Any Suspension and Any Homology Sphere Are $2H$-Spaces

D. V. Gugnin

Informatics and Automation, Toric Topology, Group Actions, Geometry, and Combinatorics. Part 2, Tome 318 (2022), pp. 51-65.

Résumé

We prove that the reduced suspension $X = \Sigma Y$ over any finite or countable connected polyhedron $Y$ can be endowed with a two-valued multiplication $\mu \colon X\times X \to \mathrm {Sym}^2 X$ satisfying the unit axiom: $\mu (e,x) = \mu (x,e) = [x,x]$ for all $x\in X$. If $X$ is a sphere $S^m$, $m = 1,3,7$, this is a classical result; for $X=S^2$, this is V. M. Buchstaber's theorem of 1990; and for $X=S^{2k+1}$, $k\ne 0,1,3$, this is our theorem of 2019. We also prove a similar statement for all $X$ that are smoothable homology spheres of arbitrary dimension and for $X=\mathbb R\mathrm P^m$, $m\ge 2$. The proof of one of the main results uses the following statement, which is of independent interest. Let $X$ and $Y$ be connected finite CW complexes and $f\colon X\to Y$ a continuous map inducing an isomorphism in integral homology. Then, for any $n\ge 2$, the map $\mathrm {Sym}^n f\colon \mathrm {Sym}^n X \to \mathrm {Sym}^n\kern 1pt Y$ also induces an isomorphism in integral homology.

Détail

Keywords: symmetric powers, $nH$-spaces, homology spheres.

Bibliographie

[1] V. M. Bukhshtaber, “Functional equations associated with addition theorems for elliptic functions and two-valued algebraic groups”, Russ. Math. Surv., 45:3 (1990), 213–215 | DOI | MR | MR | Zbl

[2] Buchstaber V.M., “$n$-Valued groups: Theory and applications”, Moscow Math. J., 6:1 (2006), 57–84 | DOI | MR | Zbl

[3] V. M. Bukhshtaber and E. G. Rees, “Multivalued groups and Hopf $n$-algebras”, Russ. Math. Surv., 51:4 (1996), 727–729 | DOI | MR | Zbl

[4] Buchstaber V.M., Rees E.G., “Multivalued groups, their representations and Hopf algebras”, Transform. Groups, 2:4 (1997), 325–349 | DOI | MR | Zbl

[5] D. V. Gugnin, “Topological applications of graded Frobenius $n$-homomorphisms”, Trans. Moscow Math. Soc., 2011 (2011), 97–142 | DOI | MR | Zbl

[6] D. V. Gugnin, “Topological applications of graded Frobenius $n$-homomorphisms. II”, Trans. Moscow Math. Soc., 2012 (2012), 167–182 | MR | Zbl

[7] D. V. Gugnin, “Branched coverings of manifolds and $nH$-spaces”, Funct. Anal. Appl., 53:2 (2019), 133–136 | DOI | MR | Zbl

[8] Dold A., “Homology of symmetric products and other functors of complexes”, Ann. Math. Ser. 2, 68 (1958), 54–80 | DOI | MR | Zbl

[9] Dold A., Lectures on algebraic topology, Grundl. Math. Wiss., 200, Springer, Berlin, 1972 | MR | Zbl

[10] Dold A., Puppe D., “Homologie nicht-additiver Funktoren. Anwendungen”, Ann. Inst. Fourier, 11 (1961), 201–312 | DOI | MR | Zbl

[11] Dold A., Thom R., “Quasifaserungen und unendliche symmetrische Produkte”, Ann. Math. Ser. 2, 67 (1958), 239–281 | DOI | MR | Zbl

[12] Hatcher A., Algebraic topology, Cambridge Univ. Press, Cambridge, 2002 | MR | Zbl

[13] Kervaire M.A., “Smooth homology spheres and their fundamental groups”, Trans. Amer. Math. Soc., 144 (1969), 67–72 | DOI | MR | Zbl

[14] Kirby R.C., Siebenmann L.C., “On the triangulation of manifolds and the Hauptvermutung”, Bull. Amer. Math. Soc., 75 (1969), 742–749 | DOI | MR | Zbl

[15] Kirby R.C., Siebenmann L.C., Foundational essays on topological manifolds, smoothings, and triangulations, Ann. Math. Stud., 88, Princeton Univ. Press, Princeton, NJ, 1977 | MR | Zbl

[16] Morton H.R., “Symmetric products of the circle”, Proc. Cambridge Philos. Soc., 63 (1967), 349–352 | DOI | MR | Zbl

[17] Nakaoka M., “Cohomology of symmetric products”, J. Inst. Polytech. Osaka City Univ. Ser. A, 8 (1957), 121–145 | MR | Zbl

[18] T. E. Panov, “On the structure of a Hopf cohomology 2-algebra of four-dimensional manifolds”, Russ. Math. Surv., 51:1 (1996), 155–157 | DOI | MR | Zbl