Questions tagged [cobordism]
Cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary of a manifold.
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Elegant proof that any closed, oriented 3-manifold is the boundary of some oriented 4-manifold?
I'm looking for an elegant proof that any closed, oriented $3$-manifold $M$ is the boundary of some oriented $4$-manifold $B$.
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What manifolds are bounded by RP^odd?
Real projective spaces $\mathbb{R}P^n$ have $\mathbb{Z}/2$ cohomology rings $\mathbb{Z}/2[x]/(x^{n+1})$ and total Stiefel-Whitney class $(1+x)^{n+1}$ which is $1$ when $n$ is odd, so it follows that ...
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Explanation for the Thom-Pontryagin construction (and its generalisations)
In 1950, Pontryagin showed that the n-th framed cobordism group of smooth manifolds was equal to n-th stable homotopy group of spheres:
$$ \lim_{k \to \infty} \pi_{n+k}(S^k) \cong \Omega_n^{\text{...
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Nilpotence of the stable Hopf map via framed cobordism
The Pontryagin-Thom construction shows that the stable homotopy groups of spheres are the same as the groups of stably framed manifolds up to cobordism. Specifically the Hopf map corresponds to the ...
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Book recommendation for cobordism theory
I am planning to organize a seminar on cobordism theory and I'm looking for a reference. Such a reference is preferably a book, but I'm open to other ideas.
The audience is familiar with ...
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Is there an explicit description of a cobordism between $\mathbb{CP}^n$ and $\mathbb{RP}^n\times\mathbb{RP}^n$?
With a little bit of work, one can show that $\mathbb{CP}^n$ and $\mathbb{RP}^n\times\mathbb{RP}^n$ have the same Stiefel-Whitney numbers, so by a theorem of Thom, they are (unorientedly) cobordant.
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Cobordism of orbifolds?
Is it possible to setup classical cobordism theory in the context of orbifolds? For example, let's consider the free abelian group generated by oriented smooth orbifolds and quotient by those which ...
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Chromatic Spectra and Cobordism
I apologize in advance, if some of the things I've written are incorrect.
The cobordism hypothesis states that $\mathbf{Bord}^\mathrm{fr}_n$ is the free symmetric monoidal $(\infty,n)$-category with ...
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Vanishing of characteristic numbers vs vanishing of characteristic classes
A famous result by Thom states that Oriented Bordism classes are determined by characteristic numbers; specifically, two closed manifolds are orientedly bordant if and only if they have the same ...
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Twistings for other cohomology theories
Twistings in cohomology theories have a long history and have been used to great effect. The classical example is cohomology with local coefficients. Using this one can formulate Poincaré duality and ...
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Does Spin cobordism vanish in dimension $4k-1$?
For the purposes of a remark in a paper in preparation, I would like to know if anyone can confirm that $\Omega^{spin}_{4k-1} = 0$.
In the Atiyah-Patodi-Singer paper, Spectral asymmetry and ...
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Every Manifold Cobordant to a Simply Connected Manifold
I am wondering if it is true that every compact, connected, oriented manifold is cobordant to a simply connected manifold.
I believe that some sort of surgery will do the trick. Roughly speaking, I ...
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Super-cobordisms
One can construct the $d$-dimensional bordism category by declaring the objects to be the $(d-1)$-dimensional compact manifolds without boundary and the morphisms the $d$-dimensional bordisms between ...
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Homology theory represented by Madsen-Tillmann spectra
The generalized homology theory of the Thom spectrum $MO=\varinjlim\Sigma^nMTO_n$ is bordism theory:\begin{equation*}\pi_k(MO\wedge X)=\Omega^O_k(X)\end{equation*}These groups form the ring of (...
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Does the bordism homology theory satisfy the weak equivalence axiom?
There is an interesting and important homology theory called bordism. Briefly speaking, a singular manifold in a space $X$ is a pair $(M, f)$ where $M$ is a closed smooth manifold and $f : M \to X$ is ...
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Oriented cobordism classes represented by rational homology spheres
Any homology sphere is stably parallelizable, hence nullcobordant. However, rational homology spheres need not be nullcobordant, as the example of the Wu manifold shows, which generates $\text{torsion}...
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Cobordism categories that don't involve manifolds
In order to capture the various flavors of cobordism into one concept, the notion of a "cobordism category" is introduced. This is an essentially small category $C$, together with finite coproducts, ...
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String Orientation and Level Structures
Atiyah, Bott and Shapiro defined orientations of real and complex K-theory that were later refined to maps of ($E_\infty$-ring) spectra
$$MSpin \to KO$$
and
$$MSpin^c \to KU.$$
Likewise, but more ...
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Explicit cobordism between Wu manifold and Dold manifold P(1,2)?
The Wu manifold $SU(3)/SO(3)$ and the Dold manifold $P(1,2)$, the latter being defined as $(S^1\times \mathbb{C}P^2) / (p, x) \sim (-p, \overline{x})$, are cobordant because they are both generators ...
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Are Thom spectra MU, MSO and K-theory spectra KU, KO modules over some truncations of the sphere spectrum?
The Thom spectrum MO is a module over the ring spectrum π≤0S=HZ, where S is the sphere spectrum.
In particular, MO is equivalent to the Eilenberg-MacLane spectrum Hπ*(MO).
On the other hand, MU and ...
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Does every $SL_2\mathbb{C}$ representation of a closed oriented surface extend over a compact oriented three-manifold?
Let $F$ be a compact oriented surface and $\rho:\pi_1(F)\rightarrow SL_2\mathbb{C}$ be a representation. Does there exist a compact oriented three-manifold $M$ with $\partial M=F$ and a homomorphism $...
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What is known about exotic spheres up to stable diffeomorphism?
In even dimensions $n=2k$ we can define two smooth manifolds $M$ and $N$ to be stably diffeomorphic if they become diffeomorphic after the connect sum with $r$ many copies of $S^k \times S^k$ for some ...
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The structure of complex cobordism cohomology of the Eilenberg-Maclane spectrum
Let $MU$ be the complex bordism spectrum and let $H\mathbb{Z}$ be the Eilenberg-Maclane spectrum.
Is it know what the structure of the complex cobordism cohomology $MU^{*}(H\mathbb{Z})$ is?
EDIT: ...
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Is there a PL, or topological, bordism hypothesis?
The bordism hypothesis says that the $(\infty, n)$-category of smooth, framed $n$-bordisms, $(n-1)$-dimensional boundaries, and corners down to points, is freely generated symmetric monoidal with ...
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Formal group law of unoriented cobordism
It is well known that the formal group law $F_U$ of complex cobordism, expressing the Euler class of a tensor product of complex line bundles, is universal.
Also, the formal group law $F_O$ of ...
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Does the signature admit a homotopy coherent refinement?
Cobordism genera can often be refined to $E_\infty$-orientations in the sense of Ando-Blumberg-Gepner-Hopkins-Rezk:
1) the mod 2 Euler characteristic $MO\to H\mathbb{F}_2$;
2) the $\widehat A$-genus ...
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Does the paper "On the cobordism ring $\Omega_*$ and a complex analogue II" exist?
I've been investigating the Milnor hypersurfaces, and every reference seems to point to the paper by Milnor, "On the cobordism ring $\Omega_*$ and a complex analogue II". Despite my best efforts, I ...
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Units of MO and MU
Real (or complex) cobordism is described by a symmetric ring spectrum MO (or MU respectively) as explained in examples 2.8 and 2.9 here. Associated to such a ring spectrum $R$, we have a unit spectrum ...
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Why Lagrangian cobordism?
There are a good number of quantum topology papers in which a TQFT-like set-up is constructed as a functor to the category of vector spaces from some category of cobordisms which satisfy some "...
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How does quotienting by a finite subgroup act on the framed-cobordism class of a group manifold?
Let $G$ be a connected simple connected compact Lie group, and $\Gamma \subset G$ a finite subgroup. Then (the underlying manifold of) $G$ can be framed by right-invariant vector fields, and this ...
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How is the differential in complex cobordism defined?
This is my first MO question...hopefully it's not a bad one...
Background: As a stable homotopy theorist, I like to think of complex cobordism $MU$ as a ring spectrum. If I needed to get my hands ...
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Cobordism and Kirby calculus
It may be a simple question but I wonder to ask:
Is it possible to draw a homology cobordism between $3$-manifolds by using the techniques of Kirby calculus?
At least, for instance, Brieskorn ...
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Are the symmetric spaces $\operatorname{SU}(n)/{\operatorname{SO}(n)}$ always nontrivial in the bordism rings for $n>2$?
In my recent research, I need to know if the symmetric spaces $\operatorname{SU}(n)/{\operatorname{SO}(n)}$ are always nontrivial in the unoriented and oriented bordism rings for $n>2$. (For the ...
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Unoriented bordism with twisted orientation
The computation of the unoriented bordism group of the point $N_*=\Omega_*^O$ is a classic result.
I would like to know a related bordism group, where we specify the twisted fundamental class $[M]\in ...
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Reference on complex cobordism
I am trying to study a little of algebraic cobordism and I lack background from the classic setting. Hence, I am looking for a textbook or expository writing covering the basics of complex cobordism.
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Can one show corbordism invariance of the Crane-Yetter state-sum using simplicial methods / are there 'Pachner-like' moves for cobordisms?
Let $\mathcal{C}$ denote some Unitary Braided Modular Fusion Category. It is well known that the Crane-Yetter state-sum, $Z_{CY}(\bullet|\mathcal{C})$ is an oriented-cobordism invariant. In other ...
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Simply connected homology cobordisms
I'm looking for interesting examples of a homology 3-sphere $Y$ for which there exists a smooth, simply connected homology cobordism from $Y$ to itself (or simply to another homology 3-sphere $Y'$, ...
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Critical dimensions D for "smooth manifolds iff triangulable manifolds"
I am aware that at least for lower dimensions,
"smooth manifolds iff triangulable manifolds"
at least for dimensions below a certain critical dimensions D.
My question is that for
For ...
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Orientable with respect to complex cobordism?
I have learned that an orientation of a manifold $M$ with respect to ordinary cohomology is an ordinary orientation, that an orientation with respect to complex K-theory is a Spin$^c$ structure, and ...
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Interdependence between A^1 homotopy theory and algebraic cobordism
I would like to learn something about $\mathbb{A}^1$-homotopy theory. I know about standard references on the subject, but before dwelling into studying them I have a doubt which some expert could ...
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Twisting bordism classes
Let $X$ be a reasonable topological space (I'd be happy to assume that $X$ is a smooth closed manifold) and let $f\colon M^n \rightarrow X$ be a continuous map from a smooth oriented $n$-manifold $M^n$...
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How to write the Thom spectrum representing cobordism as an $\Omega$-spectrum?
It is often said [e.g. Atiyah, "Bordism and Cobordism" (1961)] that the Thom spectrum $MSO(i)$ represents oriented cobordism, in the following sense:
\begin{eqnarray}
MSO^n(X,Y) &:=& \lim_{i \...
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Lower bounds for Betti numbers of a manifold given its boundary?
Let $B$ be some compact, path connected $n$-manifold without boundary such that its cobordism class is trivial, so that there exists some other $n+1$ manifold $M$ with $\partial M= B$. While there is ...
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Reference request for TQFT, functoriality
I am reading Turaev's blue book Quantum Invariants of Knots and 3-manifolds.
It is difficult for me to understand the proof of Theorem 1.9 in chapter 4, which says;
The function $(M, \partial_{-}M, \...
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Natural knot homology
All knot homology theories I've seen share a flaw: their definitions explicitly use some combinatorial choices (such as a diagram presentation). The coin, however, has two sides and the other one is ...
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Madsen-Tillmann spectrum $MTE$ of the group $E$ which is defined in Freed-Hopkins's paper
In Freed-Hopkins's paper, the group $E(d)$ is defined to be the subgroup of $O(d)\times\mathbb{Z}_4$ consisting of the pairs $(A,j)$ such that $\det A=j^2$, where $\mathbb{Z}_4=\{\pm1,\pm\sqrt{-1}\}$ ...
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Sphere spectrum, Character dual and Anderson dual
The homotopy groups of the sphere spectrum are the stable homotopy groups of spheres.
However, could you help me to appreciate the mathematical meanings of the following:
What is the significance of ...
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Is the boundary of a manifold topologically unique? [duplicate]
Let $X$ be a manifold without boundary and let $Y$ and $Z$ be two manifolds with boundary such that $X$ is homeomorphic to their interiors: $X \cong Y^\circ \cong Z^\circ$. Does it follow that $Y \...
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Which sets of Stiefel-Whitney characteristic numbers can be realized as coming from a manifold?
EDIT: The original question was answered very quickly (and very nicely!) but the answer leads to a pretty obvious subsequent question, which I will now ask. The original question is maintained for ...
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Is there a closed 5-manifold $M$ with $w_1(M)w_2(M)\ne 0$?
I'm trying to find generating manifolds for the cobordism group $\mathit{MO}_5(K(\mathbb Z/2, 2))\cong (\mathbb Z/2)^4$, which can be represented as the cobordism group of closed 5-manifolds $M$ ...