1972), 207–208 | MR | Zbl

[107] Wagoner J.B., “Delooping classifying spaces in algebraic $K$-theory”, Topology, 11:4 (1972), 349–370 | DOI | MR | Zbl

[108] Waldhausen F., “Algebraic $K$-theory of topological spaces. I”, Algebraic and geometric topology: Proc. Conf. Stanford, CA, 1976, Proc. Symp. Pure Math., 32, Pt. 1, Am. Math. Soc., Providence, RI, 1978, 35–60 | DOI | MR

[109] Wall C.T.C., “Surgery of non-simply-connected manifolds”, Ann. Math. Ser. 2, 84:2 (1966), 217–276 | DOI | MR | Zbl

[110] Wall C.T.C., “On the axiomatic foundations of the theory of Hermitian forms”, Proc. Cambridge Philos. Soc., 67:2 (1970), 243–250 | DOI | MR | Zbl

[111] Wall C.T.C., “On the classification of Hermitian forms. I: Rings of algebraic integers”, Compos. math., 22:4 (1970), 425–451 | MR | Zbl

[112] Wall C.T.C., “On the classification of Hermitian forms. II: Semisimple rings”, Invent. math., 18 (1972), 119–141 | DOI | MR

[113] Wall C.T.C., “Foundations of algebraic $L$-theory”, Algebraic $K$-theory III: Hermitian $K$-theory and geometric applications: Proc. Conf. Seattle Res. Cent. Battelle Mem. Inst., 1972, Lect. Notes Math., 343, Springer, Berlin, 1973, 266–300 | DOI | MR

[114] Wall C.T.C., “On the classification of Hermitian forms. III: Complete semilocal rings”, Invent. math., 19 (1973), 59–71 | DOI | MR | Zbl

[115] Wall C.T.C., “Some $L$ groups of finite groups”, Bull. Am. Math. Soc., 79:3 (1973), 526–529 | DOI | MR | Zbl

[116] Wall C.T.C., “On the classification of Hermitian forms. IV: Adele rings”, Invent. math., 23 (1974), 241–260 | DOI | MR | Zbl

[117] Wall C.T.C., “On the classification of Hermitian forms. V: Global rings”, Invent. math., 23 (1974), 261–288 | DOI | MR | Zbl

[118] Wall C.T.C., “Classification of Hermitian forms. VI: Group rings”, Ann. Math. Ser. 2, 103:1 (1976), 1–80 | DOI | MR | Zbl

[119] Wall C.T.C., Surgery on compact manifolds, Math. Surv. Monogr., 69, 2nd ed., Am. Math. Soc., Providence, RI, 1999 | DOI | MR | Zbl

[120] Wallace A.H., “Modifications and cobounding manifolds”, Can. J. Math., 12 (1960), 503–528 | DOI | MR

[121] Weibel C.A., The $K$-book: An introduction to algebraic $K$-theory, Grad. Stud. Math., 145, Am. Math. Soc., Providence, RI, 2013 | DOI | MR | Zbl

[122] Williams B., “Quadratic $K$-theory and geometric topology”, Handbook of $K$-theory, v. 2, Springer, Berlin, 2005, 611–651 | DOI | MR | Zbl

[123] Witt E., “Theorie der quadratischen Formen in beliebigen Körpern”, J. reine angew. Math., 176 (1937), 31–44 | DOI | MR

[124] G. Yu, “The Novikov conjecture”, Russ. Math. Surv., 74:3 (2019), 525–541 | DOI | DOI | MR | Zbl