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Hilbert-Hadamard spaces and the equivariant coarse Novikov conjecture

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Pith's one-line read The paper proves that equivariant coarse embeddability into an admissible Hilbert-Hadamard space makes the equivariant coarse Miščenko-Kasparov assembly map rationally injective, with a stronger conclusion when the group is torsion-free.

desk verdict Substantial and credible extension of Novikov-type injectivity to Hilbert-Hadamard targets, with the main risk sitting in the imported Bott-map injectivity from [GWY21]. read the letter →

arxiv 2411.18538 v2 pith:E537FTAN submitted 2024-11-27 math.KT math.OA

Liang Guo , Qin Wang , Jianchao Wu , Guoliang Yu This is my paper · ORCID
classification math.KTmath.OA MSC 19K5646L8053C2358J22
keywords NovikovconjectureequivariantcoarseHilbert-HadamardspacesembeddingshigherindextheorylocalizationalgebrasK-theoryMilnor-Ripscomplexes
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proves that one coarse-geometric condition — equivariant coarse embeddability into an admissible Hilbert-Hadamard space, a complete nonpositively curved (CAT(0)) space of possibly infinite dimension — forces a rational injectivity statement in higher index theory. Concretely, for any countable discrete group Γ acting properly by isometries on a bounded-geometry space X, if X admits a Γ-equivariant coarse embedding into such a space, then the rational analytic equivariant coarse Novikov conjecture holds: the rationalized equivariant coarse Miščenko-Kasparov assembly map is injective (Theorem 1.1). When Γ is torsion-free, the same mechanism proves the stronger rational equivariant coarse strong Novikov conjecture (Theorem 1.3), and when the action is cobounded the result reproduces the classical rational Novikov conjecture for such groups. The result matters because it replaces the earlier hypotheses of an isometric proper action or a Hilbert-space embedding with the much weaker requirement of a coarse embedding, and it yields new obstructions in geometry: a complete Riemannian manifold whose uniformly contractible universal cover coarsely embeds this way admits no metric of uniformly positive scalar curvature (Corollary 1.2).

What carries the argument

The central object is the noncommutative coefficient algebra $A(M)$ attached to an admissible Hilbert-Hadamard space $M$, generated by functional calculi of Clifford generators $C_{x_0}(x,t)=(-\log_x(x_0), t)$ at all base points $x_0$; its Bott map $\beta_{x_0}: S\to A(M)$ from $S=C_0(\mathbb{R})$ is rationally injective on K-theory, and the proof imports this injectivity as the essential property of the coefficient algebra. Around $A(M)$ the paper builds equivariant twisted Roe algebras and equivariant twisted localization algebras over the Rips complexes $P_d(X)$, so that the assembly map becomes the evaluation map from localization algebras to Roe algebras, placed inside one large commuting diagram. Two devices carry the argument: a deformation trick in which the Γ-action on the continuum product $M^{[0,1]}$ is continuously homotoped to the trivial action, making the K-theory computable by a Künneth formula for twisted localization algebras; and a cutting-and-pasting argument that slices the twisted algebras into pieces indexed by Γ-slices (balanced products $\Gamma\times_F U_0$ for finite subgroups $F$) and reassembles the resulting isomorphisms by Mayer–Vietoris, with equivariant bounded geometry guaranteeing the process terminates. For groups with torsion, the Milnor-Rips complexes $\widetilde{P}_{d,n}(X)$ — a Milnor-join-style model for free and proper Γ-spaces coarsely equivalent to $X$ — replace the Rips complexes, and rational injectivity of the comparison map $\pi_*$ is shown using a compact Γ-space with property TAF, concretely the space of all linear orders on $X$, together with a KK-product construction and a Künneth theorem for twisted localization algebras. A further result shows $A(M)$ is a direct limit of type I C$^*$-algebras, hence in the bootstrap class.

What would settle it

Compute $K_*(A(M))$ for an admissible Hilbert-Hadamard space that is neither a finite-dimensional manifold nor a Hilbert space (for instance the continuum product $M^{[0,1]}$ of a positive-dimensional Hadamard manifold with an interval) and check whether the Bott map $S \to A(M)$ is injective on rationalized K-theory, since the main proof imports exactly this fact; alternatively, exhibit a countable group Γ and a proper Γ-space X with equivariant bounded geometry and a Γ-equivariant coarse embedding into such a space for which the rationalized equivariant coarse Miščenko-Kasparov assembly map has nonzero kernel.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that the equivariant coarse Miščenko-Kasparov assembly map $\nu^\Gamma_X$, built from the Milnor-Rips complexes $\widetilde{P}_{d,n}(X)$ that classify the free and proper Γ-spaces equivariantly coarsely equivalent to $X$, becomes injective after tensoring with $\mathbb{Q}$ whenever the proper Γ-space $X$ admits a Γ-equivariant coarse embedding into an admissible Hilbert-Hadamard space. This is the statement the paper introduces as the rational analytic equivariant coarse Novikov conjecture; it generalizes the rational analytic Novikov conjecture and interpolates between the classical Miščenko-Kasparov assembly map (cobounded Γ-action) and the coarse assembly map (trivial group). For torsion-free Γ the Milnor-Rips complexes coincide with the ordinary Rips complexes, and the proof yields the stronger Theorem 1.3: the equivariant coarse assembly map $\mu^\Gamma_X$ is rationally injective, i.e., the rational equivariant coarse strong Novikov conjecture holds. Both theorems are driven by a single commuting diagram whose left column is rationally injective — via a deformation trick that continuously trivializes the Γ-action on a larger coefficient algebra — and whose bottom row is an isomorphism — via a cutting-and-pasting argument over Γ-slices whose length is controlled by equivariant bounded geometry.

Load-bearing premise

The whole argument rests on a borrowed technical property — the rational injectivity of the Bott map for the coefficient algebra A(M) — and if that property failed, the main diagram would lose its injectivity and both theorems would collapse.

Editorial extensions

If this is right

  • In the cobounded case, Theorem 1.1 recovers the rational Novikov conjecture for countable groups acting properly and isometrically on admissible Hilbert-Hadamard spaces, which in turn implies the classical Novikov conjecture and the Gromov-Lawson conjecture for such groups.
  • For complete Riemannian manifolds whose universal cover is uniformly contractible, coarse embeddability into an admissible Hilbert-Hadamard space rules out metrics of uniformly positive scalar curvature (Corollary 1.2).
  • When Γ is torsion-free, the conclusion strengthens to rational injectivity of the equivariant coarse assembly map $\mu^\Gamma_X$, i.e., the rational equivariant coarse strong Novikov conjecture.
  • With Γ trivial, every bounded-geometry space coarsely embeddable into an admissible Hilbert-Hadamard space satisfies the rational coarse Novikov conjecture, extending what is known from Hilbert-space embeddings to a much larger class of targets.
  • The coefficient algebra $A(M)$ is shown to be a direct limit of type I C$^*$-algebras, hence in the bootstrap class, a structural fact used to make the Künneth computations rigorous and of independent interest in K-theory.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Since no metric space of bounded geometry is known to fail coarse embeddability into some CAT(0) space, this theorem suggests the rational analytic equivariant coarse Novikov conjecture may hold for all bounded-geometry spaces; the paper's machinery gives a concrete route for testing that possibility case by case.
  • The linear-orders model of a property-TAF space is a general recipe for replacing a proper group action by a free one while preserving coarse geometry, a device likely reusable beyond this paper in equivariant index theory.
  • The localization-algebra KK-products and Künneth formulas developed here plausibly transfer to other coefficient algebras, such as maximal Roe algebras or $\ell^p$-geometric settings, where they could turn rational statements into integral ones if the underlying Bott map is integrally injective.
  • One concrete testable extension of Corollary 1.2 is to prove coarse embeddability into Hilbert-Hadamard spaces for universal covers of aspherical or large-scale-contractible manifolds that resist Hilbert-space embeddings, which would yield new uniformly-positive-scalar-curvature obstructions.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper proves two main theorems. Theorem 1.3 states that if a countable discrete group Γ is torsion-free and acts properly and isometrically on a bounded-geometry metric space X which admits a Γ-equivariant coarse embedding into an admissible Hilbert-Hadamard space, then the equivariant coarse assembly map is rationally injective. Theorem 1.1 removes the torsion-free assumption and instead proves rational injectivity of a newly introduced equivariant coarse Miščenko-Kasparov assembly map, whose domain is built from Milnor-Rips complexes of a free and proper Γ-space equivariantly coarsely equivalent to X. The proof uses equivariant twisted Roe algebras and localization algebras with coefficients in the C*-algebra A(M) of an admissible Hilbert-Hadamard space, a deformation trick based on the algebra A[0,1](M), a Mayer-Vietoris/cutting-and-pasting argument for the twisted assembly map, and a rational injection for the map induced by the projection from Milnor-Rips complexes to Rips complexes. The paper also constructs a concrete model Ω_X (the space of linear orders on X) of a compact Γ-space with property TAF and proves that A(M) is in the bootstrap class when M is separable.

Significance. If the proof is completed, the results constitute a substantial advance in the Novikov-conjecture literature: they extend the Hilbert-Hadamard methodology of Gong-Wu-Yu [GWY21] from the cocompact, proper-action setting to general equivariant coarse embeddings with non-free actions, and they supply a new, explicitly weaker conjecture that still implies the classical applications to the Novikov conjecture and positive scalar curvature. The paper is technically rich: it introduces new twisted Roe and localization algebras, a deformation trick, Milnor-Rips complexes for free actions, an order-space model with property TAF, and a proof that A(M) belongs to the bootstrap class. These tools are likely to be useful beyond the present theorems. The paper is also commendably transparent about the fact that its main conjecture is weaker than the equivariant coarse strong Novikov conjecture.

major comments (4)
  1. [§5, Proposition 5.7 and Theorem 2.11] The proof of the rational injectivity of (ev0)_* ∘ (βL)_* is reduced to the assertion that the bottom horizontal map 1⊗(β_m)_* is injective 'by Theorem 2.11'. Theorem 2.11(2), the injectivity of (β_{x0})_* : K_*(S) → K_*(A(M)), is imported from [GWY21] and is not proved in the present paper. This injectivity is load-bearing: tracing diagram (10), the rational injectivity of the left column, and hence the proofs of both Theorem 1.3 and Theorem 1.1, would collapse if this Bott map had a nonzero kernel. The paper should state this dependency explicitly in the introduction and verify that the space M[0,1] used in Proposition 5.7 indeed satisfies the hypotheses of Theorem 2.11(2).
  2. [§6, Lemma 6.11] The four ideal equalities (1)-(4) in Lemma 6.11 are asserted with the comment that the proof is similar to [Yu00, Lemma 6.3] and that details are left to the reader. This lemma is used directly in the Mayer-Vietoris argument in the proof of Theorem 6.4, which establishes the isomorphism of the twisted assembly map (map (11) in diagram (10)). The intersection formulas (2) and (4), in particular, involve the weak topology on M[0,1] × R+ and the coefficient algebra A(M[0,1]), and are not formally consequences of the non-twisted lemma. A complete proof or a very detailed translation of [Yu00, Lemma 6.3] to this twisted setting should be supplied.
  3. [Appendix A, Lemmas A.3 and A.4] The Künneth formula of Theorem A.10, which is used in the proof of Proposition 5.7, rests on Lemma A.3 (the quotient isomorphism i_Q) and Lemma A.4 (the vanishing of K-theory of the annihilator ideals). Lemma A.3 is proved, but Lemma A.4 is dismissed with 'an Eilenberg swindle argument similar to [WY20, Lemma 6.4.11], which we leave to the reader'. Since this Künneth formula is load-bearing for the main theorems, the authors should either give the full Eilenberg swindle argument or provide a precise statement in [WY20] together with a detailed verification that the hypotheses apply to the twisted localization algebras considered here.
  4. [§11, Lemma 11.1] The proof of the rational injectivity of π_* relies on the ordering map (11) and on the claim that the trace τ_{Ω_X} satisfies τ_{Ω_X}([1_{C(Ω_X)⋊_r Γ}]) = 1. While plausible, the continuity argument for the ordering map and the verification that the constructed trace is well-defined and Γ-invariant are only sketched. Since this lemma is the key new ingredient for the non-torsion-free case, a more detailed treatment of these points would increase confidence in the proof.
minor comments (4)
  1. [Abstract] The phrase 'a handful profound K-theoretic conjectures' should be 'a handful of profound K-theoretic conjectures'.
  2. [§2, Definition 2.3] In the display for the angle ∠(α, β), the inner product ⟨¯α_t − ¯α_0, ¯β_s − ¯β_0⟩ uses notation from the comparison triangle but the comparison points are not explicitly defined there; please add a sentence clarifying the notation.
  3. [§8, after Definition 8.2] The statement 'Since X is countable' should be justified, since X was only assumed to be a proper Γ-space with bounded geometry. The earlier reduction in Section 3 to a Γ-invariant countable dense subset should be invoked, or the construction of Ω_X should be explicitly restricted to such a net.
  4. [§9, Theorem 9.8] In the proof, the phrase 'Since C0(Mn(R)+, CliffC(Rn)) is a Type I C*-algebra' is correct, but it would be helpful to mention that B0(n) is a closed subalgebra of this Type I algebra and hence Type I, rather than leaving this inference implicit.

Circularity Check

1 steps flagged · score 2.0 of 10

The only load-bearing input from the authors' overlapping prior work is the Bott-injectivity K*(S) → K*(A(M)) imported from [GWY21, Thm 2.11] and used as the final step of Proposition 5.7; because that result is published, parameter-free, and does not assert the target conjecture, the main theorems retain independent content and the finding is mild.

  1. self citation load bearing [Section 2.2 (Theorem 2.11) and Section 5 (Proposition 5.7), feeding the left column of diagram (10)]
    "The next theorem summarizes some results we will need in the following of this paper; one can find a proof in Section 5 and Section 7 of [GWY21]. Theorem 2.11. Let M be an admissible Hilbert-Hadamard space. Then ... (2) for any x0 ∈ M, the Bott map induces an injection on K-theory, i.e., (βx0)∗ : K∗(S) → K∗(A(M)) is an injection; ... Since M is admissible, we conclude that the bottom map is injective by Theorem 2.11. This implies that (ev0)∗ ◦ (βL)∗ is an injection, which completes the proof."

    The rational injectivity of the left column of diagram (10) — the step on which both Theorem 1.3 and Theorem 1.1 depend — is finished in Proposition 5.7 by 'the bottom map is injective by Theorem 2.11'. Theorem 2.11(2) is stated without proof and referenced to [GWY21], which shares two authors (J. Wu, G. Yu) with the present paper; Proposition 2.7 (admissibility of M[0,1]) feeding it is likewise imported from [GWY21]. A self-citation is thus logically load-bearing: if (β_m)_* had nonzero kernel, (ev1)_*∘(βL)_* would not be rationally injective and both main theorems would collapse.

full rationale

Derivation chain: Theorem 1.1 follows by tracing diagram (10) — the left column is rationally injective via Proposition 5.7 (in-paper deformation trick Lemma 5.6, in-paper Künneth Theorem 5.1/Appendix A, plus the imported Bott injectivity Theorem 2.11); map (11) is an isomorphism via Theorem 6.4 (in-paper cutting-and-pasting, Section 6); map (12) is rationally injective via Lemma 11.1 (in-paper ordering/collapsing maps, Property-TAF model Section 8, bootstrap class Section 9, KK-product and Künneth Section 10). The only externally cited load-bearing step is Theorem 2.11(2) together with the admissibility of M[0,1] from [GWY21] needed to apply it. Under the stated criteria this is legitimate support: it is a published, parameter-free result about the coefficient algebra A(M), independent of any fitted values, and it does not assert the rational analytic equivariant coarse Novikov conjecture that it is used to prove. No fitted parameters are renamed as predictions, no uniqueness theorem is invoked to forbid alternatives, no known result is repackaged as new, and the weakened new conjecture is announced transparently in the introduction. Section 9 actually proves A(M) is in the bootstrap class in-paper rather than importing that ingredient. The flagged self-citation is therefore a normal reliance on overlapping prior work, and the central claim retains genuinely independent content, so the score stays in the benign 0–2 band at 2.

Assumptions & free parameters 0 free parameters · 4 assumptions · 2 invented entities

The paper introduces no fitted parameters. The main axioms are standard K-theoretic results and the explicit geometric hypotheses of the theorems. The new mathematical objects (Ω_X, twisted algebras) are constructed and verified, not postulated on faith.

assumptions (4)
  • standard math Properties of admissible Hilbert-Hadamard spaces and the associated C*-algebra A(M) from [GWY21], including Theorem 2.11 (Bott map injectivity).
    Used throughout Sections 2-6; if these properties fail, the Bott map arguments in Section 5 collapse.
  • domain assumption Equivariant bounded geometry for (X, Γ, α) as in Definition 6.1.
    Assumed in Theorem 1.1 and used for dimension control in the Mayer-Vietoris arguments and for the Milnor-Rips complexes.
  • domain assumption Existence of a Γ-equivariant coarse embedding of X into an admissible Hilbert-Hadamard space.
    This is the main geometric hypothesis of Theorems 1.1 and 1.3.
  • standard math Standard results in K-theory, KK-theory, localization algebras, and coarse geometry from [Kas88, WY20, Yu97, Yu00, RS87].
    The proof relies on these foundational results.
invented entities (2)
  • Ω_X, the space of linear orders on X independent evidence
    purpose: Used as a concrete compact Γ-space with property TAF to build a partial inverse to π* in Section 11.
    Explicitly constructed in Section 8 with the TAF property verified; it is a checkable mathematical object, not a postulated entity.
  • Equivariant twisted Roe and localization algebras with coefficients in A(M) and A[0,1](M) independent evidence
    purpose: New operator algebras that implement the Dirac-dual-Dirac method in the equivariant coarse setting.
    Defined explicitly in Section 4 with K-theoretic properties proven in later sections; they are constructed tools.

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Pith. "Pith review of Hilbert-Hadamard spaces and the equivariant coarse Novikov conjecture." pith.science (2026). https://pith.science/paper/E537FTAN

@misc{pith2026241118538,
  author       = {Pith},
  title        = {Pith review of: Hilbert-Hadamard spaces and the equivariant coarse Novikov conjecture},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/E537FTAN}},
  note         = {Machine review of arXiv:2411.18538}
}
abstract

The equivariant coarse Novikov conjectures stand among a handful profound $K$-theoretic conjectures in noncommutative geometry. Motivated by the quest to verify Novikov-type conjectures for groups of diffeomorphisms, we study in this paper the equivariant coarse Novikov conjectures for spaces that equivariantly and coarsely embed into admissible Hilbert-Hadamard spaces, which are a type of infinite-dimensional nonpositively curved spaces. The paper is split into two parts. We prove in the first part that for any metric space $X$ with bounded geometry and with a proper isometric action $\alpha$ by a countable discrete group $\Gamma$, if $X$ admits an equivariant coarse embedding into an admissible Hilbert-Hadamard space and $\Gamma$ is torsion-free, then the equivariant coarse strong Novikov conjecture holds rationally for $(X, \Gamma, \alpha)$. In the second part, we extend the result in the first part by dropping the torsion-free assumption on $\Gamma$. To this end, we introduce, for a proper $\Gamma$-space $X$ with equivariant bounded geometry, a new Novikov-type conjecture that we call the rational analytic equivariant coarse Novikov conjecture, which generalizes the rational analytic Novikov conjecture and asserts the rational injectivity of a certain assembly map associated with a coarse analog of the classifying space $E\Gamma$. We show that for a proper $\Gamma$-space $X$ with equivariant bounded geometry, if $X$ admits an equivariant coarse embedding into an admissible Hilbert-Hadamard space, then the rational analytic equivariant coarse Novikov conjecture holds for $(X,\Gamma,\alpha)$, i.e., the assembly map is a rational injection.

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Reference graph

Works this paper leans on

76 extracted references · 64 canonical work pages · cited by 2 Pith papers

  1. [1]

    The B aum- C onnes conjecture localised at the unit element of a discrete group

    Paolo Antonini, Sara Azzali, and Georges Skandalis. The B aum- C onnes conjecture localised at the unit element of a discrete group. Compos. Math. , 156(12):2536--2559, 2020

    work page 2020

  2. [2]

    Chern character for discrete groups

    Paul Baum and Alain Connes. Chern character for discrete groups. In A f\^ e te of topology , pages 163--232. Academic Press, Boston, MA, 1988

    work page 1988

  3. [3]

    Classifying space for proper G -actions and K -theory of group C^* -algebras

    Paul Baum, Alain Connes, and Nigel Higson. Classifying space for proper G -actions and K -theory of group C^* -algebras . Contemp. Math. , 167(167):241--291, 1994

    work page 1994

  4. [4]

    Bridson and Andr\'e Haefliger

    Martin R. Bridson and Andr\'e Haefliger. Metric spaces of non-positive curvature , volume 319 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] . Springer-Verlag, Berlin, 1999

    1999

  5. [5]

    Operator algebras , volume 122 of Encyclopaedia of Mathematical Sciences

    Bruce Blackadar. Operator algebras , volume 122 of Encyclopaedia of Mathematical Sciences . Springer-Verlag, Berlin, 2006. Theory of C^* -algebras and von Neumann algebras, Operator Algebras and Non-commutative Geometry, III

    work page 2006

  6. [6]

    Equivariant covers for hyperbolic groups

    Arthur Bartels, Wolfgang L\"uck, and Holger Reich. Equivariant covers for hyperbolic groups. Geom. Topol. , 12(3):1799--1882, 2008

    work page 2008

  7. [7]

    Permanence properties of the B aum- C onnes conjecture

    J\'er\^ome Chabert and Siegfried Echterhoff. Permanence properties of the B aum- C onnes conjecture. Doc. Math. , 6:127--183, 2001

    2001

  8. [8]

    Group cohomology with L ipschitz control and higher signatures

    Alain Connes, Mikhael Gromov, and Henri Moscovici. Group cohomology with L ipschitz control and higher signatures. Geom. Funct. Anal. , 3(1):1--78, 1993

    work page 1993

Show all 76 references
  1. [9]

    Cyclic cohomology, the N ovikov conjecture and hyperbolic groups

    Alain Connes and Henri Moscovici. Cyclic cohomology, the N ovikov conjecture and hyperbolic groups. Topology , 29(3):345--388, 1990

    1990

  2. [10]

    Chang and Shmuel Weinberger

    Stanley S. Chang and Shmuel Weinberger. On novikov-type conjectures. In Surveys in noncommutative geometry , volume 6 of Clay Math. Proc. , pages 43--70. Amer. Math. Soc., Providence, RI, 2006

    2006

  3. [11]

    The maximal coarse B aum- C onnes conjecture for spaces which admit a fibred coarse embedding into H ilbert space

    Xiaoman Chen, Qin Wang, and Guoliang Yu. The maximal coarse B aum- C onnes conjecture for spaces which admit a fibred coarse embedding into H ilbert space. Adv. Math. , 249:88--130, 2013

    2013

  4. [12]

    The coarse N ovikov conjecture and B anach spaces with P roperty ( H )

    Xiaoman Chen, Qin Wang, and Guoliang Yu. The coarse N ovikov conjecture and B anach spaces with P roperty ( H ). J. Funct. Anal. , 268(9):2754--2786, 2015

    2015

  5. [13]

    Dranishnikov, Steven C

    Alexander N. Dranishnikov, Steven C. Ferry, and Shmuel A. Weinberger. An etale approach to the novikov conjecture. Comm. Pure Appl. Math. , 61(2):139--155, 2008

    2008

  6. [14]

    Localization C^* -algebras and K -theoretic duality

    Marius Dadarlat, Rufus Willett, and Jianchao Wu. Localization C^* -algebras and K -theoretic duality. Ann. K-Theory , 3(4):615--630, 2018

    2018

  7. [15]

    Nonpositive curvature is not coarsely universal

    Alexandros Eskenazis, Manor Mendel, and Assaf Naor. Nonpositive curvature is not coarsely universal. Invent. Math. , 217(3):833--886, 2019

    2019

  8. [16]

    Foundations of algebraic topology

    Samuel Eilenberg and Norman Steenrod. Foundations of algebraic topology . Princeton University Press, Princeton, NJ, 1952

    1952

  9. [17]

    The coarse B aum- C onnes conjecture for relatively hyperbolic groups

    Tomohiro Fukaya and Shin-ichi Oguni. The coarse B aum- C onnes conjecture for relatively hyperbolic groups. J. Topol. Anal. , 4(1):99--113, 2012

    2012

  10. [18]

    Coronae of product spaces and the coarse B aum- C onnes conjecture

    Tomohiro Fukaya and Shin-ichi Oguni. Coronae of product spaces and the coarse B aum- C onnes conjecture. Adv. Math. , 279:201--233, 2015

    2015

  11. [19]

    A coarse C artan- H adamard theorem with application to the coarse B aum- C onnes conjecture

    Tomohiro Fukaya and Shin-ichi Oguni. A coarse C artan- H adamard theorem with application to the coarse B aum- C onnes conjecture. J. Topol. Anal. , 12(3):857--895, 2020

    2020

  12. [20]

    Curvature, tangentiality, and controlled topology

    Steven Ferry and Shmuel Weinberger. Curvature, tangentiality, and controlled topology. Invent. Math. , 105(2):401--414, 1991

    1991

  13. [21]

    The equivariant coarse B aum- C onnes conjecture for spaces which admit an equivariant coarse embedding into H ilbert space

    Benyin Fu and Xianjin Wang. The equivariant coarse B aum- C onnes conjecture for spaces which admit an equivariant coarse embedding into H ilbert space. J. Funct. Anal. , 271(4):799--832, 2016

    2016

  14. [22]

    The equivariant coarse N ovikov conjecture and coarse embedding

    Benyin Fu, Xianjin Wang, and Guoliang Yu. The equivariant coarse N ovikov conjecture and coarse embedding. Comm. Math. Phys. , 380(1):245--272, 2020

    2020

  15. [23]

    Equivariant E -theory for C^* -algebras

    Erik Guentner, Nigel Higson, and Jody Trout. Equivariant E -theory for C^* -algebras. Mem. Amer. Math. Soc. , 148(703):viii+86, 2000

    2000

  16. [24]

    The N ovikov conjecture for linear groups

    Erik Guentner, Nigel Higson, and Shmuel Weinberger. The N ovikov conjecture for linear groups. Publ. Math. Inst. Hautes \'Etudes Sci. , (101):243--268, 2005

    2005

  17. [25]

    A B ott periodicity theorem for ^p -spaces and the coarse N ovikov conjecture at infinity

    Liang Guo, Zheng Luo, Qin Wang, and Yazhou Zhang. A B ott periodicity theorem for ^p -spaces and the coarse N ovikov conjecture at infinity. J. Funct. Anal. , 286(2):Paper No. 110215, 58, 2024

    2024

  18. [26]

    Asymptotic invariants of infinite groups

    Mikhael Gromov. Asymptotic invariants of infinite groups. In Geometric group theory, V ol. 2 ( S ussex, 1991) , volume 182 of London Math. Soc. Lecture Note Ser. , pages 1--295. Cambridge Univ. Press, Cambridge, 1993

    1991

  19. [27]

    A coarse geometric approach to the generalised Green-Julg Theorem

    Liang Guo and Qin Wang. A coarse geometric approach to the generalised Green-Julg Theorem . arXiv e-prints , page arXiv:2208.11595, August 2022

    2022 arXiv

  20. [28]

    The Novikov conjecture, the group of diffeomorphisms and continuous fields of Hilbert-Hadamard spaces

    Sherry Gong , Jianchao Wu , Zhizhang Xie , and Guoliang Yu . The Novikov conjecture, the group of diffeomorphisms and continuous fields of Hilbert-Hadamard spaces . arXiv e-prints , arXiv:2310.01219, 2023

    2023 arXiv

  21. [29]

    Geometrization of the strong N ovikov conjecture for residually finite groups

    Guihua Gong, Qin Wang, and Guoliang Yu. Geometrization of the strong N ovikov conjecture for residually finite groups. J. Reine Angew. Math. , 621:159--189, 2008

    2008

  22. [30]

    The N ovikov conjecture, the group of volume preserving diffeomorphisms and H ilbert- H adamard spaces

    Sherry Gong, Jianchao Wu, and Guoliang Yu. The N ovikov conjecture, the group of volume preserving diffeomorphisms and H ilbert- H adamard spaces. Geom. Funct. Anal. , 31(2):206--267, 2021

    2021

  23. [31]

    Algebraic topology

    Allen Hatcher. Algebraic topology . Cambridge University Press, Cambridge, 2002

    2002

  24. [32]

    Bivariant K -theory and the N ovikov conjecture

    Nigel Higson. Bivariant K -theory and the N ovikov conjecture. Geom. Funct. Anal. , 10(3):563--581, 2000

    2000

  25. [33]

    E -theory and KK -theory for groups which act properly and isometrically on H ilbert space

    Nigel Higson and Gennadi Kasparov. E -theory and KK -theory for groups which act properly and isometrically on H ilbert space. Invent. Math. , 144(1):23--74, 2001

    2001

  26. [34]

    On the coarse B aum- C onnes conjecture

    Nigel Higson and John Roe. On the coarse B aum- C onnes conjecture. In Novikov conjectures, index theorems and rigidity, V ol. 2 ( O berwolfach, 1993) , volume 227 of London Math. Soc. Lecture Note Ser. , pages 227--254. Cambridge Univ. Press, Cambridge, 1995

    1993

  27. [35]

    The strong N ovikov conjecture for low degree cohomology

    Bernhard Hanke and Thomas Schick. The strong N ovikov conjecture for low degree cohomology. Geom. Dedicata , 135:119--127, 2008

    2008

  28. [36]

    Fibre bundles , volume 20 of Graduate Texts in Mathematics

    Dale Husemoller. Fibre bundles , volume 20 of Graduate Texts in Mathematics . Springer-Verlag, New York, third edition, 1994

    1994

  29. [37]

    Kasparov

    Gennadi G. Kasparov. Index of invariant elliptic operators, K -theory and representations of L ie groups. Dokl. Akad. Nauk SSSR , 268(3):533--537, 1983

    1983

  30. [38]

    Kasparov

    Gennadi G. Kasparov. Equivariant KK -theory and the N ovikov conjecture. Invent. Math. , 91(1):147--201, 1988

    1988

  31. [39]

    Groups acting on buildings, operator K -theory, and N ovikov's conjecture

    Gennadi Kasparov and Georges Skandalis. Groups acting on buildings, operator K -theory, and N ovikov's conjecture. K -Theory , 4(4):303--337, 1991

    1991

  32. [40]

    Groups acting properly on ``bolic'' spaces and the N ovikov conjecture

    Gennadi Kasparov and Georges Skandalis. Groups acting properly on ``bolic'' spaces and the N ovikov conjecture. Ann. of Math. (2) , 158(1):165--206, 2003

    2003

  33. [41]

    Hopf algebroids and secondary characteristic classes

    Jerome Kaminker and Xiang Tang. Hopf algebroids and secondary characteristic classes. J. Noncommut. Geom. , 3(1):1--25, 2009

    2009

  34. [42]

    The coarse geometric N ovikov conjecture and uniform convexity

    Gennadi Kasparov and Guoliang Yu. The coarse geometric N ovikov conjecture and uniform convexity. Adv. Math. , 206(1):1--56, 2006

    2006

  35. [43]

    The N ovikov conjecture and geometry of B anach spaces

    Gennadi Kasparov and Guoliang Yu. The N ovikov conjecture and geometry of B anach spaces. Geom. Topol. , 16(3):1859--1880, 2012

    2012

  36. [44]

    John M. Lee. Introduction to R iemannian manifolds , volume 176 of Graduate Texts in Mathematics . Springer, Cham, 2018. Second edition

    2018

  37. [45]

    Novikov's higher signature and families of elliptic operators

    Gheorghe Lusztig. Novikov's higher signature and families of elliptic operators. J. Differential Geometry , 7:229--256, 1972

    1972

  38. [46]

    The N ovikov conjecture for low degree cohomology classes

    Varghese Mathai. The N ovikov conjecture for low degree cohomology classes. Geom. Dedicata , 99:1--15, 2003

    2003

  39. [47]

    Construction of universal bundles

    John Milnor. Construction of universal bundles. II . Ann. of Math. (2) , 63:430--436, 1956

    1956

  40. [48]

    Mis s c enko

    Aleksandr S. Mis s c enko. Infinite-dimensional representations of discrete groups, and higher signatures. Izv. Akad. Nauk SSSR Ser. Mat. , 38:81--106, 1974

    1974

  41. [49]

    Homotopy invariance of higher signatures and 3-manifold groups

    Michel Matthey, Hervé Oyono-Oyono, and Wolfgang Pitsch. Homotopy invariance of higher signatures and 3-manifold groups. Bull. Soc. Math. France , 136(1):1--25, 2008

    2008

  42. [50]

    K -theory for the maximal R oe algebra of certain expanders

    Hervé Oyono-Oyono and Guoliang Yu. K -theory for the maximal R oe algebra of certain expanders. J. Funct. Anal. , 257(10):3239--3292, 2009

    2009

  43. [51]

    Richard S. Palais. On the existence of slices for actions of non-compact L ie groups. Ann. of Math. (2) , 73:295--323, 1961

    1961

  44. [52]

    On the localization algebra of G uoliang Y u

    Yu Qiao and John Roe. On the localization algebra of G uoliang Y u. Forum Math. , 22(4):657--665, 2010

    2010

  45. [53]

    Coarse cohomology and index theory on complete Riemannian manifolds

    John Roe. Coarse cohomology and index theory on complete Riemannian manifolds . Mem. Amer. Math. Soc. , 104(497):1--90, 1993

    1993

  46. [54]

    Index theory, coarse geometry, and topology of manifolds

    John Roe. Index theory, coarse geometry, and topology of manifolds . Regional conference series in mathematics 90. American Mathematical Society, 1996

    1996

  47. [55]

    Warped cones and property A

    John Roe. Warped cones and property A . Geom. Topol. , 9:163--178, 2005

    2005

  48. [56]

    C ^ * -algebras, positive scalar curvature, and the N ovikov conjecture

    Jonathan Rosenberg. C ^ * -algebras, positive scalar curvature, and the N ovikov conjecture. Publications Math \'e matiques de l'IH \'E S , 58:197--212, 1983

    1983

  49. [57]

    The K \" u nneth theorem and the universal coefficient theorem for K asparov's generalized K -functor

    Jonathan Rosenberg and Claude Schochet. The K \" u nneth theorem and the universal coefficient theorem for K asparov's generalized K -functor. Duke Math. J. , 55(2):431--474, 1987

    1987

  50. [58]

    An equivariant novikov conjecture

    Jonathan Rosenberg and Shmuel Weinberger. An equivariant novikov conjecture. K-Theory , 4(1):29--53, 1990

    1990

  51. [59]

    Topological methods for C^ -algebras

    Claude Schochet. Topological methods for C^ -algebras. II . G eometric resolutions and the K \" u nneth formula. Pacific J. Math. , 98(2):443--458, 1982

    1982

  52. [60]

    The coarse B aum- C onnes conjecture and groupoids

    Georges Skandalis, Jean-Louis Tu, and Guoliang Yu. The coarse B aum- C onnes conjecture and groupoids. Topology , 41(4):807--834, 2002

    2002

  53. [61]

    The coarse geometric N ovikov conjecture for subspaces of non-positively curved manifolds

    Lin Shan and Qin Wang. The coarse geometric N ovikov conjecture for subspaces of non-positively curved manifolds. J. Funct. Anal. , 248(2):448--471, 2007

    2007

  54. [62]

    Straightening warped cones

    Damina Sawicki and Jianchao Wu. Straightening warped cones. J. Topol. Anal. , 13(4):933--957, 2021

    2021

  55. [63]

    The B aum- C onnes conjecture and discrete group actions on trees

    Jean-Louis Tu. The B aum- C onnes conjecture and discrete group actions on trees. K -Theory , 17(4):303--318, 1999

    1999

  56. [64]

    Introduction to the Baum-Connes conjecture

    Alain Valette. Introduction to the Baum-Connes conjecture . Lectures in Mathematics. ETH Zürich. Birkhäuser Basel, 2002

    2002

  57. [65]

    Aspects of the novikov conjecture

    Shmuel Weinberger. Aspects of the novikov conjecture. In Geometric and topological invariants of elliptic operators , volume 105 of Contemp. Math. , pages 281--297. Amer. Math. Soc., Providence, RI, 1990

    1990

  58. [66]

    Variations on a theme of Borel: an essay on the role of the fundamental group in rigidity , volume 213 of Cambridge Tracts in Mathematics

    Shmuel Weinberger. Variations on a theme of Borel: an essay on the role of the fundamental group in rigidity , volume 213 of Cambridge Tracts in Mathematics . Cambridge University Press, Cambridge, 2023

    2023

  59. [67]

    Additivity of higher rho invariants and nonrigidity of topological manifolds

    Shmuel Weinberger, Zhizhang Xie, and Guoliang Yu. Additivity of higher rho invariants and nonrigidity of topological manifolds. Communications on Pure and Applied Mathematics , 74(1):3--113, 2021

    2021

  60. [68]

    ^p -coarse Baum-Connes conjecture for ^ q -coarse embeddable spaces

    Jinmin Wang , Zhizhang Xie , Guoliang Yu , and Bo Zhu . ^p -coarse Baum-Connes conjecture for ^ q -coarse embeddable spaces . arXiv e-prints , page arXiv:2411.15070, November 2024

    2024 arXiv

  61. [69]

    Higher index theory for certain expanders and G romov monster groups, I

    Rufus Willett and Guoliang Yu. Higher index theory for certain expanders and G romov monster groups, I . Adv. Math. , 229(3):1380--1416, 2012

    2012

  62. [70]

    Higher index theory for certain expanders and G romov monster groups, II

    Rufus Willett and Guoliang Yu. Higher index theory for certain expanders and G romov monster groups, II . Adv. Math. , 229(3):1762--1803, 2012

    2012

  63. [71]

    Higher index theory

    Rufus Willett and Guoliang Yu. Higher index theory . Cambridge University Press, 2020

    2020

  64. [72]

    Baum- C onnes conjecture and coarse geometry

    Guoliang Yu. Baum- C onnes conjecture and coarse geometry. K -Theory , 9(3):223--231, 1995

    1995

  65. [73]

    Coarse B aum- C onnes conjecture

    Guoliang Yu. Coarse B aum- C onnes conjecture. K -Theory , 9(3):199--221, 1995

    1995

  66. [74]

    Localization algebras and the coarse B aum- C onnes conjecture

    Guoliang Yu. Localization algebras and the coarse B aum- C onnes conjecture. K -Theory , 11(4):307--318, 1997

    1997

  67. [75]

    The N ovikov conjecture for groups with finite asymptotic dimension

    Guoliang Yu. The N ovikov conjecture for groups with finite asymptotic dimension. Ann. of Math. (2) , 147(2):325--355, 1998

    1998

  68. [76]

    The coarse B aum- C onnes conjecture for spaces which admit a uniform embedding into H ilbert space

    Guoliang Yu. The coarse B aum- C onnes conjecture for spaces which admit a uniform embedding into H ilbert space. Invent. Math. , 139(1):201--240, 2000

    2000

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