Tagged Questions
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.
5
votes
0answers
35 views
Does the theorem that genera vanishing on even-dim complex projective bundles are elliptic also apply for integral-valued genera?
Ochanine proved in this paper that for genera taking values in $\mathbb{Q}$-algebras, vanishing on even-dimensional projective bundles is equivalent to being an elliptic genus (i.e. a specialization ...
5
votes
1answer
211 views
+50
Pontryagin square and $\frac{1}{2}(\mathcal{P}(x) -x^2) =x \cup_1 Sq^1 x$
The Pontryagin square, maps $x \in H^2({B}^2\mathbb{Z}_2,\mathbb{Z}_2)$ to $ \mathcal{P}(x) \in H^4({B}^2\mathbb{Z}_2,\mathbb{Z}_4)$. Precisely,
$$
\mathcal{P}(x)= x \cup x+ x \cup_1 2 Sq^1 x.
$$
...
3
votes
0answers
86 views
Sphere spectrum, Thom spectrum, and Madsen-Tillmann bordism spectrum
This is a following up question of Sphere spectrum, Character dual and Anderson dual.
What are the differences and the significances of the following:
(1). Homotopy classes of maps from a Thom ...
6
votes
2answers
390 views
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 ...
3
votes
0answers
74 views
Twisted spin cobordism v.s. KO theory in low dimensions
Based on the background info and this this webpage, here is a more advanced problem:
Question: If we consider a different more subtle twisted structure, like
$${\Omega_d^{(\mathrm{spin} \times G)/N}},...
5
votes
0answers
114 views
Spin cobordism v.s. KO theory in low or in any dimensions
It seems that from this webpage, the spin cobordism is equivalent to KO theory in low dimension.
If we denote the $p$-torsion part (mean $\mathbb{Z}_{p^n}$ for some $n$) $$\Omega_d(BG)_p.$$
...
8
votes
0answers
88 views
Relating bordism groups of $\Omega_{d}^{Spin_c}$ and $\Omega_{d}^{(Spin \times SU(N))/\mathbb{Z}_2}$ to that of $U(N)$
I felt that the earlier question may be too challenging, so let me provide a different angle and more infos to tackle an easier and separate problem.
Let us consider a more explicit a short exact ...
6
votes
0answers
93 views
Bordism groups and a short exact sequence
Let us consider a short exact sequence:
$$
1\to N\to G\to Q \to 1,
$$
where $N$, $Q$, and $G$ can be continuous Lie groups in general (or finite groups).
Suppose I have the data and the computations ...
3
votes
0answers
112 views
Pairing the Arf with Stiefel-Whitney class
The Arf invariant is a nonsingular quadratic form over a field of characteristic 2.
The form that I looked at was:
$$
S(q)=|H^1(M^2,\mathbb{Z}_2)|^{-1/2} \sum_{x\in H^1(M^2,\mathbb{Z}_2)} \exp[\pi \;...
3
votes
0answers
83 views
Cobordant of 5d manifolds, and the generalization of bordisms
Some of the 5-dimensional manifolds are (co)bordant via oriented cobordism.
For example, if I understand correctly, 5-dimensional Dold manifold and Wu manifold are manifolds which are cobordant to ...
4
votes
1answer
128 views
Stable normal framings of parallelizable manifolds
Suppose $M$ is a compact, connected, orientable manifold ($\dim M=m$) with trivial tangent bundle and let $j \colon M \to \mathbb R^n$ be an embedding. Suppose we choose a trivialization of $TM$. Then ...
3
votes
0answers
70 views
Reference Request: Equivariant Symplectic bordism
Non-equivariantly, symplectic bordism has been developed extensively by Ray, Gorbunov, and specially S. Kochman in this memoir: http://dx.doi.org/10.1090/memo/0496 Yet the coefficients ...
3
votes
1answer
277 views
Thom space, homotopy group and cohomology group
In Thom's 1952 paper, Thom showed that the Thom class, the Stiefel–Whitney classes, and the Steenrod operations were all related. He used these ideas to prove in the 1954 paper Quelques propriétés ...
10
votes
2answers
371 views
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 \...
4
votes
1answer
107 views
2-morphisms for Bord(n)
I am currently reading in Boundary Conditions for Topological Quantum Field Theories, Anomalies and Projective Modular Functors, and have a (I guess) pretty basic question for my understanding of the (...
11
votes
1answer
364 views
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 ...
1
vote
0answers
52 views
Does this condition imply symplectic birational cobordism?
From the definition of symplectic birationality given here (https://arxiv.org/pdf/0906.3265.pdf, Definition 2.1), two compact symplectic $2n$-manifolds $(M_{1},\omega_{1}),(M_{2},\omega_{2})$ are ...
10
votes
1answer
316 views
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 $M^n$...
5
votes
1answer
280 views
Relative Steenrod's problem
Thom's theorem states that for every homology class $\alpha \in H_{*}(X)$ there exists an integer $k = k(\alpha)$ such that the class $k\, \alpha$ comes from the fundamental class of an orientable ...
11
votes
0answers
130 views
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 $Y'$, ...
12
votes
0answers
345 views
Functor of points definition of the Thom space
Let $X$ be a space (CW complex) and let $E \to X$ be a vector bundle.
Using the language of $\infty$-categories we can can define the Thom space $T(E)$ as the pointed space representing the ...
3
votes
0answers
127 views
Can the cobordism hypothesis be formulated as a statement about adjoint functors?
I would like to formulate the cobordism hypothesis for general tangential structure as a statement about adjoint $(\infty,1)$-functors.
For a space $Y$ with an action of $O(n)$ let $X=Y\times_{O(n)} ...
3
votes
0answers
89 views
Landweber Exact Functor Theorem for Cohomology
I have seen the Landweber exact functor theorem beeing used to retrieve cohomology theories, in particular singular cohomology and K-Theory.
However the statement of the theorem itself is always ...
10
votes
1answer
326 views
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 $M$ ...
2
votes
1answer
207 views
Oriented Bordism Group and Un-Oriented Bordism Group of points $pt$
Do we know, or are there any References that list down complete oriented and unoriented Bordism Group $Ω_{n,O}(pt)$ and $Ω_{n,SO}(pt)$ of points $pt$ for dimensions $n=1,2,...,10$?
Here are some ...
7
votes
1answer
209 views
What is the growth rate of the number of unoriented cobordism classes?
Let $\Omega_n^O$ denote the abelian group of cobordism classes of closed, unoriented manifolds of dimension $n$, and let $d(n) := \lvert\Omega_n^O\rvert$. What are the asymptotics of $d(n)$?
It's ...
5
votes
1answer
136 views
Manifold bounded by Sp(2) realisable inside End(H^2)?
The Lie group $Sp(2) = \{A\in GL(2,\mathbb{H})\mid A^\dagger A = I \}$ has a variety of nice geometric aspects. One of which is that it is the boundary of the disk bundle $D(V)$ of the rank-1 ...
6
votes
1answer
224 views
More on categories of modules over the algebraic cobordism spectrum
I have the following questions on monoidal model structure(s) for the motivic stable homotopy category $SH(k)$ (where $k$ is a field); certainly, I am also interested in general statements concerning ...
12
votes
0answers
124 views
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 ...
3
votes
1answer
110 views
Submanifold generator of Pontryagin-dual of the torsion subgroup of the $Pin^+$ bordism group (dimension 4th)
I am interested in figuring out the following submanifold generator of a $Pin^+$ Bordism or Cobordism group in the dimension 4, say for a $Pin^+$ cobordism group of the classifying space $BG=BSU(2)$:
...
8
votes
1answer
380 views
Cobordisms and Euler characteristics
I am trying to understand exactly which role the Euler characteristic plays in (smooth) cobordism theory, and especially why the answer seems to depend on the dimensions of the manifolds in question. ...
2
votes
1answer
138 views
Cobordism/bordism group based on orbifolds with corners
We define a geometric homology group of a topological space $X$ as follows: the chain complex $C_{\bullet}$is freely generated by the maps $f$ from a compact oriented orbifold with corners $P$ to $X$, ...
7
votes
3answers
364 views
Is the bordism from disjoint union to connected sum universal for connected manifolds?
Let $M_1$ and $M_2$ be two oriented, connected, closed $n$-manifolds. It is known that the disjoint union $M_1 \sqcup M_2$ and the connected sum $M_1 \# M_2$ are cobordant, via a bordism $\Sigma_{M_1, ...
7
votes
0answers
215 views
Building examples of elements of $\Omega_4(\xi)$ via surgery theory: how to do it?
When computing special bordism groups, I often need to determine existence of (singular) smooth $4$-manifolds with fixed fundamental group and certain properties like the spin behaviour (i.e. being ...
4
votes
1answer
248 views
Is equivariant oriented cobordism finite?
It is known that for $n \not\equiv 0 \mod 4$, the oriented cobordism ring $MSO_n$ is finite. That is, for oriented n-dimensional manifold $Y$, there exists $m\in \mathbb{N}$, such that $mY$ bounds.
...
6
votes
0answers
153 views
Variants and Generalizations of Arf (-Brown-Kervaire) invariants
(1) I encounter the Arf invariants in Kirby-Taylor, Pin structures on low-dimensional manifolds. The form that I looked at was:
$$
S(q)=|H^1(M^2,\mathbb{Z}_2)|^{-1/2} \sum_{x\in H^1(M^2,\mathbb{Z}_2)} ...
5
votes
1answer
269 views
Computation of $\Omega^{Pin-}_{d}(B\mathbb{Z}_2)$ and Smith isomorphism
question: I am looking for the literature with the result or the computation of Pin- bordism group: $\Omega^{Pin-}_{d}(B\mathbb{Z}_2)$. Can someone point out some useful ways to do this or any helpful ...
5
votes
1answer
258 views
Spin bordism group of classifying space $BG$ with a finite Abelian $G$
The spin bordism group for the classifying space $BG$ of group $G$ can be denoted as $\Omega^{Spin}_d(BG)$.
For example, $\Omega^{Spin}_d(pt)$ are computed by Anderson-Brown-Peterson (D. W. Anderson, ...
5
votes
0answers
141 views
Low dimensional homotopy fibration TOP(M) -> TOP(int(M))
In the thesis of Nancy Cardim she proves that for $M$ a topological manifold of dim $\geq 5$ with connected boundary, there exists a homotopy fiber sequence
$C(\partial M)\rightarrow TOP(M) \...
17
votes
1answer
336 views
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 ...
10
votes
3answers
381 views
Spin 4-manifold bounded by a mapping torus of tori
Consider a smooth torus endowed with the non-bounding spin structure. Pick a basis of its first homology and a diffeomorphism inducing the S-transformation
$\left(\begin{array}{cc} 0 & 1 \\-1 &...
6
votes
1answer
329 views
A proof of Theorem 9.2.12. in the Gompf-Stipsicz
I'm seeking for a proof of Theorem 9.2.12. in the Gompf-Stipsicz "4-Manifolds and Kirby Calculus" (for the statement, see the following image). But the textbook omits any proofs and only gives a ...
9
votes
1answer
222 views
Cohomology operations on unoriented cobordism
In unoriented cobordism there exist stable cohomology operations looking similar to Steenrod squares (they were used by Quillen to compute the unoriented cobordism ring with its formal group law ...
5
votes
1answer
391 views
Under what condition is a fiber bundle cobordant to the trivial bundle?
Let $E$ be the total space of a fiber bundle with base $B$ and fiber $F$, where $B$ and $F$ are smooth manifolds.
Under what condition is $E$ unoriented cobordant to $B\times F$?
And what happens ...
6
votes
1answer
337 views
Is $MGL$ an $H\mathbb{Z}$-algebra?
Let $\mathrm{MGL}$ be the $\mathbb{P}^1$-ring spectrum over a field $k$ representing algebraic cobordism. Suppose, for simplicity, that $k$ is of characteristic 0. Let $H\mathbb{Z}$ be the motivic ...
4
votes
0answers
158 views
For which cobordism theories framed manifolds not bound?
If $X$ is a complex and $\xi:X\to BO$ is a map, when is the natural map from the stable stem $\pi_*^S\to \pi_{*}( M\xi)$ injective, where $M\xi$ is the associated Thom spectrum?
For $MO$ or $MU$, ...
6
votes
1answer
184 views
Algebraic cobordism (of a point) outside the geometric diagonal
This question is about the state of current knowledge regarding Voevodsky's algebraic cobordism of a point $\mathrm{MGL}^{*,*}(\mathrm{Spec}\,k)$. That the geometric diagonal $\mathrm{MGL}^{2*,*}(\...
27
votes
1answer
592 views
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.
...
23
votes
6answers
2k views
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 ...
4
votes
1answer
151 views
Constructing a “nice” cobordism
Denote by $\Sigma_g$ the closed, orientable surface of genus $g$. I want to construct a cobordism $M_g$ between $\Sigma_g$ and $\Sigma_{g+1}$ with the following two nice properties:
1) $M_g$ is an ...
