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Poincar\'e duality for loop spaces

Alexandru Oancea, Kai Cieliebak, Nancy Hingston

Rabinowitz Floer homology and cohomology satisfy Poincaré duality that preserves their graded Frobenius algebra structure.

arxiv:2008.13161 v3 · 2020-08-30 · math.SG · math.AT

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Claims

C1strongest claim

We prove a Poincaré duality theorem between homology and cohomology that preserves this structure. This lifts to a duality theorem between graded open-closed TQFTs.

C2weakest assumption

The constructions and duality rely on the systematic use of the formalism of Tate vector spaces to manage the infinite-dimensional aspects of the loop spaces and Floer complexes.

C3one line summary

Poincaré duality holds for Rabinowitz Floer homology and cohomology as graded Frobenius algebras, extending to open-closed TQFT duality, with applications to cotangent bundles and loop spaces.

References

68 extracted · 68 resolved · 1 Pith anchors

[1] A. Abbondandolo, A. Portaluri, and M. Schwarz. The homology of path spaces and Floer homology with conormal boundary conditions. J. Fixed Point Theory Appl., 4(2):263–293, 2008 2008
[2] A. Abbondandolo and M. Schwarz. On the Floer homology of cotan gent bun- dles. Comm. Pure Appl. Math. , 59(2):254–316, 2006 2006
[3] A. Abbondandolo and M. Schwarz. Floer homology of cotangent b undles and the loop product. Geom. Topol., 14(3):1569–1722, 2010 2010
[4] A. Abbondandolo and M. Schwarz. On product structures in Floe r homology of cotangent bundles. In Global differential geometry , volume 17 of Springer Proc. Math., pages 491–521. Springer, Heidelberg, 2012
[5] A. Abbondandolo and M. Schwarz. Corrigendum: On the Floer hom ology of cotangent bundles. Comm. Pure Appl. Math. , 67(4):670–691, 2014 2014

Cited by

1 paper in Pith

Reduced symplectic homology and string topology
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Canonical hash

147503b1768d6fff10f0ccbd0b19c6b948ecb7accf3175cd7d76adc805dbc6de

Aliases

arxiv: 2008.13161 · arxiv_version: 2008.13161v3 · doi: 10.48550/arxiv.2008.13161 · pith_short_12: CR2QHMLWRVX7 · pith_short_16: CR2QHMLWRVX76EHQ · pith_short_8: CR2QHMLW
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Verify this Pith Number yourself 
curl -sH 'Accept: application/ld+json' https://pith.science/pith/CR2QHMLWRVX76EHQZS6QWGOGXF \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 147503b1768d6fff10f0ccbd0b19c6b948ecb7accf3175cd7d76adc805dbc6de
Canonical record JSON 
{
  "metadata": {
    "abstract_canon_sha256": "6493f7666cbf7f33b6a68f3ef5d431490f5bd805314d7debee7ca2d239564b97",
    "cross_cats_sorted": [
      "math.AT"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.SG",
    "submitted_at": "2020-08-30T13:13:13Z",
    "title_canon_sha256": "d766312226ac393645683d75741d38a9bc88983dafd3e6cf1568f11f22e3e3c0"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2008.13161",
    "kind": "arxiv",
    "version": 3
  }
}