A complex in Morse theory computing intersection homology
Annales de l'Institut Fourier, Volume 67 (2017) no. 1, pp. 197-236.

Let X be a space with isolated conical singularities. The aim of this article is to establish, using anti-radial Morse functions on X, a combinatorial complex which computes the intersection homology of X. The complex constructed here, is generated by the smooth critical points of the Morse function and representatives of the de Rham cohomology (in low degree) of the link manifolds of the singularities of X. It can be seen as an analogue of the famous Thom-Smale complex for smooth Morse functions and singular homology on a compact manifold. The article also discusses the homotopy principle familiar in smooth Morse homology in this singular context.

Dans cet article on associe à une fonction de Morse f anti-radiale sur un espace singulier X à singularités coniques un complexe généré par les points critiques de f et par certaines formes sur le link de la singularité. Ce complexe calcule de façon canonique l’homologie d’intersection. Également on discute le comportement de ce complexe par rapport aux homotopies. Le complexe construit dans cet article est un analogue du complexe de Thom-Smale sur une variété lisse pour une fonction de Morse lisse et l’homologie singulière.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.3079
Classification: 55N33, 58A35, 58K05, 57R70
Keywords: intersection homology, Morse theory, radial vector fields, Thom-Smale complex
Mot clés : homologie d’intersection, théorie de Morse, champs radiaux, complexe de Thom-Smale

Ludwig, Ursula 1

1 Universität Duisburg-Essen Fakultät für Mathematik Thea-Leymann-Str. 9 D-45127 Essen (Germany)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{AIF_2017__67_1_197_0,
     author = {Ludwig, Ursula},
     title = {A complex in {Morse} theory computing intersection homology},
     journal = {Annales de l'Institut Fourier},
     pages = {197--236},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {67},
     number = {1},
     year = {2017},
     doi = {10.5802/aif.3079},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3079/}
}
TY  - JOUR
AU  - Ludwig, Ursula
TI  - A complex in Morse theory computing intersection homology
JO  - Annales de l'Institut Fourier
PY  - 2017
SP  - 197
EP  - 236
VL  - 67
IS  - 1
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3079/
DO  - 10.5802/aif.3079
LA  - en
ID  - AIF_2017__67_1_197_0
ER  - 
%0 Journal Article
%A Ludwig, Ursula
%T A complex in Morse theory computing intersection homology
%J Annales de l'Institut Fourier
%D 2017
%P 197-236
%V 67
%N 1
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3079/
%R 10.5802/aif.3079
%G en
%F AIF_2017__67_1_197_0
Ludwig, Ursula. A complex in Morse theory computing intersection homology. Annales de l'Institut Fourier, Volume 67 (2017) no. 1, pp. 197-236. doi : 10.5802/aif.3079. https://aif.centre-mersenne.org/articles/10.5802/aif.3079/

[1] Akaho, Manabu Morse homology and manifolds with boundary, Commun. Contemp. Math., Volume 9 (2007) no. 3, pp. 301-334 | DOI

[2] Atiyah, M. F.; Bott, R. The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A, Volume 308 (1983) no. 1505, pp. 523-615 | DOI

[3] Austin, D. M.; Braam, P. J. Morse-Bott theory and equivariant cohomology, The Floer memorial volume (Progr. Math.), Volume 133, Birkhäuser, Basel, 1995, pp. 123-183

[4] Bismut, Jean-Michel; Zhang, Weiping An extension of a theorem by Cheeger and Müller, Astérisque (1992) no. 205, 235 pages (With an appendix by François Laudenbach)

[5] Bismut, Jean-Michel; Zhang, Weiping Milnor and Ray-Singer metrics on the equivariant determinant of a flat vector bundle, Geom. Funct. Anal., Volume 4 (1994) no. 2, pp. 136-212

[6] Bott, Raoul Lectures on Morse theory, old and new, Bull. Amer. Math. Soc. (N.S.), Volume 7 (1982) no. 2, pp. 331-358 | DOI

[7] Bott, Raoul Morse theory indomitable, Inst. Hautes Études Sci. Publ. Math. (1988) no. 68, p. 99-114 (1989)

[8] Chang, Kung-Ching Infinite dimensional Morse theory and multiple solution problems., Boston: Birkhäuser, 1993

[9] Cheeger, Jeff Analytic torsion and the heat equation, Ann. of Math. (2), Volume 109 (1979) no. 2, pp. 259-322 | DOI

[10] Cheeger, Jeff On the Hodge theory of Riemannian pseudomanifolds, Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979) (Proc. Sympos. Pure Math., XXXVI), Amer. Math. Soc., Providence, R.I., 1980, pp. 91-146

[11] Cheeger, Jeff; Goresky, Mark; MacPherson, Robert L 2 -cohomology and intersection homology of singular algebraic varieties, Seminar on Differential Geometry (Ann. of Math. Stud.), Volume 102, Princeton Univ. Press, Princeton, N.J., 1982, pp. 303-340

[12] Dar, Aparna Intersection R-torsion and analytic torsion for pseudomanifolds, Math. Z., Volume 194 (1987) no. 2, pp. 193-216 | DOI

[13] Floer, Andreas Witten’s complex and infinite-dimensional Morse theory, J. Differential Geom., Volume 30 (1989) no. 1, pp. 207-221 http://projecteuclid.org.revues.math.u-psud.fr:2048/euclid.jdg/1214443291

[14] Goresky, Mark Triangulation of stratified objects, Proc. Amer. Math. Soc., Volume 72 (1978) no. 1, pp. 193-200

[15] Goresky, Mark Whitney stratified chains and cochains, Trans. Amer. Math. Soc., Volume 267 (1981) no. 1, pp. 175-196 | DOI

[16] Goresky, Mark; MacPherson, Robert Intersection homology theory, Topology, Volume 19 (1980) no. 2, pp. 135-162 | DOI

[17] Goresky, Mark; MacPherson, Robert Intersection homology. II, Invent. Math., Volume 72 (1983) no. 1, pp. 77-129 | DOI

[18] Goresky, Mark; MacPherson, Robert Simplicial Intersection Homology, Invent. Math., Volume 84 (1986) no. 2, pp. 432-433 (Appendix to MacPherson, Robert and Vilonen, Kari “Elementary construction of perverse sheaves”) | DOI

[19] Goresky, Mark; MacPherson, Robert Stratified Morse theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 14, Springer-Verlag, Berlin, 1988

[20] Grinberg, Mikhail Gradient-like flows and self-indexing in stratified Morse theory, Topology, Volume 44 (2005) no. 1, pp. 175-202 | DOI

[21] Hartmann, L.; Spreafico, M. The analytic torsion of a cone over a sphere, J. Math. Pures Appl. (9), Volume 93 (2010) no. 4, pp. 408-435 | DOI

[22] Hartmann, L.; Spreafico, M. The analytic torsion of a cone over an odd dimensional manifold, J. Geom. Phys., Volume 61 (2011) no. 3, pp. 624-657 | DOI

[23] Helffer, Bernard; Sjöstrand, Johannes Puits multiples en mécanique semi-classique. IV. Étude du complexe de Witten, Comm. Partial Differential Equations, Volume 10 (1985) no. 3, pp. 245-340

[24] Johnson, F. E. A. On the triangulation of stratified sets and singular varieties, Trans. Amer. Math. Soc., Volume 275 (1983) no. 1, pp. 333-343 | DOI

[25] Kirwan, Frances; Woolf, Jonathan An introduction to intersection homology theory, Chapman & Hall/CRC, Boca Raton, FL, 2006, xiv+229 pages | DOI

[26] Kronheimer, Peter; Mrowka, Tomasz Monopoles and three-manifolds, New Mathematical Monographs, 10, Cambridge University Press, Cambridge, 2007, xii+796 pages | DOI

[27] Laudenbach, François Appendix: On the Thom-Smale complex, Astérisque (1992) no. 205, 235 pages

[28] Laudenbach, François A Morse complex on manifolds with boundary, Geom. Dedicata, Volume 153 (2011), pp. 47-57 | DOI

[29] Laudenbach, François Transversalité, courants et théorie de Morse, Éditions de l’École Polytechnique, Palaiseau, 2012, x+182 pages (Un cours de topologie différentielle. [A course of differential topology], Exercises proposed by François Labourie)

[30] Ludwig, Ursula Comparison between two complexes on a singular space to appear in Journal für die reine und angewandte Mathematik (Crelle) | DOI

[31] Ludwig, Ursula Morse-Smale-Witten complex for gradient-like vector fields on stratified spaces, Singularity theory, World Sci. Publ., Hackensack, NJ, 2007, pp. 683-713 | DOI

[32] Milnor, J. Morse theory, Based on lecture notes by M. Spivak and R. Wells. Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, N.J., 1963, vi+153 pages

[33] Milnor, John Lectures on the h-cobordism theorem, Notes by L. Siebenmann and J. Sondow, Princeton University Press, Princeton, N.J., 1965

[34] Müller, Werner Analytic torsion and R-torsion of Riemannian manifolds, Adv. in Math., Volume 28 (1978) no. 3, pp. 233-305 | DOI

[35] Müller, Werner Analytic torsion and R-torsion for unimodular representations, J. Amer. Math. Soc., Volume 6 (1993) no. 3, pp. 721-753 | DOI

[36] Munkres, James R. Concordance is equivalent to smoothability, Topology, Volume 5 (1966), pp. 371-389

[37] Nicolaescu, Liviu An invitation to Morse theory, Universitext, Springer, New York, 2011, xvi+353 pages | DOI

[38] Pflaum, Markus J. Analytic and geometric study of stratified spaces, Lecture Notes in Mathematics, 1768, Springer-Verlag, Berlin, 2001, viii+230 pages

[39] Ray, D. B.; Singer, I. M. R-torsion and the Laplacian on Riemannian manifolds, Advances in Math., Volume 7 (1971), pp. 145-210

[40] Schwartz, Marie-Hélène Classes caractéristiques définies par une stratification d’une variété analytique complexe. I, C. R. Acad. Sci. Paris, Volume 260 (1965), pp. 3262-3264

[41] Schwartz, Marie-Hélène Champs radiaux sur une stratification analytique, Travaux en Cours [Works in Progress], 39, Hermann, Paris, 1991, x+185 pages

[42] Schwarz, Matthias Morse homology, Progress in Mathematics, 111, Birkhäuser Verlag, Basel, 1993, x+235 pages | DOI

[43] Smale, Stephen On gradient dynamical systems, Ann. of Math. (2), Volume 74 (1961), pp. 199-206

[44] Thom, René Sur une partition en cellules associée à une fonction sur une variété, C. R. Acad. Sci. Paris, Volume 228 (1949), pp. 973-975

[45] Trotman, David J. A. Geometric versions of Whitney regularity for smooth stratifications, Ann. Sci. École Norm. Sup. (4), Volume 12 (1979) no. 4, pp. 453-463

[46] Verona, Andrei Stratified mappings—structure and triangulability, Lecture Notes in Mathematics, 1102, Springer-Verlag, Berlin, 1984, ix+160 pages

[47] Vertman, Boris Analytic torsion of a bounded generalized cone, Comm. Math. Phys., Volume 290 (2009) no. 3, pp. 813-860 | DOI

[48] Wasserman, Arthur G. Equivariant differential topology, Topology, Volume 8 (1969), pp. 127-150

[49] Weber, Joa The Morse-Witten complex via dynamical systems, Expo. Math., Volume 24 (2006) no. 2, pp. 127-159 | DOI

[50] Whitney, Hassler Local properties of analytic varieties, Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), Princeton Univ. Press, Princeton, N. J., 1965, pp. 205-244

[51] Witten, Edward Supersymmetry and Morse theory, J. Differential Geom., Volume 17 (1982) no. 4, p. 661-692 (1983)

Cited by Sources: