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Products in spin$^c$-cobordism

Hassan Abdallah, Andrew Salch

Abstract

We calculate the mod $2$ spin$^c$-cobordism ring up to uniform $F$-isomorphism (i.e., inseparable isogeny). As a consequence we get the prime ideal spectrum of the mod $2$ spin$^c$-cobordism ring. We also calculate the mod $2$ spin$^c$-cobordism ring ``on the nose'' in degrees $\leq 33$. We construct an infinitely generated nonunital subring of the $2$-torsion in the spin$^c$-cobordism ring. We use our calculations of product structure in the spin and spin$^c$ cobordism rings to give an explicit example, up to cobordism, of a compact $24$-dimensional spin manifold which is not cobordant to a sum of squares, which was asked about in a 1965 question of Milnor.

Products in spin$^c$-cobordism

Abstract

We calculate the mod spin-cobordism ring up to uniform -isomorphism (i.e., inseparable isogeny). As a consequence we get the prime ideal spectrum of the mod spin-cobordism ring. We also calculate the mod spin-cobordism ring ``on the nose'' in degrees . We construct an infinitely generated nonunital subring of the -torsion in the spin-cobordism ring. We use our calculations of product structure in the spin and spin cobordism rings to give an explicit example, up to cobordism, of a compact -dimensional spin manifold which is not cobordant to a sum of squares, which was asked about in a 1965 question of Milnor.
Paper Structure (16 sections, 18 theorems, 35 equations, 2 tables)

This paper contains 16 sections, 18 theorems, 35 equations, 2 tables.

Table of Contents

  1. Introduction and summary of results
  2. Spin$^c$ cobordism
  3. The mod $2$ spin${}^c$-cobordism ring in low degrees
  4. Milnor's $24$-dimensional spin manifold.
  5. Determination of the mod $2$ spin${}^c$-cobordism ring up to inseparable isogeny
  6. Conventions
  7. Funding
  8. Acknowledgements
  9. Preliminaries
  10. Review of the cohomology of the spectra $MSpin^{c}$ and $MSpin$
  11. Review of $MO_*$ and symmetric polynomials in the Stiefel--Whitney classes.
  12. Maps between bordism theories
  13. The $Spin^{c}$ bordism ring in low degrees
  14. The spin${}^c$-cobordism ring in all degrees.
  15. A nonunital subring of the $2$-torsion in the spin${}^c$-cobordism ring.
  16. ...and 1 more sections

Key Result

Theorem A

The subring of the mod $2$ spin${}^c$-cobordism ring $\Omega^{Spin^{c}}_{*}\otimes_{\mathbb{Z}}\mathbb{F}_2$ generated by all homogeneous elements of degree $\leq 33$ is isomorphic to where $I$ is the ideal generated by the relations: The degrees of the generators are as follows: $\beta = [\mathbb{C}P^1]$ is in degree $2$, while $Z_i$ and $T_i$ are each in degree $i$.

Theorems & Definitions (28)

  • Remark 1.1
  • Theorem A: Theorem \ref{['mspinc_subring in main text']}
  • Theorem B: Theorem \ref{['large nonunital subring thm']}
  • Definition 1.2
  • Theorem C: Theorem \ref{['f-iso thm']}
  • Corollary D
  • Corollary E: Corollary \ref{['f-iso cor']}
  • Theorem 2.1
  • Theorem 2.2
  • Definition 2.3
  • ...and 18 more