nLab 4-manifold

Idea

A 4-manifold is a manifold of dimension 4.

Examples

• 4-sphere

• $U(2)$

• spacetime

• $E_8$ manifold

Properties

Cohomotopy

Let $X$ be a 4-manifold which is connected and oriented.

The Pontryagin-Thom construction as above gives for $n \in \mathbb{Z}$ the commuting diagram of sets

$$\array{ \pi^n(X) &\overset{\simeq}{\longrightarrow}& \mathbb{F}_{4-n}(X) \\ {}^{ \mathllap{h^n} } \downarrow && \downarrow^{ h_{4-n} } \\ H^n(X,\mathbb{Z}) &\underset{\simeq}{\longrightarrow}& H_{4-n}(X,\mathbb{Z}) \,, }$$

where $\pi^\bullet$ denotes cohomotopy sets, $H^\bullet$ denotes ordinary cohomology, $H_\bullet$ denotes ordinary homology and $\mathbb{F}_\bullet$ is normally framed cobordism classes of normally framed submanifolds. Finally $h^n$ is the operation of pullback of the generating integral cohomology class on $S^n$ (by the nature of Eilenberg-MacLane spaces):

$$h^n(\alpha) \;\colon\; X \overset{\alpha}{\longrightarrow} S^n \overset{generator}{\longrightarrow} B^n \mathbb{Z} \,.$$

Now

• $h^0$, $h^1$, $h^4$ are isomorphisms

• $h^3$ is an isomorphism if $X$ is “odd” in that it contains at least one closed oriented surface of odd self-intersection, otherwise $h^3$ becomes an isomorphism on a $\mathbb{Z}/2$-quotient group of $\pi^3(X)$ (which is a group via the group-structure of the 3-sphere (SU(2)))

(Kirby-Melvin-Teichner 12)

Related concepts

• Yang-Mills instanton

• Donaldson-Thomas invariants

manifolds in low dimension:

• 2-manifolds

• 3-manifolds

• 4-manifolds

• 5-manifolds

• 8-manifolds

References

General

• Neil Strickland, section 13 of: A Bestiary of Topological Objects [pdf, pdf]

All PL 4-manifolds are simple branched covers of the 4-sphere:

• Riccardo Piergallini, Four-manifolds as 4-fold branched covers of $S^4$, Topology Volume 34, Issue 3, July 1995 (doi:10.1016/0040-9383(94)00034-I, pdf)

• Massimiliano Iori, Riccardo Piergallini, 4-manifolds as covers of the 4-sphere branched over non-singular surfaces, Geom. Topol. 6 (2002) 393-401 (arXiv:math/0203087)

On cohomotopy of 4-manifolds:

• Daniel Freed, Karen Uhlenbeck, Appendix B of: Instantons and Four-Manifolds, Mathematical Sciences Research Institute Publications, Springer 1991 (doi:10.1007/978-1-4613-9703-8)

• Robion Kirby, Paul Melvin, Peter Teichner: Cohomotopy sets of 4-manifolds, Geometry & Topology Monographs 18 (2012) 161-190 [arXiv:1203.1608, doi:10.2140/gtm.2012.18.161]

Relation to 2d CFTs via 6d CFT

On KK-compactification of D=6 N=(2,0) SCFT on 4-manifolds to 2d CFTs:

• Abhijit Gadde, Sergei Gukov, Pavel Putrov, Fivebranes and 4-manifolds, in: Arbeitstagung Bonn 2013, Progress in Mathematics 319, Birkhäuser (2016) [arXiv:1306.4320, doi:10.1007/978-3-319-43648-7_7]

• Mykola Dedushenko, Sergei Gukov, Pavel Putrov, Vertex algebras and 4-manifold invariants, Chapter 11 in: Geometry and Physics: Volume I (2018) 249–318 [arXiv:1705.01645, doi:10.1093/oso/9780198802013.003.0011]

• Boris Feigin, Sergei Gukov, $VOA[M_4]$, J. Math. Phys. 61 012302 (2020) [arXiv:1806.02470, doi:10.1063/1.5100059]

In relation to M5-brane elliptic genus:

• Sergei Gukov, Du Pei, Pavel Putrov, Cumrun Vafa, 4-manifolds and topological modular forms, J. High Energ. Phys. 2021 84 (2021) [arXiv:1811.07884, doi:10.1007/JHEP05(2021)084, spire:1704312]

and in relation to defects:

• Jin Chen, Wei Cui, Babak Haghighat, Yi-Nan Wang, SymTFTs and Duality Defects from 6d SCFTs on 4-manifolds, JHEP 2023 208 (2023) [arXiv:2305.09734, doi:10.1007/JHEP11(2023)208]