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The homotopy invariance of the string topology loop product and string bracket
Author(s) -
Cohen Ralph L.,
Klein John R.,
Sullivan Dennis
Publication year - 2008
Publication title -
journal of topology
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.447
H-Index - 31
eISSN - 1753-8424
pISSN - 1753-8416
DOI - 10.1112/jtopol/jtn001
Subject(s) - mathematics , homotopy , homology (biology) , pure mathematics , equivariant map , loop space , morita equivalence , biochemistry , chemistry , gene
Let M n be a closed, oriented, n ‐manifold, and LM its free loop space. In [Chas and Sullivan, ‘String topology’, Ann. of Math ., to appear] a commutative algebra structure in homology, H * ( LM ), and a Lie algebra structure in equivariant homologyH * S 1, were defined. In this paper, we prove that these structures are homotopy invariants in the following sense. Let f : M 1 → M 2 be a homotopy equivalence of closed, oriented n ‐manifolds. Then the induced equivalence, Lf : LM 1 → LM 2 induces a ring isomorphism in homology, and an isomorphism of Lie algebras in equivariant homology. The analogous statement also holds true for any generalized homology theory h * that supports an orientation of the M i .