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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeDec 6th 2020
   • PermaLink
   Author: Urs
   Format: MarkdownItexam giving this theorem its own stand-alone entry, for ease of hyperlinking between _[[MUFr]]_, _[[Todd class]]_ and _[[e-invariant]]_. <a href="https://ncatlab.org/nlab/revision/e-invariant+is+Todd+class+of+cobounding+%28U%2Cfr%29-manifold/1">v1</a>, <a href="https://ncatlab.org/nlab/show/e-invariant+is+Todd+class+of+cobounding+%28U%2Cfr%29-manifold">current</a>

   am giving this theorem its own stand-alone entry, for ease of hyperlinking between MUFr, Todd class and e-invariant.

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeDec 6th 2020
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have added the proof ([here](https://ncatlab.org/nlab/show/e-invariant+is+Todd+class+of+cobounding+(U%2Cfr)-manifold#TheProof)), following Conner-Floyd's proof from 1966 (is there any more recent discussion anywhere ?!) but re-casting their argument elegantly as one single homotopy pasting diagram <a href="https://ncatlab.org/nlab/revision/diff/e-invariant+is+Todd+class+of+cobounding+%28U%2Cfr%29-manifold/3">diff</a>, <a href="https://ncatlab.org/nlab/revision/e-invariant+is+Todd+class+of+cobounding+%28U%2Cfr%29-manifold/3">v3</a>, <a href="https://ncatlab.org/nlab/show/e-invariant+is+Todd+class+of+cobounding+%28U%2Cfr%29-manifold">current</a>

   I have added the proof (here), following Conner-Floyd’s proof from 1966 (is there any more recent discussion anywhere ?!) but re-casting their argument elegantly as one single homotopy pasting diagram

   diff, v3, current

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeDec 18th 2020
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have added mentioning of the version of the construction on manifold with SU-structure (instead of just U-structure) and how this gives Adams's finer $e_{\mathbb{R}}$-invariant in degrees $8 k + 3$ and hence in particular the full [[third stable homotopy group of spheres]]. For the time being I made this a new subsection "Variants" ([here](https://ncatlab.org/nlab/show/e-invariant+is+Todd+class+of+cobounding+(U%2Cfr)-manifold#Variants)). <a href="https://ncatlab.org/nlab/revision/diff/e-invariant+is+Todd+class+of+cobounding+%28U%2Cfr%29-manifold/4">diff</a>, <a href="https://ncatlab.org/nlab/revision/e-invariant+is+Todd+class+of+cobounding+%28U%2Cfr%29-manifold/4">v4</a>, <a href="https://ncatlab.org/nlab/show/e-invariant+is+Todd+class+of+cobounding+%28U%2Cfr%29-manifold">current</a>

   I have added mentioning of the version of the construction on manifold with SU-structure (instead of just U-structure) and how this gives Adams’s finer $e_{\mathbb{R}}$-invariant in degrees $8 k + 3$ and hence in particular the full third stable homotopy group of spheres.

   For the time being I made this a new subsection “Variants” (here).

   diff, v4, current

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeDec 22nd 2020
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded these references: *** An alternative formulation via [[index theory]]: * [[Michael Atiyah]], [[Vijay Patodi]], [[Isadore Singer]], Theorem 4.14 of: _Spectral asymmetry and Riemannian geometry. II_, Mathematical Proceedings of the Cambridge Philosophical Society , Volume 78, Issue 3, 1975, pp. 405-432 ([doi:10.1017/S0305004100051872](https://doi.org/10.1017/S0305004100051872)) Review in: * [[Ulrich Bunke]], [[Niko Naumann]], Section 2 of: _The f-invariant and index theory_, Manuscripta Math. 132, 365–397 (2010) ([arXiv:0808.0257](https://arxiv.org/abs/0808.0257), [https://doi.org/10.1007/s00229-010-0351-7](https://doi.org/10.1007/s00229-010-0351-7)) <a href="https://ncatlab.org/nlab/revision/diff/e-invariant+is+Todd+class+of+cobounding+%28U%2Cfr%29-manifold/6">diff</a>, <a href="https://ncatlab.org/nlab/revision/e-invariant+is+Todd+class+of+cobounding+%28U%2Cfr%29-manifold/6">v6</a>, <a href="https://ncatlab.org/nlab/show/e-invariant+is+Todd+class+of+cobounding+%28U%2Cfr%29-manifold">current</a>

   added these references:

   ---

   An alternative formulation via index theory:

   • Michael Atiyah, Vijay Patodi, Isadore Singer, Theorem 4.14 of: Spectral asymmetry and Riemannian geometry. II, Mathematical Proceedings of the Cambridge Philosophical Society , Volume 78, Issue 3, 1975, pp. 405-432 (doi:10.1017/S0305004100051872)

   Review in:

   • Ulrich Bunke, Niko Naumann, Section 2 of: The f-invariant and index theory, Manuscripta Math. 132, 365–397 (2010) (arXiv:0808.0257, https://doi.org/10.1007/s00229-010-0351-7)

   diff, v6, current