Equivariant geometric K-homology for compact Lie group actions

Baum, Paul; Oyono‐Oyono, Hervé; Schick, Thomas; Walter, Michael

doi:10.1007/s12188-010-0034-z

Cited by 26 publications

(56 citation statements)

References 22 publications

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“…Proof Based on the proof of Lemma 2.1 of , there is a

P U (n)

‐equivariant embedding of P into a complex

P U (n)

‐representation W and for a small enough

P U (n)

‐invariant neighborhood U of P there is a retraction

U \to P

. Furthermore, there is a

V \subseteq U

that is a

P U (n)

‐invariant manifold with boundary.…”

Section: Projective K‐homologymentioning

confidence: 99%

“…We will not prove Lemma since the proof is the same as that of Theorem 4.1 in mutatis mutandis; the same bordism can be used due to Lemma . Lemma If P is a principal spin c ‐

P U (n)

‐bundle and

(M, E, f)

is a projective cycle over P ,

false[M, E, f false] = false[P, f_{!} E, id false] .

…”

Section: Projective K‐homologymentioning

confidence: 99%

“…The proof is mutatis mutandis the same as the final paragraphs of Section of but since there is a small difference in the cycles, we include a sketch of this proof.…”

Section: Projective K‐homologymentioning

confidence: 99%

“…The mapping

h : = f \times j : M \to P \times V^{+}

is a

P U (n)

‐equivariant embedding and

{pr}_{P} \circ h = f

. The subtle difference to Section of is the requirement that the

P U (n)

‐action on

P \times V^{+}

is free; this is of course true since the action on P is free. The embedding j is homotopic to the south pole mapping

c : M \to V^{+}

h_{!} = {(f \times c)}_{!}

.…”

Section: Projective K‐homologymentioning

confidence: 99%

“…We will not prove Lemma 4.10 since the proof is the same as that of Theorem 4.1 in [6] mutatis mutandis; the same bordism can be used due to Lemma 3.7. The proof is mutatis mutandis the same as the final paragraphs of Section 4 of [6] but since there is a small difference in the cycles, we include a sketch of this proof.…”

Section: Theorem 113]) □mentioning

confidence: 99%

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Applying geometricK-cycles to fractional indices

Deeley

Goffeng

2017

Math. Nachr.

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A geometric model for twisted K‐homology is introduced. It is modeled after the Mathai–Melrose–Singer fractional analytic index theorem in the same way as the Baum–Douglas model of K‐homology was modeled after the Atiyah–Singer index theorem. A natural transformation from twisted geometric K‐homology to the new geometric model is constructed. The analytic assembly mapping to analytic twisted K‐homology in this model is an isomorphism for torsion twists on a finite CW‐complex. For a general twist on a smooth manifold the analytic assembly mapping is a surjection. Beyond the aforementioned fractional invariants, we study T‐duality for geometric cycles.

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“…Proof Based on the proof of Lemma 2.1 of , there is a

P U (n)

‐equivariant embedding of P into a complex

P U (n)

‐representation W and for a small enough

P U (n)

‐invariant neighborhood U of P there is a retraction

U \to P

. Furthermore, there is a

V \subseteq U

that is a

P U (n)

‐invariant manifold with boundary.…”

Section: Projective K‐homologymentioning

confidence: 99%

“…We will not prove Lemma since the proof is the same as that of Theorem 4.1 in mutatis mutandis; the same bordism can be used due to Lemma . Lemma If P is a principal spin c ‐

P U (n)

‐bundle and

(M, E, f)

is a projective cycle over P ,

false[M, E, f false] = false[P, f_{!} E, id false] .

…”

Section: Projective K‐homologymentioning

confidence: 99%

“…The proof is mutatis mutandis the same as the final paragraphs of Section of but since there is a small difference in the cycles, we include a sketch of this proof.…”

Section: Projective K‐homologymentioning

confidence: 99%

“…The mapping