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window.loswp.shouldDetectTimezone = true; window.loswp.shouldShowBulkDownload = true; window.loswp.showSignupCaptcha = false window.loswp.willEdgeCache = false; window.loswp.work = {"work":{"id":16831269,"created_at":"2015-10-15T06:47:03.886-07:00","from_world_paper_id":142834829,"updated_at":"2024-11-16T06:59:11.825-08:00","_data":{"grobid_abstract":"For a normal covering over a closed oriented topological manifold we give a proof of the L 2 -signature theorem with twisted coefficients, using Lipschitz structures and the Lipschitz signature operator introduced by Teleman. We also prove that the L-theory isomorphism conjecture as well as the C * max -version of the Baum-Connes conjecture imply the L 2signature theorem for a normal covering over a Poincaré space, provided that the group of deck transformations is torsion-free.","publication_date":"2003,,","grobid_abstract_attachment_id":"39206292"},"document_type":"paper","pre_hit_view_count_baseline":null,"quality":"high","language":"en","title":"Various L2-signatures and a topological L2-signature theorem","broadcastable":false,"draft":null,"has_indexable_attachment":true,"indexable":true}}["work"]; window.loswp.workCoauthors = [36282427]; window.loswp.locale = "en"; window.loswp.countryCode = "SG"; window.loswp.cwvAbTestBucket = ""; window.loswp.designVariant = "ds_vanilla"; window.loswp.fullPageMobileSutdModalVariant = "full_page_mobile_sutd_modal"; window.loswp.useOptimizedScribd4genScript = false; window.loginModal = {}; window.loginModal.appleClientId = 'edu.academia.applesignon'; window.userInChina = "false";</script><script defer="" src="https://accounts.google.com/gsi/client"></script><div class="ds-loswp-container"><div class="ds-work-card--grid-container"><div class="ds-work-card--container js-loswp-work-card"><div class="ds-work-card--cover"><div class="ds-work-cover--wrapper"><div class="ds-work-cover--container"><button class="ds-work-cover--clickable js-swp-download-button" data-signup-modal="{"location":"swp-splash-paper-cover","attachmentId":39206292,"attachmentType":"pdf"}"><img alt="First page of “Various L2-signatures and a topological L2-signature theorem”" class="ds-work-cover--cover-thumbnail" src="https://0.academia-photos.com/attachment_thumbnails/39206292/mini_magick20190223-4576-r4v8zm.png?1550930594" /><img alt="PDF Icon" class="ds-work-cover--file-icon" src="//a.academia-assets.com/images/single_work_splash/adobe_icon.svg" /><div class="ds-work-cover--hover-container"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span><p>Download Free PDF</p></div><div class="ds-work-cover--ribbon-container">Download Free PDF</div><div class="ds-work-cover--ribbon-triangle"></div></button></div></div></div><div class="ds-work-card--work-information"><h1 class="ds-work-card--work-title">Various L2-signatures and a topological L2-signature theorem</h1><div class="ds-work-card--work-authors ds-work-card--detail"><a class="ds-work-card--author js-wsj-grid-card-author ds2-5-body-md ds2-5-body-link" data-author-id="36282427" href="https://independent.academia.edu/SchickT"><img alt="Profile image of Thomas Schick" class="ds-work-card--author-avatar" src="//a.academia-assets.com/images/s65_no_pic.png" />Thomas Schick</a></div><div class="ds-work-card--detail"><p class="ds-work-card--detail ds2-5-body-sm">2003</p><div class="ds-work-card--work-metadata"><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">visibility</span><p class="ds2-5-body-sm" id="work-metadata-view-count">…</p></div><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">description</span><p class="ds2-5-body-sm">36 pages</p></div><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">link</span><p class="ds2-5-body-sm">1 file</p></div></div><script>(async () => { const workId = 16831269; const worksViewsPath = "/v0/works/views?subdomain_param=api&work_ids%5B%5D=16831269"; const getWorkViews = async (workId) => { const response = await fetch(worksViewsPath); if (!response.ok) { throw new Error('Failed to load work views'); } const data = await response.json(); return data.views[workId]; }; // Get the view count for the work - we send this immediately rather than waiting for // the DOM to load, so it can be available as soon as possible (but without holding up // the backend or other resource requests, because it's a bit expensive and not critical). const viewCount = await getWorkViews(workId); const updateViewCount = (viewCount) => { try { const viewCountNumber = parseInt(viewCount, 10); if (viewCountNumber === 0) { // Remove the whole views element if there are zero views. document.getElementById('work-metadata-view-count')?.parentNode?.remove(); return; } const commaizedViewCount = viewCountNumber.toLocaleString(); const viewCountBody = document.getElementById('work-metadata-view-count'); if (!viewCountBody) { throw new Error('Failed to find work views element'); } viewCountBody.textContent = `${commaizedViewCount} views`; } catch (error) { // Remove the whole views element if there was some issue parsing. document.getElementById('work-metadata-view-count')?.parentNode?.remove(); throw new Error(`Failed to parse view count: ${viewCount}`, error); } }; // If the DOM is still loading, wait for it to be ready before updating the view count. if (document.readyState === "loading") { document.addEventListener('DOMContentLoaded', () => { updateViewCount(viewCount); }); // Otherwise, just update it immediately. } else { updateViewCount(viewCount); } })();</script></div><p class="ds-work-card--work-abstract ds-work-card--detail ds2-5-body-md">For a normal covering over a closed oriented topological manifold we give a proof of the L 2 -signature theorem with twisted coefficients, using Lipschitz structures and the Lipschitz signature operator introduced by Teleman. We also prove that the L-theory isomorphism conjecture as well as the C * max -version of the Baum-Connes conjecture imply the L 2signature theorem for a normal covering over a Poincaré space, provided that the group of deck transformations is torsion-free.</p><div class="ds-work-card--button-container"><button class="ds2-5-button js-swp-download-button" data-signup-modal="{"location":"continue-reading-button--work-card","attachmentId":39206292,"attachmentType":"pdf","workUrl":"https://www.academia.edu/16831269/Various_L2_signatures_and_a_topological_L2_signature_theorem"}">See full PDF</button><button class="ds2-5-button ds2-5-button--secondary js-swp-download-button" data-signup-modal="{"location":"download-pdf-button--work-card","attachmentId":39206292,"attachmentType":"pdf","workUrl":"https://www.academia.edu/16831269/Various_L2_signatures_and_a_topological_L2_signature_theorem"}"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span>Download PDF</button></div><div class="ds-signup-banner-trigger-container"><div class="ds-signup-banner-trigger ds-signup-banner-trigger-control"></div></div><div class="ds-signup-banner ds-signup-banner-control"><div id="ds-signup-banner-close-button"><button class="ds2-5-button ds2-5-button--secondary ds2-5-button--inverse"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">close</span></button></div><div class="ds-signup-banner-ctas"><img src="//a.academia-assets.com/images/academia-logo-capital-white.svg" /><h4 class="ds2-5-heading-serif-sm">Sign up for access to the world's latest research</h4><button class="ds2-5-button ds2-5-button--inverse ds2-5-button--full-width js-swp-download-button" data-signup-modal="{"location":"signup-banner"}">Sign up for free<span class="material-symbols-outlined" style="font-size: 20px" translate="no">arrow_forward</span></button></div><div class="ds-signup-banner-divider"></div><div class="ds-signup-banner-reasons"><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Get notified about relevant papers</span></div><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Save papers to use in your research</span></div><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Join the discussion with peers</span></div><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Track your impact</span></div></div></div><script>(() => { // Set up signup banner show/hide behavior: // 1. 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We get associated Γ/Γ k -coverings (X k , Y k ) → (X, Y ). We prove that</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Approximating L 2-Signatures by Their Compact Analogues","attachmentId":39158560,"attachmentType":"pdf","work_url":"https://www.academia.edu/16744152/Approximating_L_2_Signatures_by_Their_Compact_Analogues","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/16744152/Approximating_L_2_Signatures_by_Their_Compact_Analogues"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="1" data-entity-id="100024431" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/100024431/Index_type_invariants_for_twisted_signature_complexes_and_homotopy_invariance">Index type invariants for twisted signature complexes and homotopy invariance</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="102338704" href="https://independent.academia.edu/MoulayBenameur">Moulay Benameur</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Mathematical Proceedings of the Cambridge Philosophical Society, 2014</p><p class="ds-related-work--abstract ds2-5-body-sm">For a closed, oriented, odd dimensional manifold X, we define the rho invariant ρ(X,${\cal E}$,H) for the twisted odd signature operator valued in a flat hermitian vector bundle ${\cal E}$, where H = ∑ ij+1H2j+1 is an odd-degree closed differential form on X and H2j+1 is a real-valued differential form of degree 2j+1. We show that ρ(X,${\cal E}$,H) is independent of the choice of metrics on X and ${\cal E}$ and of the representative H in the cohomology class [H]. We establish some basic functorial properties of the twisted rho invariant. We express the twisted eta invariant in terms of spectral flow and the usual eta invariant. In particular, we get a simple expression for it on closed oriented 3-dimensional manifolds with a degree three flux form. A core technique used is our analogue of the Atiyah–Patodi–Singer theorem, which we establish for the twisted signature operator on a compact, oriented manifold with boundary. The homotopy invariance of the rho invariant ρ(X,${\cal E}$,H)...</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Index type invariants for twisted signature complexes and homotopy invariance","attachmentId":100959040,"attachmentType":"pdf","work_url":"https://www.academia.edu/100024431/Index_type_invariants_for_twisted_signature_complexes_and_homotopy_invariance","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/100024431/Index_type_invariants_for_twisted_signature_complexes_and_homotopy_invariance"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="2" data-entity-id="16744174" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/16744174/On_the_cut_and_paste_property_of_higher_signatures_of_a_closed_oriented_manifold">On the cut-and-paste property of higher signatures of a closed oriented manifold</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="36153155" href="https://independent.academia.edu/WolfgangLueck">Wolfgang Lueck</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Topology, 2002</p><p class="ds-related-work--abstract ds2-5-body-sm">We extend the notion of the symmetric signature (M M , r)3¸L(R) for a compact n-dimensional manifold M without boundary, a reference map r : MPBG and a homomorphism of rings with involutions : 9GPR to the case with boundary *M, where (M M , *M)P(M, *M) is the G-covering associated to r. We need the assumption that C H (*M) 9 % R is R-chain homotopy equivalent to a R-chain complex D H with trivial mth di!erential for n"2m resp. n"2m#1. We prove a glueing formula, homotopy invariance and additivity for this new notion. Let Z be a closed oriented manifold with reference map ZPBG. Let FLZ be a cutting codimension one submanifold FLZ and let F M PF be the associated G-covering. Denote by K (F M ) the mth Novikov}Shubin invariant and by b K (F M ) the mth¸-Betti number. If for the discrete group G the Baum}Connes assembly map is rationally injective, then we use (M M , r) to prove the additivity (or cut and paste property) of the higher signatures of Z, if we have K (F M )"R> in the case n"2m and, in the case n"2m#1, if we have K (F M )"R> and b K (F M )"0. This additivity result had been proved (by a di!erent method) in (On the Homotopy Invariance of Higher Signatures for Mainfolds with Boundary, preprint, 1999, Corollary 0.4) when G is Gromov hyperbolic or virtually nilpotent. We give new examples, where these conditions are not satis"ed and additivity fails.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"On the cut-and-paste property of higher signatures of a closed oriented manifold","attachmentId":39158574,"attachmentType":"pdf","work_url":"https://www.academia.edu/16744174/On_the_cut_and_paste_property_of_higher_signatures_of_a_closed_oriented_manifold","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/16744174/On_the_cut_and_paste_property_of_higher_signatures_of_a_closed_oriented_manifold"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="3" data-entity-id="63085938" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/63085938/A_signature_theorem_for_disk_bundles_and_the_eta_invariant">A signature theorem for disk bundles and the eta invariant</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="60627828" href="https://independent.academia.edu/SanjayTiwariCapeCodMass">Sanjay Tiwari, Cape Cod, Mass.</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Differential Geometry and its Applications, 1993</p><p class="ds-related-work--abstract ds2-5-body-sm">We apply the Signature Theorem of Atiyah, Patodi and Singer [2] to disk bundles over compact Riemannian manifolds. We show that the signature of a disk bundle can be expressed as the integral of a characteristic class over the manifold and a limiting eta invariant of the bounding sphere bundle. The characteristic class can be explicitly described and is modelled after the Hirzebruch L-class. The limiting process is essential and we show by a counter example that in the absence of the limit, the eta invariant is not a topological invariant. If the base manifold is Hodge, we obtain an expression for the limiting eta invariant in terms of our characteristic class and the bigraded Betti numbers of the base.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"A signature theorem for disk bundles and the eta invariant","attachmentId":75628855,"attachmentType":"pdf","work_url":"https://www.academia.edu/63085938/A_signature_theorem_for_disk_bundles_and_the_eta_invariant","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/63085938/A_signature_theorem_for_disk_bundles_and_the_eta_invariant"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="4" data-entity-id="112757783" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/112757783/Separating_topology_and_number_theory_in_the_Atiyah_Singer_g_signature_formula">Separating topology and number theory in the Atiyah-Singer $g$ -signature formula</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="53299976" href="https://independent.academia.edu/DanielBerend">Daniel Berend</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Duke Mathematical Journal, 1990</p><p class="ds-related-work--abstract ds2-5-body-sm">0. Introduction. One of the most useful results in equivariant differential topology is the celebrated Atiyah-Singer #-signature formula [AS]. In this paper we present a new version of this formula, "separating" its number theoretical and topological ingredients. This enables us to clarify the geometrical meaning of the latter ones and, at the same time, makes the interplay between the topology of smooth cyclic actions and the number theory of the corresponding cyclotomic fields more transparent. For other interactions of this sort see [HZ]. Moreover, we improve (Proposition 2.1 and Theorem 2.2) the g-signature theorem [AS] by proving some integrality results concerning its topological ingredients, the so-called normal quasi-signatures (see Section 2, especially (2.5) and (2.6), for definitions). A distinguishing feature of our approach is that the number-theoretical aspects of the g-signature formula depend only on the representation normal to the fixed point set of the action, and not on the totality of the normal bundles. On the other hand, we show that the topological complexity of the bundles normal to the g-fixed point set contributes to the g-signature only via some special integers {S,}mthe (normal) quasi-signatures of these bundles (see Theorem 2.2). Related ideas, in the special case of semifree cyclic actions with real codimension 2 fixed point set, may be found in [H2, Section 4, Theorem]. Although the derivation of our formula is based on the original Atiyah-Singer theorem, the new version may suggest an alternate, more geometrical, proof of this theorem. Such a proof may, in turn, be extended to a broader category of actions, not necessarily smooth. Let us introduce a few notations. Cm is the cyclic group of order m, and g an arbitrary fixed generator thereof. Cm acts on an even-dimensional, closed manifold M by orientation-preserving diffeomorphisms. Denote by M g the set of Cm-fixed points in M. Throughout the paper, it is assumed that the normal bundle v(M , M) admits an equivariant complex structure (for odd m this assumption is automatically fulfilled). Denote by It an isomorphism class of a typical complex Cm-representation normal to M. Let M be the components of M a having It as the complex slice-type of their generic point. Sign(g, M) stands for the g-signature of M [AS]. This invariant lies in the ring ofintegers Z[2-1 ofthe cyclotomic field Q(2), 2 exp(2rri/m). We shall now describe the plan of the paper. In Section 1 we introduce a notion of g-signature for equivariant, symmetric or antisymmetric, unimodular bilinear</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Separating topology and number theory in the Atiyah-Singer $g$ -signature formula","attachmentId":109889832,"attachmentType":"pdf","work_url":"https://www.academia.edu/112757783/Separating_topology_and_number_theory_in_the_Atiyah_Singer_g_signature_formula","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/112757783/Separating_topology_and_number_theory_in_the_Atiyah_Singer_g_signature_formula"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="5" data-entity-id="53860602" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/53860602/L2_Invariants_Theory_and_Applications_to_Geometry_and_K_Theory">L2-Invariants: Theory and Applications to Geometry and K-Theory</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="36153155" href="https://independent.academia.edu/WolfgangLueck">Wolfgang Lueck</a></div><p class="ds-related-work--metadata ds2-5-body-xs">L2-Invariants: Theory and Applications to Geometry and K-Theory, 2002</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"L2-Invariants: Theory and Applications to Geometry and K-Theory","attachmentId":70503749,"attachmentType":"pdf","work_url":"https://www.academia.edu/53860602/L2_Invariants_Theory_and_Applications_to_Geometry_and_K_Theory","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/53860602/L2_Invariants_Theory_and_Applications_to_Geometry_and_K_Theory"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="6" data-entity-id="25559427" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/25559427/The_signature_with_local_coefficients_of_locally_symmetric_spaces">The signature with local coefficients of locally symmetric spaces</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="49048701" href="https://osu1.academia.edu/MoscoviciHenri">Henri Moscovici</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Tohoku Mathematical Journal, 1985</p><p class="ds-related-work--abstract ds2-5-body-sm">We obtain explicit formulae for the (L 2 -) signature with local coefficients of certain locally symmetric spaces and then apply them to derive non-vanishing criteria for the middle dimensional cohomology.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"The signature with local coefficients of locally symmetric spaces","attachmentId":45890736,"attachmentType":"pdf","work_url":"https://www.academia.edu/25559427/The_signature_with_local_coefficients_of_locally_symmetric_spaces","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/25559427/The_signature_with_local_coefficients_of_locally_symmetric_spaces"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="7" data-entity-id="102569589" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/102569589/The_signature_and_the_elliptic_genus_of_%CF%802_finite_manifolds_with_circle_actions">The signature and the elliptic genus of π2-finite manifolds with circle actions</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="271763399" href="https://independent.academia.edu/rafaelBernalherrera2">rafael Bernal herrera</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Topology and its Applications, 2004</p><p class="ds-related-work--abstract ds2-5-body-sm">We prove the vanishing of various characteristic numbers, such as the signature and the A-genus, on manifolds with finite second homotopy group and which admit smooth S 1-actions. More precisely, we prove the vanishing of various coefficientes of the elliptic genus on non-spin π 2-finite manifolds when the S 1-action either satisfies a "parity" condition or has isolated fixed points only.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"The signature and the elliptic genus of π2-finite manifolds with circle actions","attachmentId":102806227,"attachmentType":"pdf","work_url":"https://www.academia.edu/102569589/The_signature_and_the_elliptic_genus_of_%CF%802_finite_manifolds_with_circle_actions","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/102569589/The_signature_and_the_elliptic_genus_of_%CF%802_finite_manifolds_with_circle_actions"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="8" data-entity-id="55067778" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/55067778/The_surgery_exact_sequence_K_theory_and_the_signature_operator">The surgery exact sequence, K-theory and the signature operator</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="36282427" href="https://independent.academia.edu/SchickT">Thomas Schick</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Annals of K-Theory, 2016</p><p class="ds-related-work--abstract ds2-5-body-sm">In this paper, we study the space of metrics of positive scalar curvature using methods from coarse geometry. Given a closed spin manifold M with fundamental group Γ, Stephan Stolz introduced the positive scalar curvature exact sequence, in analogy to Wall's surgery exact sequence in topology. It calculates a structure group of metrics of positive scalar curvature on M (the object we want to understand) in terms of spin-bordism of BΓ (the classifying space of Γ) and a further group R spin (Γ). Higson and Roe introduced a K-theory exact sequence → K * (BΓ) Remark 1.19. It is important to point out that in contrast to the ρ-class ρ(g) ∈ K n+1 (D * (X) Γ), the ρ Γ-class ρ Γ (g) ∈ K n+1 (D * Γ) vanishes for groups with torsion, at least for those for which the Baum-Connes conjecture holds. See the fundamental remark appearing in (1.14). This means we expect ρ Γ (g) to be different from zero only for groups Γ with torsion. Basic non-trivial examples of ρ Γ (g) for Γ with torsion are considered in [14]. Notice that the ρ-class is well defined whenever the Dirac operator D X is L 2-invertible; we denote it ρ(D X) in this more general case. In fact, we will sometime employ this notation also for the spin Dirac operator associated to a positive scalar curvature metric. 1.3. Delocalized APS-index theorem Geometric setup 1.20. Let now (W, g W) be a Riemannian spin manifold with boundary of dimension n > 0, complete as metric space ‡. We denote its boundary (M, g M), and we assume † In [33], it is only required that U covers id X. However, as pointed out by Ulrich Bunke, to make sure that U χ(D X) + ∈ D * (X) Γ one needs the stronger assumption. ‡ i.e. every Cauchy sequence converges</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"The surgery exact sequence, K-theory and the signature operator","attachmentId":71119191,"attachmentType":"pdf","work_url":"https://www.academia.edu/55067778/The_surgery_exact_sequence_K_theory_and_the_signature_operator","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/55067778/The_surgery_exact_sequence_K_theory_and_the_signature_operator"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="9" data-entity-id="16943960" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/16943960/De_Rham_Hodge_theory_for_L2_cohomology_of_infinite_coverings">De Rham-Hodge theory for L2-cohomology of infinite coverings</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="36367941" href="https://gc-cuny.academia.edu/J%C3%B3zefDodziuk">Józef Dodziuk</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Topology, 1977</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"De Rham-Hodge theory for L2-cohomology of infinite coverings","attachmentId":39269982,"attachmentType":"pdf","work_url":"https://www.academia.edu/16943960/De_Rham_Hodge_theory_for_L2_cohomology_of_infinite_coverings","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/16943960/De_Rham_Hodge_theory_for_L2_cohomology_of_infinite_coverings"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div></div></div><div class="ds-sticky-ctas--wrapper js-loswp-sticky-ctas hidden"><div class="ds-sticky-ctas--grid-container"><div class="ds-sticky-ctas--container"><button class="ds2-5-button js-swp-download-button" data-signup-modal="{"location":"continue-reading-button--sticky-ctas","attachmentId":39206292,"attachmentType":"pdf","workUrl":null}">See full PDF</button><button class="ds2-5-button ds2-5-button--secondary js-swp-download-button" data-signup-modal="{"location":"download-pdf-button--sticky-ctas","attachmentId":39206292,"attachmentType":"pdf","workUrl":null}"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span>Download PDF</button></div></div></div><div class="ds-below-fold--grid-container"><div class="ds-work--container js-loswp-embedded-document"><div class="attachment_preview" data-attachment="Attachment_39206292" style="display: none"><div class="js-scribd-document-container"><div class="scribd--document-loading js-scribd-document-loader" style="display: block;"><img alt="Loading..." src="//a.academia-assets.com/images/loaders/paper-load.gif" /><p>Loading Preview</p></div></div><div style="text-align: center;"><div class="scribd--no-preview-alert js-preview-unavailable"><p>Sorry, preview is currently unavailable. 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Higher signatures</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="5772910" href="https://illinois.academia.edu/PierreAlbin">Pierre Albin</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2009</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"The signature package on Witt spaces, II. 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I</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="6163290" href="https://independent.academia.edu/PatyIh">Paty Ih</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Inventiones Mathematicae, 1985</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"L 2 -cohomology of normal algebraic surfaces. 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cohomology</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="44268024" href="https://independent.academia.edu/NilsBaas">Nils Baas</a></div><p class="ds-related-work--metadata ds2-5-body-xs">arXiv (Cornell University), 2004</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Topology, Geometry and Quantum Field Theory: Two-vector bundles and forms of elliptic cohomology","attachmentId":109384851,"attachmentType":"pdf","work_url":"https://www.academia.edu/112025518/Topology_Geometry_and_Quantum_Field_Theory_Two_vector_bundles_and_forms_of_elliptic_cohomology","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a 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Chess</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Bulletin of the American Mathematical Society, 1980</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"A Poincaré-Hopf type theorem for the de Rham invariant","attachmentId":108233610,"attachmentType":"pdf","work_url":"https://www.academia.edu/110406656/A_Poincar%C3%A9_Hopf_type_theorem_for_the_de_Rham_invariant","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/110406656/A_Poincar%C3%A9_Hopf_type_theorem_for_the_de_Rham_invariant"><span class="ds2-5-text-link__content">View PDF</span><span 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