+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Manifolds and cobordisms +-- {: .hide} [[!include manifolds and cobordisms - contents]] =-- #### Geometry +--{: .hide} [[!include higher geometry - contents]] =-- =-- =-- \tableofcontents ## Idea A *4-manifold* is a [[manifold]] of [[dimension of a manifold|dimension]] 4. ## Examples * [[4-sphere]] * $U(2)$ * [[spacetime]] * [[E8-manifold|$E_8$ manifold]] ## Properties #### Cohomotopy Let $X$ be a [[4-manifold]] which is [[connected topological space|connected]] and [[orientation|oriented]]. The [[Pontryagin-Thom construction]] as [above](cohomotopy#RelationToCobordismGroup) 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 [[normal bundle|normally]] [[framing|framed]] [[cobordism classes]] of [[normal bundle|normally]] [[framing|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]] ([[special unitary group|SU(2)]])) ([Kirby-Melvin-Teichner 12](#KirbyMelvinTeichner12)) ## Related concepts * [[Wall's theorem]] * [[Yang-Mills instanton]] * [[Donaldson-Thomas invariants]] [[!include low dimensional manifolds -- table]] ## References ### General * {#DonaldsonKronheimer97} [[Simon Donaldson]], [[Peter Kronheimer]]: _The Geometry of Four-Manifolds_ (1990, revised 1997), Oxford University Press and Claredon Press, [[oup:52942](https://academic.oup.com/book/52942), [doi:10.1093/oso/9780198535539.001.0001](https://doi.org/10.1093/oso/9780198535539.001.0001), ISBN:978-0198502692, ISSN:0964-9174] * [[Neil Strickland]], section 13 of: *A Bestiary of Topological Objects* \[<a href="https://strickland1.org/courses/bestiary/bestiary.pdf">pdf</a>, [[Strickland-BestiaryTopological.pdf:file]]\] On 4-manifolds and [[Kirby calculus]]: * {#GompfStipsicz99} [[Robert Gompf]], [[András Stipsicz]]: _4-Manifolds and Kirby Calculus_, Graduate Studies in Mathematics **20**, AMS (1999) [[ISBN:978-0-8218-0994-5](https://www.ams.org/books/gsm/020), [doi:10.1090/gsm/020](https://www.ams.org/books/gsm/020)] All [[PL manifold|PL]] [[4-manifolds]] are _simple_ [[branched covers]] of the [[4-sphere]]: * {#Piergallini95} [[Riccardo Piergallini]], _Four-manifolds as 4-fold branched covers of $S^4$_, Topology Volume 34, Issue 3, July 1995 (<a href="https://doi.org/10.1016/0040-9383(94)00034-I">doi:10.1016/0040-9383(94)00034-I</a>, [pdf](https://core.ac.uk/download/pdf/82379618.pdf)) * {#IoriPiergallini02} 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](https://arxiv.org/abs/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](https://link.springer.com/book/10.1007/978-1-4613-9703-8)) * {#KirbyMelvinTeichner12} [[Robion Kirby]], [[Paul Melvin]], [[Peter Teichner]]: *Cohomotopy sets of 4-manifolds*, Geometry & Topology Monographs **18** (2012) 161-190 [[arXiv:1203.1608](https://arxiv.org/abs/1203.1608), [doi:10.2140/gtm.2012.18.161](http://dx.doi.org/10.2140/gtm.2012.18.161)] Via [[topologically twisted D=4 super Yang-Mills theory]]: * Niklas Garner, Martijn Kool, Jan Manschot, Yuuji Tanaka: *Quantum field theory and four-manifolds: Perspectives from physics and geometry* [[arXiv:2610.30288](https://arxiv.org/abs/2610.30288)] ### 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](https://arxiv.org/abs/1306.4320), [doi:10.1007/978-3-319-43648-7_7](https://doi.org/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](https://arxiv.org/abs/1705.01645), [doi:10.1093/oso/9780198802013.003.0011](https://doi.org/10.1093/oso/9780198802013.003.0011)] * [[Boris Feigin]], [[Sergei Gukov]], $VOA[M_4]$, J. Math. Phys. **61** 012302 (2020) [[arXiv:1806.02470](https://arxiv.org/abs/1806.02470), [doi:10.1063/1.5100059](https://doi.org/10.1063/1.5100059)] In relation to [[M5-brane elliptic genus]]: * {#GukovPeiPutrovVafa18} [[Sergei Gukov]], [[Du Pei]], [[Pavel Putrov]], [[Cumrun Vafa]], *4-manifolds and topological modular forms*, J. High Energ. Phys. **2021** 84 (2021) [[arXiv:1811.07884](https://arxiv.org/abs/1811.07884), <a href="https://doi.org/10.1007/JHEP05(2021)084">doi:10.1007/JHEP05(2021)084</a>, [spire:1704312](https://inspirehep.net/literature/1704312)] and in relation to [[QFT with defects|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](https://arxiv.org/abs/2305.09734), <a href="https://doi.org/10.1007/JHEP11(2023)208">doi:10.1007/JHEP11(2023)208</a>] [[!redirects 4-manifolds]]