Geodesic
A perturbation of the de Rham complex
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 13 (2020) no. 5, pp. 519-532.

Voir la notice de l'article provenant de la source Math-Net.Ru

We consider a perturbation of the de Rham complex on a compact manifold with boundary. This perturbation goes beyond the framework of complexes, and so cohomology does not apply to it. On the other hand, its curvature is "small" hence there is a natural way to introduce an Euler characteristic and develop a Lefschetz theory for the perturbation. This work is intended as an attempt to develop a cohomology theory for arbitrary sequences of linear mappings.
Keywords: De Rham complex, cohomology, Hodge theory, Neumann problem. 
@article{JSFU_2020_13_5_a0,
     author = {Ihsane Malass and Nikolai Tarkhanov},
     title = {A perturbation of the de {Rham} complex},
     journal = {\v{Z}urnal Sibirskogo federalʹnogo universiteta. Matematika i fizika},
     pages = {519--532},
     publisher = {mathdoc},
     volume = {13},
     number = {5},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JSFU_2020_13_5_a0/}
}
TY  - JOUR
AU  - Ihsane Malass
AU  - Nikolai Tarkhanov
TI  - A perturbation of the de Rham complex
JO  - Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika
PY  - 2020
SP  - 519
EP  - 532
VL  - 13
IS  - 5
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/JSFU_2020_13_5_a0/
LA  - en
ID  - JSFU_2020_13_5_a0
ER  - 
%0 Journal Article
%A Ihsane Malass
%A Nikolai Tarkhanov
%T A perturbation of the de Rham complex
%J Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika
%D 2020
%P 519-532
%V 13
%N 5
%I mathdoc
%U http://geodesic.mathdoc.fr/item/JSFU_2020_13_5_a0/
%G en
%F JSFU_2020_13_5_a0
Ihsane Malass; Nikolai Tarkhanov. A perturbation of the de Rham complex. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 13 (2020) no. 5, pp. 519-532. http://geodesic.mathdoc.fr/item/JSFU_2020_13_5_a0/

[1] S. Agmon, A. Douglis, L. Nirenberg, “Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. Pt. I”, Comm. Pure Appl. Math., 12 (1959), 623–727 | DOI | MR | Zbl

[2] C.-G. Ambrozie, F.-H. Vasilescu, Banach Space Complexes, Mathematics and its Applications, 334, Kluwer Academic Publishers, Dordrecht, NL, 1995 | MR | Zbl

[3] L. Boutet de Monvel, “Boundary problems for pseudodifferential operators”, Acta Math., 126 (1971), 11–51 | DOI | MR | Zbl

[4] L.Hörmander, “$L^2$ estimates and existence theorems for the $\overline{\partial}$ operator”, Acta Math., 113:1–2 (1965), 89–152 | DOI | MR

[5] I. Malass, N. Tarkhanov, J. Siber. Fed. Univ., Math. and Phys., 12:4 (2019), 455–465 | DOI | MR

[6] S.P. Novikov, “On Metric-Independent Exotic Homology”, Proc. Steklov Inst. Math., 251 (2005), 206–212 | MR | Zbl

[7] M. Putinar, “Some invariants for semi- Fredholm systems of essentially commuting operators”, J. Operator Theory, 8 (1982), 65–90 | MR | Zbl

[8] D.B. Ray, I.M. Singer, “Reidemeister torsion and the Laplacian on Riemannian manifolds”, Adv. in Math., 7 (1971), 145–210 | DOI | MR | Zbl

[9] D.C. Spencer, “Harmonic integrals and Neumann problems associated with linear partial differential equations”, Outlines of Joint Soviet-American Symposium on Partial Differential Equations, Novosibirsk, 1963, 253–260 | MR

[10] N. Tarkhanov, Complexes of Differential Operators, Kluwer Academic Publishers, Dordrecht, NL, 1995 | MR | Zbl

[11] N. Tarkhanov, “Euler characteristic of Fredholm quasicomplexes”, Funct. Anal. and its Appl., 41 (2007), 318–322 | DOI | MR | Zbl

[12] K. Krupchyk, N. Tarkhanov, J. Tuomela, “Elliptic quasicomplexes in Boutet de Monvel algebra”, J. of Funct. Anal., 247 (2007), 202–230 | DOI | MR | Zbl

[13] D. Wallenta, “A Lefschetz fixed point formula for elliptic quasicomplexes”, Integr. Equ. Oper. Theory, 78:4 (2014), 577–587 | DOI | MR | Zbl

[14] R.O. Wells, Differential Analysis on Complex Manifolds, Springer-Verlag, New York, 1980 | MR | Zbl

[15] E. Witten, “Supersymmetry and Morse theory”, J. Diff. Geom., 17 (1982), 661–692 | DOI | MR | Zbl