On the rational homotopy type of intersection spaces

Dominik J. Wrazidlo

Journal of Singularities
volume 20 (2020), 251-273

Received: 26 March 2020. Received in revised form: 5 August 2020.

DOI: 10.5427/jsing.2020.20k

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Abstract:

Banagl's method of intersection spaces allows modification of certain types of stratified pseudomanifolds near the singular set in such a way that the rational Betti numbers of the modified spaces satisfy generalized Poincaré duality in analogy with Goresky-MacPherson's intersection homology. In the case of one isolated singularity, we show that the duality isomorphism comes from a nondegenerate intersection pairing which depends on the choice of a chain representative of the fundamental class of the regular stratum. On the technical side, we use piecewise linear polynomial differential forms due to Sullivan to define a suitable commutative cochain algebra model for intersection spaces. We show that the weak equivalence class of our model does not depend on choices, which implies uniqueness of the rational intersection space cohomology ring. Our construction parallels Banagl's commutative cochain algebra of smooth differential forms modeling intersection space cohomology, and we show that both algebras are weakly equivalent.

2010 Mathematical Subject Classification:

55N33, 57P10, 55P62

Key words and phrases:

Stratified spaces, pseudomanifolds, intersection spaces, Poincar\'{e} duality, rational homotopy, de Rham theorem, smooth differential forms

Author(s) information:

Dominik J. Wrazidlo
Institute of Mathematics for Industry
Kyushu University
Motooka 744, Nishi-ku
Fukuoka 819-0395, Japan
email: d-wrazidlo@imi.kyushu-u.ac.jp