The cobordism tag has no wiki summary.
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Cobordism modulated by a cohomology operation
I've recently encountered the following cobordism theory modulated by a class $\sigma \in H^{d+1}(B^2\mathbb{Z}/2,U(1))$.
My objects are $d$-dimensional spin manifolds with chosen spin structure. ...
3
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0answers
100 views
Computing the spin cobordism groups of a CW complex from its cohomology groups
It is shown in Conner and Floyd's book "Differentiable periodic maps", Theorem 14.2, that the oriented bordism group of a CW-complex $X$ can be computed by
$\Omega_p(X) = \sum_{q = 0}^p ...
3
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1answer
186 views
$(BU,f)$ structures on manifolds via stable normal bundles and stable tangent bundles
Background
Consider $BU=colim \, BU_k$ where we take $BU_k$ to be the specific model of classifying space for the group $U(k)\subseteq O(2k)$ given by the quotient space of the infinite real Stiefel ...
5
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1answer
224 views
Construction of Thom-Spectrum for G_2-Structures
The motivation to this question is the paper of Crowley and Nordstrøm "A New Invariant of $G_2$-Structures". I am trying to find a homotopy theoretic interpretation of the following geometric ...
3
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0answers
160 views
(∞,n)-category of triangulated cobordisms
What is an accepted definition of a (∞,n)-category of triangulated cobordisms?
Is there one that has a forgetful functor to (Rezk - Hopkins -) Lurie's smooth cobordisms? Does it shed light on how ...
10
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2answers
432 views
Does a smooth homeomorphism of closed manifolds preserve cobordism fundamental class?
Let $f:M\to N$ be a smooth map of closed oriented smooth manifolds which is also a homeomorphism. Let $[M]\in H_\bullet(M;\mathbb Z)$ denote the fundamental class (and similarly for $N$). It is ...
7
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2answers
278 views
Computing a cobordism group of manifolds endowed with a real vector bundle with constraints on the Stiefel-Whitney classes
I am interested in computing the cobordism group of oriented manifolds $M$ of dimension 7 endowed with real vector bundles $N$ of rank 5 with the following conditions on the Siefel-Whitney classes:
$ ...
11
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1answer
280 views
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 ...
10
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1answer
328 views
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 ...
13
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1answer
467 views
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 ...
6
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1answer
529 views
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 ...
3
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1answer
296 views
Connection between complex orientations and R-orientations for a ring spectrum R?
We have a well defined notion of complex orientation for a spectrum (coh. theory) $E$, that is, we have a class $x_E\in \tilde{E}^2(\mathbb{C}P^\infty)$ which restricts to identity along the inclusion ...
4
votes
1answer
179 views
Seifert surfaces via Alexander duality
If we take a knot $K$ in $S^3$, there are several ways to construct the associated Seifert surface. One way, which I am not familiar with, I just came across in a paper I am reading. It goes like ...
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1answer
283 views
Diffeomorphism coming from the s-cobordism theorem
Let $f:M_0\rightarrow M_1$ be a diffeomorphism between two compact $n$-dimensional manifolds. Let $W^{n+1}$ be an $h$-cobordism between $M_0$ and $M_1$. Assume that the cobordism has no torsion and ...
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207 views
Is the pushout of smooth varieties along a smooth closed subvariety again a variety?
The following question is motivated by a desire to find a rough analog in algebraic geometry of the usual notion of gluing of smooth bordisms.
Suppose k is an algebraically closed field of ...
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2answers
612 views
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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138 views
For a finite flat (etale?) morphism $f:Y\to X$, is $f_*1_Y-\deg f . 1_X$ nilpotent in $A^0(X)$, where $A^*$ is the algebraic cobordism?
Let $A^\ast$ be an algebraic oriented cohomology theory (i.e. it is equipped with certain push-forwards for projective morphisms of smooth varieties over the base field $k$; see section 2 of ...
4
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2answers
306 views
(Infinite) Suspension Functor on the Pontryagin-Thom Construction
This is a slightly revamped version of a question I asked on the stackexchange forum. That question was asking if the Pontryagin-Thom constructon respects the suspension operation, alluding to stable ...
1
vote
1answer
287 views
Geometry Realization of Homology Class
Hello!
My question is about the realization of homology class.
The definition of the realizaion of homology class is: for manifold M and a homology class $z\in H_k(M)$, k is an integer. If we find a ...
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2answers
269 views
Coherent MU_*-Modules
It is proven by Thom that for a finite cw-complex $X$, its $MU$-homology, which, in honor of the authors I'm currently reading, I'll denote by $\Omega_\ast^U(X)$, is a coherent module over ...
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2answers
347 views
Orientation of complex bordism spectrum
I have the following question: If $E$ is a ring spectrum, then a complex orientation of $E$ is an element of $E^2(\mathbb{C}P^{\infty})$ that is mapped to $1$ in $E^2(\mathbb{C}P^{1})$.
I have read ...
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0answers
87 views
Extending psc metrics
Let $S^1$ denote the circle with the non-trivial spin structure, i.e. $0\neq[S^1]\in\Omega^{Spin}_1$. Considering characteristic numbers it is easy so see that $S^1\times\mathbb{H}P^3$ is spin null ...
5
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1answer
419 views
Atiyah-Bott-Shapiro Orientation
Dear community,
there are so-called orientation maps $a:MSpin\to ko$ and $b:MSpin^c \to k$, "defined" in ABS's paper "Clifford modules". Unfortunately I am not familiar with representation theory.
...
4
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1answer
179 views
Computation of KO characteristic classes/numbers
How to compute KO characteristic classes/numbers?
They were introduced by Anderson/Brown/Peterson to study the structure of the spin cobordism ring. I looked through the literature but I did not find ...
4
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4answers
501 views
Characteristic classes detecting nontrivial fiberwise homotopy of sphere bundles
I am looking for characteristic classes of vector bundles (either real or complex) with values in generalized multiplicative cohomology theories such that:
i) they vanish if the bundle of unit ...
9
votes
1answer
567 views
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 ...
8
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273 views
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, ...
3
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0answers
167 views
Extension of homeomorphism of boundaries to a homeomorphism of a cobordism
Suppos we have a cobordism $(M, \partial_{-}M, \partial_{+}M)$, where $M$ is a oriented compact (topological) 3-manifold.
Assume we have orientation preserving homeomorphism $f_{\pm}: \Sigma \to ...
1
vote
1answer
233 views
Mapping class group and cylindrical structure
Let us fix a torus $\Sigma=S^1 \times S^1$. We consider a cylinder $\Sigma \times I$ and a data $(\Sigma\times I, \Sigma\times 0, \Sigma\times 1, f_{0},f_{1})$. Here $f_{i}$, called parametrization, ...
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Isomorphism of cobordisms
Let $(M, \partial_{-}M, \partial_{+}M)$ be a decorated 3-cobordism, where $M$ be a (topological) 3-manifold and $\partial M=-\partial_{-}M \cup \partial_{+}M$.
(decorated in a sense of Turaev, Quantum ...
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1answer
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Embedded (framed) cobordisms
[The title initially was "Actions of gauge groups on framed cobordisms. This has been changed.]
This question is a follow-up to my answer to When is a submanifold of $\mathbf R^n$ given by global ...
12
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1answer
390 views
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 ...
6
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1answer
408 views
Does a fixed-point free “homotopy involution” imply that a manifold bounds?
Let $M^n$ be a closed (compact, connected, without boundary) smooth manifold. It is known that if there exists a fixed point free involution $\tau:M \rightarrow M$, then M bounds. That is, there ...
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2answers
661 views
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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3answers
367 views
Question concerning h-cobordisms
Suppose we have a cobordism $W$ of manifolds $M_0$ and $M_1$ and suppose the inclusion of $M_0$ into $W$ is a homotopy equivalence. Is the same true for the inclusion of $M_1$ (ie. is $W$ already an ...
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The signature of a mapping torus
Consider a manifold $M$ of dimension $4k + 2$, $k$ an integer. Pick a diffeomorphism $\phi$ of $M$ and construct the mapping torus $T$ of $\phi$. Suppose that there is a $4k+4$ dimensional manifold ...
6
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1answer
303 views
Third bordism group of BG, where G is an arbitrary compact Lie group.
Is anything known about $\Omega_3(BG)$, where $G$ is an arbitrary compact Lie group; i.e., is it possible to describe the structure of $\Omega_3(BG)$ for any compact Lie group? I know that $H_3(BG)$ ...
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3answers
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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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3answers
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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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1answer
339 views
Fill in the blanks: “1Cob is the free ____ category on a ____”
This is probably straightforward, but I'm having trouble writing down a precise statement. "Everyone knows" that the cobordism category $\text{2Cob}$ (all manifolds compact and oriented) is the free ...
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2answers
664 views
Simple examples of equivariant homology and bordism
I'm looking for simple examples of calculations of equivariant homology and of equivariant bordism.
I have a finite group G acting on an CW-complex X. I would like to calculate the equivariant ...
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1answer
176 views
what does the coefficients ring of generalized cohomology defined by the unitary Thom spectrum like?
Let $MU$ be the unitary Thom spectrum, then it gives a generalized cohomology,
so what is the coefficients $MU^*(point)$ like?
Is it just the complex cobordism ring $\Omega_U^*?$
5
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2answers
484 views
Definition of the Kervaire invariant for normal maps (as in Browder's book)
Browder's book "Surgery on simply-connected manifolds" defines the Kervaire invariant in a very general setting. My question is: how does one get the more usual definition of the invariant for a ...
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1answer
769 views
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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4answers
602 views
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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1answer
662 views
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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351 views
Casson Gordon paper - Cobordism of classical knots
It is given in Progress in mathematics 62, Guillou and Marin book. In the proof of Lemma 4, They choose $\alpha$ and $r\in \mathbb{N}$ such that $h^r_*\colon H_1(X;Z_p)\to H_1(X;Z_p)$ satisfies ...
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1answer
561 views
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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454 views
Reference request for relative bordism coinciding with homology in low dimensions
It's a standard fact that, for finite CW complexes, the relative (edit: oriented) bordism group $\Omega_n(X,A)$ coincides with the homology $[H_\ast(X,A;\Omega_\ast(pt))]_n\simeq H_n(X,A)$ for ...
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1answer
904 views
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 ...
