# On the chain-level intersection pairing for PL manifolds

@inproceedings{McClure2004OnTC,
title={On the chain-level intersection pairing for PL manifolds},
author={James E. McClure},
year={2004}
}
Let M be a compact oriented PL manifold and let C M be its PL chain complex. The domain of the chain-level intersection pairing is a subcomplex of C M ̋C M . We prove that the inclusion map from this subcomplex to C M ̋C M is a quasiisomorphism. An analogous result is true for the domain of the iterated intersection pairing. Using this, we show that the intersection pairing gives C M a structure of partially defined commutative DGA, which in particular implies that C M is canonically quasi… Expand
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#### References

SHOWING 1-10 OF 37 REFERENCES
On the chain-level intersection pairing for PL manifolds
Let M be a compact oriented PL manifold and let C M be its PL chain complex. The domain of the chain-level intersection pairing is a subcomplex of C M C M . We prove that the inclusion map from thisExpand
Partial Algebras over Operads of Complexes and Applications
The properties of a manifold's chains which are in general position , and their intersections, leads naturally to a theory of algebraic structures which are only partially defined. In this paper weExpand
INTERSECTION HOMOLOGY THEORY
• Mathematics
• 1980
INTRODUCTION WE DEVELOP here a generalization to singular spaces of the Poincare-Lefschetz theory of intersections of homology cycles on manifolds, as announced in [6]. Poincart, in his 1895 paperExpand
String Topology
• Mathematics
• 1999
There is a diffeomorphism invariant structure in the free loop space of a manifold defined (with Moira Chas) by considering transversal intersections in families of collections of closed curves. AtExpand
We present a definition of homotopy algebra for an operad, and explore its consequences. The paper should be accessible to topologists, category theorists, and anyone acquainted with operads. AfterExpand