Loop products and closed geodesics. (English) Zbl 1181.53036
The closed geodesics on a compact Riemannian manifold \(M\) are the critical points of the length function on the free loop space \(\Lambda(M)\). The length function provides a filtration of the homology of \(\Lambda(M)\). The authors prove that the Chas-Sullivan product
\[ H_i(\Lambda)\times H_j(\Lambda)@>*>> H_{i+j-n}(\Lambda) \]
is compatible with this filtration. Furthermore, they also interprete Sullivan’s coproduct \(\vee\) on \(C_*(\Lambda)\) as a product in cohomology
\[ H^i(\Lambda,\Lambda_0)\times H^j(\Lambda,\Lambda_0)@>\circledast>> H^{i+j+n-1}(\Lambda, \Lambda_0), \]
where \(\Lambda_0= M)\) is the constant loop. They show that \(\circledast\) is also compatible with the length filtration and provide a similar expression for the ring \(\text{Gr\,}H^*(\Lambda, \Lambda_0)\). Among other results they also determine the full ring structure \((H^*(\Lambda, \Lambda_0),\circledast)\) for spheres \(M= S^n\), where \(n\geq 3\). They obtain a significant contribution on loop products and closed geodesics.
\[ H_i(\Lambda)\times H_j(\Lambda)@>*>> H_{i+j-n}(\Lambda) \]
is compatible with this filtration. Furthermore, they also interprete Sullivan’s coproduct \(\vee\) on \(C_*(\Lambda)\) as a product in cohomology
\[ H^i(\Lambda,\Lambda_0)\times H^j(\Lambda,\Lambda_0)@>\circledast>> H^{i+j+n-1}(\Lambda, \Lambda_0), \]
where \(\Lambda_0= M)\) is the constant loop. They show that \(\circledast\) is also compatible with the length filtration and provide a similar expression for the ring \(\text{Gr\,}H^*(\Lambda, \Lambda_0)\). Among other results they also determine the full ring structure \((H^*(\Lambda, \Lambda_0),\circledast)\) for spheres \(M= S^n\), where \(n\geq 3\). They obtain a significant contribution on loop products and closed geodesics.
Reviewer: Themistocles M. Rassias (Athens)
MSC:
| 53C22 | Geodesics in global differential geometry |
| 58E05 | Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces |
| 58E10 | Variational problems in applications to the theory of geodesics (problems in one independent variable) |
| 55N45 | Products and intersections in homology and cohomology |
| 55P50 | String topology |
Keywords:
loop products; closed geodesics; Morse index; geodesic; Sullivan’s coproduct; ring structure; cohomologyReferences:
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