46
votes
Accepted
Is there an explicit description of a cobordism between $\mathbb{CP}^n$ and $\mathbb{RP}^n\times\mathbb{RP}^n$?
An explicit cobordism is given by Stong:
R. E. Stong, A Cobordism, Proceedings of the American Mathematical Society
Vol. 35, No. 2 (Oct. 1972), pp. 584-586
I do like the short title "A Cobordism".
...
- 10.7k
25
votes
Accepted
Critical dimensions D for "smooth manifolds iff triangulable manifolds"
All smooth manifolds are triangulable, as you say. This follows from Morse theory, which dictates that you only need to know how to triangulate (PL) handle-attachments, which one can do by hand. The ...
- 9,388
24
votes
Elegant proof that any closed, oriented 3-manifold is the boundary of some oriented 4-manifold?
Thom wrote two notes in the proceedings of the "Colloque de Topologie de Strasbourg", which was a topology seminar organized by Ehresmann at that time:
"Quelques propriétés des ...
18
votes
Accepted
What is known about exotic spheres up to stable diffeomorphism?
The inertia group $I_M$ of a closed oriented $d$-manifold $M$ is the subgroup of $\theta_d$ of h-cobordism classes of homotopy spheres $\Sigma$ such that $\Sigma \# M$ is diffeomorphic to $M$.
Wall ...
- 11.9k
17
votes
Does Spin cobordism vanish in dimension $4k-1$?
I believe the bordism groups are nonzero in every dimension after some relatively small finite dimension, just by looking at the Poincaré polynomial in Anderson-Brown-Peterson's earlier paper "Spin ...
- 13.2k
17
votes
Accepted
Sphere spectrum, Character dual and Anderson dual
The Anderson dualizing spectrum $I_\mathbf{Z}$ can be defined as follows. Consider the functor $X\mapsto \mathrm{Hom}(\pi_{-\ast} X,\mathbf{Q/Z})$ from the homotopy category of spectra to graded ...
- 5,550
16
votes
Accepted
Cobordisms and Euler characteristics
René Thom proved that two closed $n$-manifolds $M$ and $N$ are (unoriented) cobordant iff their Stiefel-Whitney numbers agree: for any partition $i_1 + \dotsb + i_k = n$,
$$[M]\frown w_{i-1}(M)w_{i_2}...
- 6,766
16
votes
Accepted
The structure of complex cobordism cohomology of the Eilenberg-Maclane spectrum
One can prove that $\mathrm{Map}(H\mathbf{F}_p,MU)$ is contractible. We know that $H\mathbf{F}_p$ is dissonant (Theorem 4.7 of Ravenel's "Localization with Respect to Certain Periodic Homology ...
- 5,550
16
votes
Accepted
Does every $SL_2\mathbb{C}$ representation of a closed oriented surface extend over a compact oriented three-manifold?
Here is an argument that "most" points in the $SL(2, {\mathbb C})$-character variety $X(F)$ of the surface $F$ do not correspond to representations extendible to 3-manifold groups (as in the question)....
- 9,748
16
votes
Accepted
Oriented bordism in higher dimensions (e.g. $12 \leq d \leq 28$)
Most of the main results needed for this calculation can be found in Wall's paper "Determination of the oriented cobordism ring", but this note by Gwynne might be helpful to express this in ...
- 51.5k
15
votes
Orientable with respect to complex cobordism?
Let me expand a bit on my comments. If $E$ is a nice enough ring spectrum (e.g. an $\mathbb{E}_2$-ring spectrum; there is also a slightly modified version that works for an $\mathbb{E}_1$-ring) then ...
- 13.2k
15
votes
Accepted
Are the symmetric spaces $\operatorname{SU}(n)/{\operatorname{SO}(n)}$ always nontrivial in the bordism rings for $n>2$?
There is a fibration $SU(n) \overset p\to SU(n)/SO(n) \overset j\to BSO(n)$, where the $j$ is the classifying map of $p$, viewed as (the projection of) a principal $SO(n)$-bundle. The Stiefel–Whitney ...
- 2,984
15
votes
Cobordism cohomology of Lie groups
Firstly, for any space $X$ we have an Atiyah-Hirzebruch spectral sequence
$$ H^p(X;MU^q) \Longrightarrow MU^{p+q}(X). $$
The differentials are always torsion-valued, essentially because the higher ...
- 55k
14
votes
Accepted
Twisting bordism classes
The correct definition of bordism should have this built in. More precisely, two singular manifolds $(M_0^n,f_0)$ and $(M_1^n,f_1)$ in a space $X$ are oriented bordant if there exists a smooth ...
- 35k
13
votes
Accepted
Is there a closed 5-manifold $M$ with $w_1(M)w_2(M)\ne 0$?
Note that the third Wu class is $\nu_3 = w_1w_2$, so on a closed connected smooth $n$-manifold $M$, $\operatorname{Sq}^3 : H^{n-3}(M; \mathbb{Z}_2) \to H^n(M; \mathbb{Z}_2)$ is given by $\operatorname{...
- 18.8k
13
votes
Accepted
Cobordism and Kirby calculus
As Golla pointed out that since every smooth $4$-manifold has a handle decomposition, you can draw a Kirby diagram. See the following pretty nice picture from Akbulut's lecture notes (now it is a ...
- 1,292
13
votes
Accepted
About the cohomology of $BG^\delta$. Making a Lie group discrete
I will only attempt to answer your first question. The reason there is no contradiction is that it is not true for arbitrary spaces that $H^{\ast}(X;\mathbb Q) = H^{\ast}(X;\mathbb Z) \otimes \mathbb ...
- 11.9k
12
votes
Accepted
Oriented Bordism Group and Un-Oriented Bordism Group of points $pt$
Unoriented cobordism: can be read off from the structure of the unoriented cobordism ring (calculated in Thom's thesis): $\Omega_6^O = (\mathbb Z/2)^3$, $\Omega_7^O = \mathbb Z/2$, $\Omega_8^O = (\...
- 6,766
12
votes
Accepted
Unoriented bordism with twisted orientation
I think this theory is the same as unoriented bordism, when one tries to make sense of it.
As it stands I don't think it makes sense, because I think that the expression "$\mathbb{Z}^w$" does not ...
- 17.7k
12
votes
Does every $SL_2\mathbb{C}$ representation of a closed oriented surface extend over a compact oriented three-manifold?
In addition to Moishe Kohan's geometric argument, there's also a
bordism-theoretic proof.
$\newcommand{\BDel}{B\mathrm{SL}_2(\mathbb C)^\delta}$
Let $\Omega_*^{\mathrm{SO}}(-)$ denote oriented ...
- 6,766
12
votes
Accepted
Generators and relations for the 2-dimensional unoriented cobordism category
My initial answer was wrong, here's the correct version plus a reference: Turaev-Turner
New generating morphisms: The Mobius strip $\emptyset \rightarrow S^1$ and the "orientation reversing" ...
- 27.8k
11
votes
Accepted
Is there a PL, or topological, bordism hypothesis?
This is addressed in Remark 2.4.30 of Jacob's paper. The PL case has a very nice description but the topological case does not. In particular, there's no difference between framed bordisms in the PL ...
- 27.8k
11
votes
Künneth formulas/theorem for bordism groups and cobordisms?
The Künneth formula for ordinary homology as you present it works only when $R$ is a PID (or more generally of cohomological dimension 1).
For a general well-behaved homology theory[1] (this includes ...
- 16.2k
11
votes
Accepted
Reference on complex cobordism
This is worked out in part 2 of
Adams, J. F., Stable homotopy and generalised homology, Chicago Lectures in Mathematics. Chicago - London: The University of Chicago Press. X, 373 p. £ 3.00 (1974). ...
- 16.2k
11
votes
Accepted
Oriented cobordism classes represented by rational homology spheres
The necessary condition pointed out by Jens Reinhold is also sufficient: any torsion class $x = [M] \in \Omega^{SO}_d$ admits a representative where $M$ is a rational homology sphere.
EDIT: This is ...
- 411
11
votes
Bordism for oriented triangulable manifolds without smooth differentiable structures
I am not sure whether this answers your question in full, but I can suggest an idea. It is based on triangulated, oriented, labelled manifolds. You can drop the orientation and then move from ...
- 6,787
11
votes
Accepted
What motivated Thom to relate the cobordism groups with some homotopy groups?
I did not know Thom so I can't speak to all his personal motivations. But I can speak as someone that has read much of his work carefully and I think I have some insights into this.
The primary ...
- 43.1k
10
votes
Accepted
Is $MGL$ an $H\mathbb{Z}$-algebra?
$MGL$ does not admit a structure of $H\mathbb Z$-module. There are many ways to prove this. As Sean said in the comments, if it were true over $\mathbb C$, topological realization would imply that $MU$...
- 8,702
10
votes
Accepted
Which bordism classes fiber over the circle?
A manifold $M$ fibres over $S^1$ with fibre $F$ if and only if it is isomorphic to the mapping torus $$T(h)=F \times [0,1]/\{(x,0) \sim (h(x),1)\vert x \in F\}$$ of an automorphism $h:F \to F$. ...
- 3,831
10
votes
String Orientation and Level Structures
Dylan's nice paper at https://arxiv.org/abs/1507.05116 answers this question.
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