The Homotopy Types of Sp(2)-Gauge Groups over Closed Simply Connected Four-Manifolds
Informatics and Automation, Algebraic topology, combinatorics, and mathematical physics, Tome 305 (2019), pp. 309-329.

Voir la notice de l'article provenant de la source Math-Net.Ru

We determine the number of distinct homotopy types for the gauge groups of principal $\mathrm {Sp}(2)$-bundles over a closed simply connected four-manifold.
Keywords: gauge group, simply connected four-manifold
Mots-clés : homotopy type.
@article{TRSPY_2019_305_a16,
     author = {Tseleung So and Stephen Theriault},
     title = {The {Homotopy} {Types} of {Sp(2)-Gauge} {Groups} over {Closed} {Simply} {Connected} {Four-Manifolds}},
     journal = {Informatics and Automation},
     pages = {309--329},
     publisher = {mathdoc},
     volume = {305},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2019_305_a16/}
}
TY  - JOUR
AU  - Tseleung So
AU  - Stephen Theriault
TI  - The Homotopy Types of Sp(2)-Gauge Groups over Closed Simply Connected Four-Manifolds
JO  - Informatics and Automation
PY  - 2019
SP  - 309
EP  - 329
VL  - 305
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TRSPY_2019_305_a16/
LA  - ru
ID  - TRSPY_2019_305_a16
ER  - 
%0 Journal Article
%A Tseleung So
%A Stephen Theriault
%T The Homotopy Types of Sp(2)-Gauge Groups over Closed Simply Connected Four-Manifolds
%J Informatics and Automation
%D 2019
%P 309-329
%V 305
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TRSPY_2019_305_a16/
%G ru
%F TRSPY_2019_305_a16
Tseleung So; Stephen Theriault. The Homotopy Types of Sp(2)-Gauge Groups over Closed Simply Connected Four-Manifolds. Informatics and Automation, Algebraic topology, combinatorics, and mathematical physics, Tome 305 (2019), pp. 309-329. http://geodesic.mathdoc.fr/item/TRSPY_2019_305_a16/

[1] Atiyah M.F., Bott R., “The Yang–Mills equations over Riemann surfaces”, Philos. Trans. R. Soc. London A, 308 (1983), 523–615 | DOI | Zbl

[2] Bradlow S.B., García-Prada O., Gothen P.B., “Homotopy groups of moduli spaces of representations”, Topology, 47:4 (2008), 203–224 | DOI | Zbl

[3] Choi Y., Hirato Y., Mimura M., “Composition methods and homotopy types of the gauge groups of $Sp(2)$ and $SU(3)$”, Bull. Belg. Math. Soc. - Simon Stevin, 15:3 (2008), 409–417 | Zbl

[4] Cohen F.R., “A course in some aspects of classical homotopy theory”, Algebraic topology, Proc. Workshop (Seattle, WA, 1985), Lect. Notes Math., 1286, Berlin, Springer, 1987, 1–92 | DOI

[5] Crabb M.C., Sutherland W.A., “Counting homotopy types of gauge groups”, Proc. London Math. Soc. Ser. 3, 81:3 (2000), 747–768 | DOI | Zbl

[6] Cutler T., “The homotopy types of $Sp(3)$-gauge groups”, Topology Appl., 236 (2018), 44–58 | DOI | Zbl

[7] Cutler T., Theriault S., The homotopy types of $SU(4)$-gauge groups, E-print, 2019, arXiv: 1909.04643 [math.AT]

[8] Donaldson S.K., “Connections, cohomology and the intersection forms on 4-manifolds”, J. Diff. Geom., 24 (1986), 275–341 | DOI | Zbl

[9] Gottlieb D.H., “Applications of bundle map theory”, Trans. Amer. Math. Soc., 171 (1972), 23–50 | DOI | Zbl

[10] Hamanaka H., Kono A., “Unstable $K^1$-group and homotopy type of certain gauge groups”, Proc. R. Soc. Edinb. Sect. A: Math., 136:1 (2006), 149–155 | DOI | Zbl

[11] Hasui S., Kishimoto D., Kono A., Sato T., “The homotopy types of $\mathrm {PU}(3)$- and $\mathrm {PSp}(2)$-gauge groups”, Algebr. Geom. Topol., 16:3 (2016), 1813–1825 | DOI | Zbl

[12] Hasui S., Kisihmoto D., So T., Theriault S., “Odd primary homotopy types of the gauge groups of exceptional Lie groups”, Proc. Amer. Math. Soc., 147:4 (2019), 1751–1762 | DOI | Zbl

[13] Huang R., Wu J., “Cancellation and homotopy rigidity of classical functors”, J. London Math. Soc. Ser. 2, 99:1 (2019), 225–248 ; arXiv: 1706.02917 [math.AT] | DOI | Zbl

[14] James I.M., “On sphere-bundles over spheres”, Comment. math. Helv., 35 (1961), 126–135 | DOI | Zbl

[15] Kamiyama Y., Kishimoto D., Kono A., Tsukuda S., “Samelson products of $SO(3)$ and applications”, Glasg. Math. J., 49:2 (2007), 405–409 | DOI | Zbl

[16] Kishimoto D., Kono A., “On the homotopy types of $\mathrm {Sp}(n)$ gauge groups”, Algebr. Geom. Topol., 19:1 (2019), 491–502 ; arXiv: 1803.06477 [math.AT] | DOI | Zbl

[17] Kishimoto D., Kono A., Tsutaya M., “On $p$-local homotopy types of gauge groups”, Proc. R. Soc. Edinb. Sect. A: Math., 144:1 (2014), 149–160 | DOI | Zbl

[18] Kishimoto D., Theriault S., Tsutaya M., “The homotopy types of $G_2$-gauge groups”, Topol. Appl., 228 (2017), 92–107 | DOI | Zbl

[19] Kono A., “A note on the homotopy type of certain gauge groups”, Proc. R. Soc. Edinb. Sect. A: Math., 117:3–4 (1991), 295–297 | DOI | Zbl

[20] Kono A., Tsukuda S., “A remark on the homotopy type of certain gauge groups”, J. Math. Kyoto Univ., 36:1 (1996), 115–121 | DOI | Zbl

[21] Lang G.E., \textup {Jr.}, “The evaluation map and $EHP$ sequences”, Pac. J. Math., 44 (1973), 201–210 | DOI

[22] Mimura M., “On the number of multiplications on $\mathrm {SU}(3)$ and $\mathrm {Sp}(2)$”, Trans. Amer. Math. Soc., 146 (1969), 473–492 | Zbl

[23] Mimura M., Toda H., “Homotopy groups of $SU(3)$, $SU(4)$, and $Sp(2)$”, J. Math. Kyoto Univ., 3 (1964), 217–250 | DOI | Zbl

[24] So T., “Homotopy types of $SU(n)$-gauge groups over non-spin $4$-manifolds”, J. Homotopy Relat. Struct., 14:3 (2019), 787–811 ; arXiv: 1807.01931 [math.AT] | DOI | Zbl

[25] So T., “Homotopy types of gauge groups over non-simplyconnected closed 4-manifolds”, Glasg. Math. J., 61:2 (2019), 349–371 | DOI

[26] Sutherland W.A., “Function spaces related to gauge groups”, Proc. R. Soc. Edinb. Sect. A, 121:1–2 (1992), 185–190 | DOI | Zbl

[27] Theriault S.D., “Odd primary homotopy decompositions of gauge groups”, Algebr. Geom. Topol., 10:1 (2010), 535–564 | DOI | Zbl

[28] Theriault S.D., “The homotopy types of $\mathrm {Sp}(2)$-gauge groups”, Kyoto J. Math., 50:3 (2010), 591–605 | DOI | Zbl

[29] Theriault S., “The homotopy types of $SU(3)$-gauge groups over simply connected 4-manifolds”, Publ. Res. Inst. Math. Sci., 48:3 (2012), 543–563 | DOI | Zbl

[30] Theriault S., “The homotopy types of $SU(5)$-gauge groups”, Osaka J. Math., 52:1 (2015), 15–29 | Zbl

[31] Theriault S., “Odd primary homotopy types of $SU(n)$-gauge groups”, Algebr. Geom. Topol., 17:2 (2017), 1131–1150 ; Toda Kh., Kompozitsionnye metody v teorii gomotopicheskikh grupp sfer, Nauka, M., 1982 | DOI | Zbl

[32] H. Toda, Composition Methods in Homotopy Groups of Spheres, Ann. Math. Stud., 49, Princeton Univ. Press, Princeton, NJ, 1962 | Zbl