Dirac spectra and the topology of spaces of metrics

Nifa, Anass

doi:10.5281/zenodo.23546912

Preprint · v3

Dirac spectra and the topology of spaces of metrics

Anass NifaPublished: 24 November 2025Updated: 21 October 2026

Read free PDF Read on Zenodo Zenodo record

Free PDF · 126 pages · 1.61 MB

Abstract

Let $M$ be a closed connected spin manifold of dimension $n\ge3$ . We prescribe arbitrary admissible finite Dirac spectra in bounded open symmetric intervals, including multiplicities and nullity. Kernel dimensions are unbounded on each $M$ . Nonzero eigenvalues generically have the least multiplicity allowed by the spinor structures, also at fixed density.

Compatible multiplication perturbations of Dirac operators are approximated in norm resolvent by metric Dirac operators, including first parameter derivatives for compact smooth families. The metrics are arbitrarily $C^0$ -close, preserve density, and agree with the original metric outside a prescribed neighborhood of the support. In dimensions four and five, such changes in any nonempty open set realize the first admissible nullity above the index minimum.

If invertible metrics exist, ordinary real index difference identifies their components at fixed density with $KO_{n+1}$ ; for $n\ge5$ , each contains a positive-scalar-curvature metric whenever one exists. With density and the metric outside any nonempty open set fixed, the Dirac-minimal space has weak homotopy type $\mathrm{Sp}/\mathrm{Sp}(a)$ for $n\equiv4\pmod8$ , and $\Omega\mathrm{Sp}$ for $n\equiv5\pmod8$ , where $a=|\operatorname{ind}_{\mathbb H}D^+|$ and $\mathrm{Sp}$ is the stable compact symplectic group. On compact connected spin $n$ -manifolds, invertible boundary metrics admit invertible cylindrical extensions exactly when the relative real index vanishes.

MSC 2020

58J50: Spectral problems; spectral geometry; scattering theory on manifolds
53C27: Spin and Spin^c geometry
58J20: Index theory and related fixed-point theorems on manifolds
58D17: Manifolds of metrics (especially Riemannian)
57R65: Surgery and handlebodies

Mathematics Subject Classification 2020

Record

Version DOI: 10.5281/zenodo.23546912
All versions: 10.5281/zenodo.23546911
Subjects: Spin Dirac operator; norm-resolvent approximation; generic simplicity; harmonic spinor; spectral prescription; real index difference; spaces of metrics; positive scalar curvature
Language: English

Cite this paper

Anass Nifa. Dirac spectra and the topology of spaces of metrics. Preprint, Zenodo, v3 (2025). doi:10.5281/zenodo.23546912

Download BibTeX Download RIS

References

As listed in the manuscript bibliography.

[1]
I. Agricola and T. Friedrich, The Casimir operator of a metric connection with skew-symmetric torsion, J. Geom. Phys. 50 (2004), 188–204. doi:10.1016/j.geomphys.2003.11.001.
DOI
[2]
B. Ammann and M. Dahl, The space of Dirac-minimal metrics is connected in dimensions 2 and 4, SIGMA Symmetry Integrability Geom. Methods Appl. 21 (2025), Paper No. 102, 18 pp. doi:10.3842/SIGMA.2025.102.
DOI
[3]
B. Ammann, M. Dahl, and E. Humbert, Surgery and harmonic spinors, Adv. Math. 220 (2009), 523–539. arXiv:math/0606224.
arXiv
[4]
B. Ammann, M. Dahl, and E. Humbert, Harmonic spinors and local deformations of the metric, Math. Res. Lett. 18 (2011), 927–936. doi:10.4310/MRL.2011.v18.n5.a10. Theorem locators refer to arXiv:0903.4544v3.
DOI · arXiv v3
[5]
B. Ammann and N. Ginoux, Dirac-harmonic maps from index theory, Calc. Var. Partial Differential Equations 47 (2013), 739–762. doi:10.1007/s00526-012-0534-z. The conjecture is cited from arXiv:1110.1309v2.
DOI · arXiv v2
[6]
T. M. Apostol, Introduction to Analytic Number Theory, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1976. doi:10.1007/978-1-4757-5579-4.
DOI
[7]
N. Aronszajn, A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order, J. Math. Pures Appl. (9) 36 (1957), 235–249.
[8]
T. Aschieri, B. Ruba, and J. P. Solovej, SU(2)-equivariant quantum channels: semiclassical analysis, Comm. Math. Phys. 405 (2024), Article No. 298. doi:10.1007/s00220-024-05178-1.
DOI
[9]
M. F. Atiyah, Riemann surfaces and spin structures, Ann. Sci. École Norm. Sup. (4) 4 (1971), 47–62. DOI: 10.24033/asens.1205.
DOI
[10]
M. F. Atiyah, R. Bott, and A. Shapiro, Clifford modules, Topology 3, Suppl. 1 (1964), 3–38. doi:10.1016/0040-9383(64)90003-5.
DOI
[11]
M. F. Atiyah, V. K. Patodi, and I. M. Singer, Spectral asymmetry and Riemannian geometry. I, Math. Proc. Cambridge Philos. Soc. 77 (1975), 43–69. doi:10.1017/S0305004100049410.
DOI
[12]
M. F. Atiyah and I. M. Singer, The index of elliptic operators: III, Ann. of Math. (2) 87 (1968), 546–604. doi:10.2307/1970717.
DOI
[13]
M. F. Atiyah and I. M. Singer, Index theory for skew-adjoint Fredholm operators, Publ. Math. IHÉS 37 (1969), 5–26. doi:10.1007/BF02684885.
DOI
[14]
M. F. Atiyah and I. M. Singer, The index of elliptic operators: V, Ann. of Math. (2) 93 (1971), 139–149. doi:10.2307/1970757.
DOI
[15]
L. Bandara, A. McIntosh, and A. Rosén, Riesz continuity of the Atiyah–Singer Dirac operator under perturbations of the metric, Math. Ann. 370 (2018), 863–915. doi:10.1007/s00208-017-1610-7. Theorem locators refer to arXiv:1603.03647v2.
DOI · arXiv v2
[16]
C. Bär, Metrics with harmonic spinors, Geom. Funct. Anal. 6 (1996), 899–942. doi:10.1007/BF02246994.
DOI
[17]
C. Bär, Harmonic spinors for twisted Dirac operators, Math. Ann. 309 (1997), 225–246. doi:10.1007/s002080050111. Theorem locators refer to arXiv:dg-ga/9706016.
DOI · arXiv
[18]
C. Bär and W. Ballmann, Guide to elliptic boundary value problems for Dirac-type operators, in Arbeitstagung Bonn 2013, Progr. Math. 319, Birkhäuser, Cham, 2016, pp. 43–80. doi:10.1007/978-3-319-43648-7_3. Theorem locators refer to arXiv:1307.3021v2.
DOI · arXiv v2
[19]
C. Bär and M. Dahl, Surgery and the spectrum of the Dirac operator, J. Reine Angew. Math. 552 (2002), 53–76. arXiv:math/0201195.
arXiv
[20]
C. Bär, P. Gauduchon, and A. Moroianu, Generalized cylinders in semi-Riemannian and spin geometry, Math. Z. 249 (2005), 545–580. doi:10.1007/s00209-004-0718-0. Equation numbers refer to arXiv:math/0303095v1.
DOI · arXiv v1
[21]
G. Bazzoni, L. Martín-Merchán, and V. Muñoz, Spin-harmonic structures and nilmanifolds, Comm. Anal. Geom. 32 (2024), 153–201. arXiv:1904.01462v2.
arXiv v2
[22]
E. Bierstone and P. D. Milman, Semianalytic and subanalytic sets, Publ. Math. IHÉS 67 (1988), 5–42. doi:10.1007/BF02699126.
DOI
[23]
R. Bott, The stable homotopy of the classical groups, Ann. of Math. (2) 70 (1959), 313–337. doi:10.2307/1970106.
DOI
[24]
B. Botvinnik, J. Ebert, and O. Randal-Williams, Infinite loop spaces and positive scalar curvature, Invent. Math. 209 (2017), 749–835. doi:10.1007/s00222-017-0719-3. Theorem locators refer to arXiv:1411.7408v3.
DOI · arXiv v3
[25]
J.-P. Bourguignon and P. Gauduchon, Spineurs, opérateurs de Dirac et variations de métriques, Comm. Math. Phys. 144 (1992), 581–599. doi:10.1007/BF02099184.
DOI
[26]
C. Bourne, A. L. Carey, M. Lesch, and A. Rennie, The KO-valued spectral flow for skew-adjoint Fredholm operators, J. Topol. Anal. 14 (2022), no. 2, 505–556. doi:10.1142/S1793525320500557. Section and theorem locators refer to arXiv:1907.04981v2 (6 April 2020).
DOI · arXiv v2
[27]
G. C. Brito and M. A. M. Marrocos, On the generic simplicity of the spectrum for connection Laplacian and G-simplicity on principal bundles, preprint, 13 February 2026. arXiv:2602.12884v1.
arXiv v1
[28]
A. Calderon and N. Salter, Higher spin mapping class groups and strata of Abelian differentials over Teichmüller space, Adv. Math. 389 (2021), 107926. doi:10.1016/j.aim.2021.107926.
DOI
[29]
Y. Canzani, On the multiplicity of eigenvalues of conformally covariant operators, Ann. Inst. Fourier 64 (2014), 947–970. doi:10.5802/aif.2870.
DOI
[30]
A. L. Carey, J. Phillips, and H. Schulz-Baldes, Spectral flow for skew-adjoint Fredholm operators, J. Spectr. Theory 9 (2019), 137–170. doi:10.4171/JST/243.
DOI
[31]
M. Chatzakou and B. Helffer, Generic simplicity for self-adjoint operators under bounded potential perturbations, preprint, version 2 (16 July 2026). arXiv:2605.31368v2.
arXiv v2
[32]
Y. Colin de Verdière, Construction de laplaciens dont une partie finie du spectre est donnée, Ann. Sci. École Norm. Sup. (4) 20 (1987), 599–615. doi:10.24033/asens.1546.
DOI
[33]
D. Crowley, T. Schick, and W. Steimle, Harmonic spinors and metrics of positive curvature via the Gromoll filtration and Toda brackets, J. Topol. 11 (2018), 1077–1099. doi:10.1112/topo.12081.
DOI
[34]
G. M. D’Ariano, P. Perinotti, and M. F. Sacchi, Informationally complete measurements and group representation, J. Opt. B: Quantum Semiclass. Opt. 6 (2004), S487–S491. doi:10.1088/1464-4266/6/6/005.
DOI
[35]
M. Dahl, Dirac eigenvalues for generic metrics on three-manifolds, Ann. Global Anal. Geom. 24 (2003), 95–100. doi:10.1023/A:1024231524848. The correction and theorem locators refer to arXiv:math/0204232v2 (3 July 2013).
DOI · arXiv v2
[36]
M. Dahl, Prescribing eigenvalues of the Dirac operator, Manuscripta Math. 118 (2005), 191–199. arXiv:math/0311172v3. doi:10.1007/s00229-005-0583-0.
arXiv v3 · DOI
[37]
M. Dahl, On the space of metrics with invertible Dirac operator, Comment. Math. Helv. 83 (2008), 451–469. doi:10.4171/CMH/132.
DOI
[38]
M. Dahl and N. Große, Invertible Dirac operators and handle attachments on manifolds with boundary, J. Topol. Anal. 6 (2014), 339–382. arXiv:1203.3637v2.
arXiv v2
[39]
R. J. Downes, M. Levitin, and D. Vassiliev, Spectral asymmetry of the massless Dirac operator on a 3-torus, J. Math. Phys. 54 (2013), 111503. arXiv:1306.5689.
arXiv
[40]
S. Dyatlov, Lecture notes for 18.155: distributions, elliptic regularity, and applications to PDEs, Massachusetts Institute of Technology, version of 20 October 2026. Lecture notes.
Source
[41]
J. Ebert, The two definitions of the index difference, Trans. Amer. Math. Soc. 369 (2017), 7469–7507. doi:10.1090/tran/7133. Theorem locators refer to arXiv:1308.4998v3.
DOI · arXiv v3
[42]
J. Ebert and O. Randal-Williams, Infinite loop spaces and positive scalar curvature in the presence of a fundamental group, Geom. Topol. 23 (2019), 1549–1610. doi:10.2140/gt.2019.23.1549.
DOI
[43]
Y. Eliashberg and N. Mishachev, Introduction to the h-Principle, Graduate Studies in Mathematics 48, American Mathematical Society, Providence, RI, 2002. doi:10.1090/gsm/048.
DOI
[44]
S. Farinelli, Riemannian metrics and harmonic sections of spinor bundles, preprint, 2012, version 10 (2024). arXiv:1204.3248v10. doi:10.48550/arXiv.1204.3248.
arXiv v10 · DOI
[45]
D. Ferus, H. Karcher, and H.-F. Münzner, Cliffordalgebren und neue isoparametrische Hyperflächen, Math. Z. 177 (1981), 479–502. doi:10.1007/BF01219082. English translation by T. E. Cecil, arXiv:1112.2780.
DOI · arXiv
[46]
D. S. Freed, Multivariable analysis, lecture notes, version of 2 February 2021. Author notes.
Source
[47]
W. Fulton and J. Harris, Representation Theory: A First Course, Graduate Texts in Mathematics 129, Springer-Verlag, New York, 1991. doi:10.1007/978-1-4612-0979-9.
DOI
[48]
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer, Berlin, 2001, reprint of the 1998 edition. doi:10.1007/978-3-642-61798-0.
DOI
[49]
H. Gimperlein and G. Grubb, Heat kernel estimates for pseudodifferential operators, fractional Laplacians and Dirichlet-to-Neumann operators, J. Evol. Equ. 14 (2014), 49–83. doi:10.1007/s00028-013-0206-2.
DOI
[50]
G. H. Golub and C. F. Van Loan, Matrix Computations, third edition, Johns Hopkins University Press, Baltimore, 1996.
[51]
M. Gromov and H. B. Lawson, Jr., The classification of simply connected manifolds of positive scalar curvature, Ann. of Math. (2) 111 (1980), 423–434. doi:10.2307/1971103.
DOI
[52]
N. Große and N. Pederzani, Homotopy equivalence of spaces of metrics with invertible Dirac operator, preprint, 2019, version 2 (2022). arXiv:1910.09167v2. doi:10.48550/arXiv.1910.09167.
arXiv v2 · DOI
[53]
V. Guillemin and A. Pollack, Differential Topology, Prentice-Hall, Englewood Cliffs, NJ, 1974.
[54]
B. C. Hall, Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, second edition, Graduate Texts in Mathematics 222, Springer, Cham, 2015. doi:10.1007/978-3-319-13467-3.
DOI
[55]
A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002. Author text.
Source
[56]
A. Hermann, Dirac eigenspinors for generic metrics, doctoral dissertation, Universität Regensburg, 2012. arXiv:1201.5771v3.
arXiv v3
[57]
T. Hertl, Concordances in positive scalar curvature and index theory, doctoral dissertation, Georg-August-Universität Göttingen, 2022; arXiv:2303.07844v1 (2023). doi:10.48550/arXiv.2303.07844.
arXiv v1 · DOI
[58]
N. Higson and G. Kasparov, E-theory and KK-theory for groups which act properly and isometrically on Hilbert space, Invent. Math. 144 (2001), 23–74. doi:10.1007/s002220000118.
DOI
[59]
M. W. Hirsch, Differential Topology, Graduate Texts in Mathematics 33, Springer, New York, 1976. doi:10.1007/978-1-4684-9449-5.
DOI
[60]
N. Hitchin, Harmonic spinors, Adv. Math. 14 (1974), 1–55. doi:10.1016/0001-8708(74)90021-8.
DOI
[61]
L. Hörmander, The Analysis of Linear Partial Differential Operators III: Pseudo-Differential Operators, Classics in Mathematics, Springer, Berlin, 2007, reprint of the 1994 edition. doi:10.1007/978-3-540-49938-1.
DOI
[62]
T. Kato, Perturbation Theory for Linear Operators, second edition, corrected printing, Grundlehren der mathematischen Wissenschaften 132, Springer-Verlag, Berlin, 1980; reprinted in Classics in Mathematics, Springer-Verlag, Berlin, 1995. doi:10.1007/978-3-642-66282-9.
DOI
[63]
A. Khrabustovskyi and V. Lotoreichik, Homogenization of the Dirac operator with position-dependent mass, Rev. Mat. Complut. 39 (2026), 381–405. doi:10.1007/s13163-025-00543-7.
DOI
[64]
F. Lin, Divergence-free framings of three-manifolds via eigenspinors, Doc. Math. 31 (2026), 445–451. doi:10.4171/DM/989.
DOI
[65]
S. Lockman, A non-trivial index difference on surfaces of genus at least 3, preprint, version 2 (7 August 2026). arXiv:2606.12283v2. doi:10.48550/arXiv.2606.12283.
arXiv v2 · DOI
[66]
S. Maier, Generic metrics and connections on Spin- and Spinᶜ-manifolds, Comm. Math. Phys. 188 (1997), 407–437. doi:10.1007/s002200050171.
DOI
[67]
J. Martínez-Aguinaga and Á. del Pino, Convex integration with avoidance and hyperbolic (4, 6) distributions, 2021, arXiv:2112.14632v1. Theorem 3.5 is the open-ample theorem.
arXiv v1
[68]
J. Milnor, Spin structures on manifolds, Enseign. Math. (2) 9 (1963), 198–203. doi:10.5169/seals-38784.
DOI
[69]
J. Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc. 120 (1965), 286–294. doi:10.1090/S0002-9947-1965-0182927-5.
DOI
[70]
K.-H. Neeb, Classical Hilbert–Lie groups, their extensions and their homotopy groups, Banach Center Publ. 55 (2002), 87–151. doi:10.4064/bc55-0-6.
DOI
[71]
A. Nifa, Local realization of spectral clusters, preprint, version 2, 2026. doi:10.5281/zenodo.23224622.
DOI
[72]
N. Nowaczyk, Existence of Dirac eigenvalues of higher multiplicity, Math. Z. 284 (2016), 285–307. Theorem locators refer to arXiv:1504.00961v1. doi:10.1007/s00209-016-1657-2.
arXiv v1 · DOI
[73]
L. Ornea, M. Parton, P. Piccinni, and V. Vuletescu, Spin(9) geometry of the octonionic Hopf fibration, Transform. Groups 18 (2013), no. 3, 845–864. doi:10.1007/s00031-013-9233-x. See also arXiv:1208.0899v2, Section 4.
DOI · arXiv v2
[74]
N. Pederzani, On the spin cobordism invariance of the homotopy type of the space ℛⁱⁿᵛ(M), doctoral dissertation, Universität Leipzig, 2018. urn:nbn:de:bsz:15-qucosa2-213917.
Source
[75]
C. Qian, Z. Tang, and W. Yan, Clifford systems, harmonic maps and metrics with non-negative curvature, Pacific J. Math. 320 (2022), no. 2, 391–424. doi:10.2140/pjm.2022.320.391. Section locators refer to arXiv:2202.09078v2 (23 August 2022).
DOI · arXiv v2
[76]
T. Schick, Real versus complex K-theory using Kasparov’s bivariant KK-theory, Algebr. Geom. Topol. 4 (2004), 333–346. doi:10.2140/agt.2004.4.333.
DOI
[77]
S. Smale, An infinite dimensional version of Sard’s theorem, Amer. J. Math. 87 (1965), 861–866.
[78]
S. Stolz, Simply connected manifolds of positive scalar curvature, Ann. of Math. (2) 136 (1992), 511–540. doi:10.2307/2946598.
DOI
[79]
M. E. Taylor, Multidimensional analytic Fredholm theory, author notes, Theorem 1.1, p. 2.
Source
[80]
N. Waterstraat, A remark on the space of metrics having non-trivial harmonic spinors, J. Fixed Point Theory Appl. 13 (2013), 143–149. arXiv:1206.0499v2.
arXiv v2
[81]
C. Wockel, A generalization of Steenrod’s approximation theorem, Arch. Math. (Brno) 45 (2009), no. 2, 95–104. DML-CZ.
Source
[82]
A. Zinger, The determinant line bundle for Fredholm operators: construction, properties, and classification, Math. Scand. 118 (2016), 203–268. The orientation discussion in Section 5.3 refers to the post-publication update of 30 May 2022, arXiv:1304.6368v4.
arXiv v4

[reference-1] [1]
I. Agricola and T. Friedrich, The Casimir operator of a metric connection with skew-symmetric torsion, J. Geom. Phys. 50 (2004), 188–204. doi:10.1016/j.geomphys.2003.11.001.
DOI

[reference-2] [2]
B. Ammann and M. Dahl, The space of Dirac-minimal metrics is connected in dimensions 2 and 4, SIGMA Symmetry Integrability Geom. Methods Appl. 21 (2025), Paper No. 102, 18 pp. doi:10.3842/SIGMA.2025.102.
DOI

[reference-3] [3]
B. Ammann, M. Dahl, and E. Humbert, Surgery and harmonic spinors, Adv. Math. 220 (2009), 523–539. arXiv:math/0606224.
arXiv

[reference-4] [4]
B. Ammann, M. Dahl, and E. Humbert, Harmonic spinors and local deformations of the metric, Math. Res. Lett. 18 (2011), 927–936. doi:10.4310/MRL.2011.v18.n5.a10. Theorem locators refer to arXiv:0903.4544v3.
DOI · arXiv v3

[reference-5] [5]
B. Ammann and N. Ginoux, Dirac-harmonic maps from index theory, Calc. Var. Partial Differential Equations 47 (2013), 739–762. doi:10.1007/s00526-012-0534-z. The conjecture is cited from arXiv:1110.1309v2.
DOI · arXiv v2

[reference-6] [6]
T. M. Apostol, Introduction to Analytic Number Theory, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1976. doi:10.1007/978-1-4757-5579-4.
DOI

[reference-7] [7]
N. Aronszajn, A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order, J. Math. Pures Appl. (9) 36 (1957), 235–249.

[reference-8] [8]
T. Aschieri, B. Ruba, and J. P. Solovej, SU(2)-equivariant quantum channels: semiclassical analysis, Comm. Math. Phys. 405 (2024), Article No. 298. doi:10.1007/s00220-024-05178-1.
DOI

[reference-9] [9]
M. F. Atiyah, Riemann surfaces and spin structures, Ann. Sci. École Norm. Sup. (4) 4 (1971), 47–62. DOI: 10.24033/asens.1205.
DOI

[reference-10] [10]
M. F. Atiyah, R. Bott, and A. Shapiro, Clifford modules, Topology 3, Suppl. 1 (1964), 3–38. doi:10.1016/0040-9383(64)90003-5.
DOI

[reference-11] [11]
M. F. Atiyah, V. K. Patodi, and I. M. Singer, Spectral asymmetry and Riemannian geometry. I, Math. Proc. Cambridge Philos. Soc. 77 (1975), 43–69. doi:10.1017/S0305004100049410.
DOI

[reference-12] [12]
M. F. Atiyah and I. M. Singer, The index of elliptic operators: III, Ann. of Math. (2) 87 (1968), 546–604. doi:10.2307/1970717.
DOI

[reference-13] [13]
M. F. Atiyah and I. M. Singer, Index theory for skew-adjoint Fredholm operators, Publ. Math. IHÉS 37 (1969), 5–26. doi:10.1007/BF02684885.
DOI

[reference-14] [14]
M. F. Atiyah and I. M. Singer, The index of elliptic operators: V, Ann. of Math. (2) 93 (1971), 139–149. doi:10.2307/1970757.
DOI

[reference-15] [15]
L. Bandara, A. McIntosh, and A. Rosén, Riesz continuity of the Atiyah–Singer Dirac operator under perturbations of the metric, Math. Ann. 370 (2018), 863–915. doi:10.1007/s00208-017-1610-7. Theorem locators refer to arXiv:1603.03647v2.
DOI · arXiv v2

[reference-16] [16]
C. Bär, Metrics with harmonic spinors, Geom. Funct. Anal. 6 (1996), 899–942. doi:10.1007/BF02246994.
DOI

[reference-17] [17]
C. Bär, Harmonic spinors for twisted Dirac operators, Math. Ann. 309 (1997), 225–246. doi:10.1007/s002080050111. Theorem locators refer to arXiv:dg-ga/9706016.
DOI · arXiv

[reference-18] [18]
C. Bär and W. Ballmann, Guide to elliptic boundary value problems for Dirac-type operators, in Arbeitstagung Bonn 2013, Progr. Math. 319, Birkhäuser, Cham, 2016, pp. 43–80. doi:10.1007/978-3-319-43648-7_3. Theorem locators refer to arXiv:1307.3021v2.
DOI · arXiv v2

[reference-19] [19]
C. Bär and M. Dahl, Surgery and the spectrum of the Dirac operator, J. Reine Angew. Math. 552 (2002), 53–76. arXiv:math/0201195.
arXiv

[reference-20] [20]
C. Bär, P. Gauduchon, and A. Moroianu, Generalized cylinders in semi-Riemannian and spin geometry, Math. Z. 249 (2005), 545–580. doi:10.1007/s00209-004-0718-0. Equation numbers refer to arXiv:math/0303095v1.
DOI · arXiv v1

[reference-21] [21]
G. Bazzoni, L. Martín-Merchán, and V. Muñoz, Spin-harmonic structures and nilmanifolds, Comm. Anal. Geom. 32 (2024), 153–201. arXiv:1904.01462v2.
arXiv v2

[reference-22] [22]
E. Bierstone and P. D. Milman, Semianalytic and subanalytic sets, Publ. Math. IHÉS 67 (1988), 5–42. doi:10.1007/BF02699126.
DOI

[reference-23] [23]
R. Bott, The stable homotopy of the classical groups, Ann. of Math. (2) 70 (1959), 313–337. doi:10.2307/1970106.
DOI

[reference-24] [24]
B. Botvinnik, J. Ebert, and O. Randal-Williams, Infinite loop spaces and positive scalar curvature, Invent. Math. 209 (2017), 749–835. doi:10.1007/s00222-017-0719-3. Theorem locators refer to arXiv:1411.7408v3.
DOI · arXiv v3

[reference-25] [25]
J.-P. Bourguignon and P. Gauduchon, Spineurs, opérateurs de Dirac et variations de métriques, Comm. Math. Phys. 144 (1992), 581–599. doi:10.1007/BF02099184.
DOI

[reference-26] [26]
C. Bourne, A. L. Carey, M. Lesch, and A. Rennie, The KO-valued spectral flow for skew-adjoint Fredholm operators, J. Topol. Anal. 14 (2022), no. 2, 505–556. doi:10.1142/S1793525320500557. Section and theorem locators refer to arXiv:1907.04981v2 (6 April 2020).
DOI · arXiv v2

[reference-27] [27]
G. C. Brito and M. A. M. Marrocos, On the generic simplicity of the spectrum for connection Laplacian and G-simplicity on principal bundles, preprint, 13 February 2026. arXiv:2602.12884v1.
arXiv v1

[reference-28] [28]
A. Calderon and N. Salter, Higher spin mapping class groups and strata of Abelian differentials over Teichmüller space, Adv. Math. 389 (2021), 107926. doi:10.1016/j.aim.2021.107926.
DOI

[reference-29] [29]
Y. Canzani, On the multiplicity of eigenvalues of conformally covariant operators, Ann. Inst. Fourier 64 (2014), 947–970. doi:10.5802/aif.2870.
DOI

[reference-30] [30]
A. L. Carey, J. Phillips, and H. Schulz-Baldes, Spectral flow for skew-adjoint Fredholm operators, J. Spectr. Theory 9 (2019), 137–170. doi:10.4171/JST/243.
DOI

[reference-31] [31]
M. Chatzakou and B. Helffer, Generic simplicity for self-adjoint operators under bounded potential perturbations, preprint, version 2 (16 July 2026). arXiv:2605.31368v2.
arXiv v2

[reference-32] [32]
Y. Colin de Verdière, Construction de laplaciens dont une partie finie du spectre est donnée, Ann. Sci. École Norm. Sup. (4) 20 (1987), 599–615. doi:10.24033/asens.1546.
DOI

[reference-33] [33]
D. Crowley, T. Schick, and W. Steimle, Harmonic spinors and metrics of positive curvature via the Gromoll filtration and Toda brackets, J. Topol. 11 (2018), 1077–1099. doi:10.1112/topo.12081.
DOI

[reference-34] [34]
G. M. D’Ariano, P. Perinotti, and M. F. Sacchi, Informationally complete measurements and group representation, J. Opt. B: Quantum Semiclass. Opt. 6 (2004), S487–S491. doi:10.1088/1464-4266/6/6/005.
DOI

[reference-35] [35]
M. Dahl, Dirac eigenvalues for generic metrics on three-manifolds, Ann. Global Anal. Geom. 24 (2003), 95–100. doi:10.1023/A:1024231524848. The correction and theorem locators refer to arXiv:math/0204232v2 (3 July 2013).
DOI · arXiv v2

[reference-36] [36]
M. Dahl, Prescribing eigenvalues of the Dirac operator, Manuscripta Math. 118 (2005), 191–199. arXiv:math/0311172v3. doi:10.1007/s00229-005-0583-0.
arXiv v3 · DOI

[reference-37] [37]
M. Dahl, On the space of metrics with invertible Dirac operator, Comment. Math. Helv. 83 (2008), 451–469. doi:10.4171/CMH/132.
DOI

[reference-38] [38]
M. Dahl and N. Große, Invertible Dirac operators and handle attachments on manifolds with boundary, J. Topol. Anal. 6 (2014), 339–382. arXiv:1203.3637v2.
arXiv v2

[reference-39] [39]
R. J. Downes, M. Levitin, and D. Vassiliev, Spectral asymmetry of the massless Dirac operator on a 3-torus, J. Math. Phys. 54 (2013), 111503. arXiv:1306.5689.
arXiv

[reference-40] [40]
S. Dyatlov, Lecture notes for 18.155: distributions, elliptic regularity, and applications to PDEs, Massachusetts Institute of Technology, version of 20 October 2026. Lecture notes.
Source

[reference-41] [41]
J. Ebert, The two definitions of the index difference, Trans. Amer. Math. Soc. 369 (2017), 7469–7507. doi:10.1090/tran/7133. Theorem locators refer to arXiv:1308.4998v3.
DOI · arXiv v3

[reference-42] [42]
J. Ebert and O. Randal-Williams, Infinite loop spaces and positive scalar curvature in the presence of a fundamental group, Geom. Topol. 23 (2019), 1549–1610. doi:10.2140/gt.2019.23.1549.
DOI

[reference-43] [43]
Y. Eliashberg and N. Mishachev, Introduction to the h-Principle, Graduate Studies in Mathematics 48, American Mathematical Society, Providence, RI, 2002. doi:10.1090/gsm/048.
DOI

[reference-44] [44]
S. Farinelli, Riemannian metrics and harmonic sections of spinor bundles, preprint, 2012, version 10 (2024). arXiv:1204.3248v10. doi:10.48550/arXiv.1204.3248.
arXiv v10 · DOI

[reference-45] [45]
D. Ferus, H. Karcher, and H.-F. Münzner, Cliffordalgebren und neue isoparametrische Hyperflächen, Math. Z. 177 (1981), 479–502. doi:10.1007/BF01219082. English translation by T. E. Cecil, arXiv:1112.2780.
DOI · arXiv

[reference-46] [46]
D. S. Freed, Multivariable analysis, lecture notes, version of 2 February 2021. Author notes.
Source

[reference-47] [47]
W. Fulton and J. Harris, Representation Theory: A First Course, Graduate Texts in Mathematics 129, Springer-Verlag, New York, 1991. doi:10.1007/978-1-4612-0979-9.
DOI

[reference-48] [48]
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer, Berlin, 2001, reprint of the 1998 edition. doi:10.1007/978-3-642-61798-0.
DOI

[reference-49] [49]
H. Gimperlein and G. Grubb, Heat kernel estimates for pseudodifferential operators, fractional Laplacians and Dirichlet-to-Neumann operators, J. Evol. Equ. 14 (2014), 49–83. doi:10.1007/s00028-013-0206-2.
DOI

[reference-50] [50]
G. H. Golub and C. F. Van Loan, Matrix Computations, third edition, Johns Hopkins University Press, Baltimore, 1996.

[reference-51] [51]
M. Gromov and H. B. Lawson, Jr., The classification of simply connected manifolds of positive scalar curvature, Ann. of Math. (2) 111 (1980), 423–434. doi:10.2307/1971103.
DOI

[reference-52] [52]
N. Große and N. Pederzani, Homotopy equivalence of spaces of metrics with invertible Dirac operator, preprint, 2019, version 2 (2022). arXiv:1910.09167v2. doi:10.48550/arXiv.1910.09167.
arXiv v2 · DOI

[reference-53] [53]
V. Guillemin and A. Pollack, Differential Topology, Prentice-Hall, Englewood Cliffs, NJ, 1974.

[reference-54] [54]
B. C. Hall, Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, second edition, Graduate Texts in Mathematics 222, Springer, Cham, 2015. doi:10.1007/978-3-319-13467-3.
DOI

[reference-55] [55]
A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002. Author text.
Source

[reference-56] [56]
A. Hermann, Dirac eigenspinors for generic metrics, doctoral dissertation, Universität Regensburg, 2012. arXiv:1201.5771v3.
arXiv v3

[reference-57] [57]
T. Hertl, Concordances in positive scalar curvature and index theory, doctoral dissertation, Georg-August-Universität Göttingen, 2022; arXiv:2303.07844v1 (2023). doi:10.48550/arXiv.2303.07844.
arXiv v1 · DOI

[reference-58] [58]
N. Higson and G. Kasparov, E-theory and KK-theory for groups which act properly and isometrically on Hilbert space, Invent. Math. 144 (2001), 23–74. doi:10.1007/s002220000118.
DOI

[reference-59] [59]
M. W. Hirsch, Differential Topology, Graduate Texts in Mathematics 33, Springer, New York, 1976. doi:10.1007/978-1-4684-9449-5.
DOI

[reference-60] [60]
N. Hitchin, Harmonic spinors, Adv. Math. 14 (1974), 1–55. doi:10.1016/0001-8708(74)90021-8.
DOI

[reference-61] [61]
L. Hörmander, The Analysis of Linear Partial Differential Operators III: Pseudo-Differential Operators, Classics in Mathematics, Springer, Berlin, 2007, reprint of the 1994 edition. doi:10.1007/978-3-540-49938-1.
DOI

[reference-62] [62]
T. Kato, Perturbation Theory for Linear Operators, second edition, corrected printing, Grundlehren der mathematischen Wissenschaften 132, Springer-Verlag, Berlin, 1980; reprinted in Classics in Mathematics, Springer-Verlag, Berlin, 1995. doi:10.1007/978-3-642-66282-9.
DOI

[reference-63] [63]
A. Khrabustovskyi and V. Lotoreichik, Homogenization of the Dirac operator with position-dependent mass, Rev. Mat. Complut. 39 (2026), 381–405. doi:10.1007/s13163-025-00543-7.
DOI

[reference-64] [64]
F. Lin, Divergence-free framings of three-manifolds via eigenspinors, Doc. Math. 31 (2026), 445–451. doi:10.4171/DM/989.
DOI

[reference-65] [65]
S. Lockman, A non-trivial index difference on surfaces of genus at least 3, preprint, version 2 (7 August 2026). arXiv:2606.12283v2. doi:10.48550/arXiv.2606.12283.
arXiv v2 · DOI

[reference-66] [66]
S. Maier, Generic metrics and connections on Spin- and Spinᶜ-manifolds, Comm. Math. Phys. 188 (1997), 407–437. doi:10.1007/s002200050171.
DOI

[reference-67] [67]
J. Martínez-Aguinaga and Á. del Pino, Convex integration with avoidance and hyperbolic (4, 6) distributions, 2021, arXiv:2112.14632v1. Theorem 3.5 is the open-ample theorem.
arXiv v1

[reference-68] [68]
J. Milnor, Spin structures on manifolds, Enseign. Math. (2) 9 (1963), 198–203. doi:10.5169/seals-38784.
DOI

[reference-69] [69]
J. Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc. 120 (1965), 286–294. doi:10.1090/S0002-9947-1965-0182927-5.
DOI

[reference-70] [70]
K.-H. Neeb, Classical Hilbert–Lie groups, their extensions and their homotopy groups, Banach Center Publ. 55 (2002), 87–151. doi:10.4064/bc55-0-6.
DOI

[reference-71] [71]
A. Nifa, Local realization of spectral clusters, preprint, version 2, 2026. doi:10.5281/zenodo.23224622.
DOI

[reference-72] [72]
N. Nowaczyk, Existence of Dirac eigenvalues of higher multiplicity, Math. Z. 284 (2016), 285–307. Theorem locators refer to arXiv:1504.00961v1. doi:10.1007/s00209-016-1657-2.
arXiv v1 · DOI

[reference-73] [73]
L. Ornea, M. Parton, P. Piccinni, and V. Vuletescu, Spin(9) geometry of the octonionic Hopf fibration, Transform. Groups 18 (2013), no. 3, 845–864. doi:10.1007/s00031-013-9233-x. See also arXiv:1208.0899v2, Section 4.
DOI · arXiv v2

[reference-74] [74]
N. Pederzani, On the spin cobordism invariance of the homotopy type of the space ℛⁱⁿᵛ(M), doctoral dissertation, Universität Leipzig, 2018. urn:nbn:de:bsz:15-qucosa2-213917.
Source

[reference-75] [75]
C. Qian, Z. Tang, and W. Yan, Clifford systems, harmonic maps and metrics with non-negative curvature, Pacific J. Math. 320 (2022), no. 2, 391–424. doi:10.2140/pjm.2022.320.391. Section locators refer to arXiv:2202.09078v2 (23 August 2022).
DOI · arXiv v2

[reference-76] [76]
T. Schick, Real versus complex K-theory using Kasparov’s bivariant KK-theory, Algebr. Geom. Topol. 4 (2004), 333–346. doi:10.2140/agt.2004.4.333.
DOI

[reference-77] [77]
S. Smale, An infinite dimensional version of Sard’s theorem, Amer. J. Math. 87 (1965), 861–866.

[reference-78] [78]
S. Stolz, Simply connected manifolds of positive scalar curvature, Ann. of Math. (2) 136 (1992), 511–540. doi:10.2307/2946598.
DOI

[reference-79] [79]
M. E. Taylor, Multidimensional analytic Fredholm theory, author notes, Theorem 1.1, p. 2.
Source

[reference-80] [80]
N. Waterstraat, A remark on the space of metrics having non-trivial harmonic spinors, J. Fixed Point Theory Appl. 13 (2013), 143–149. arXiv:1206.0499v2.
arXiv v2

[reference-81] [81]
C. Wockel, A generalization of Steenrod’s approximation theorem, Arch. Math. (Brno) 45 (2009), no. 2, 95–104. DML-CZ.
Source

[reference-82] [82]
A. Zinger, The determinant line bundle for Fredholm operators: construction, properties, and classification, Math. Scand. 118 (2016), 203–268. The orientation discussion in Section 5.3 refers to the post-publication update of 30 May 2022, arXiv:1304.6368v4.
arXiv v4