46 votes
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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". ...
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24 votes
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Nilpotence of the stable Hopf map via framed cobordism

Answer Summary Let $\eta$ be the framed 1-manifold which is the Lie group framing on the circle and let $\nu$ be the Lie group framing on $S^3 = Spin(3)$. I am probably going to conflate these framed ...
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23 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 ...
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23 votes
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Homology theory represented by Madsen-Tillmann spectra

This is an exercise in understanding the Pontrjagin--Thom correspondence. The group $\pi_k(MTO(n) \wedge X_+)$ is in bijection with tuples of a $(n+k)$-manifold $M$, an $n$-dimensional vector ...
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22 votes
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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 ...
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  • 7,891
18 votes
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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 ...
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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 ...
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  • 12.7k
16 votes
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Super-cobordisms

There are a number of technical issues with making what you describe precise, for example: what precisely is a supermanifold with boundary? how can you glue/compose bordisms? etc. I am going to ignore ...
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16 votes
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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 ...
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  • 4,773
16 votes
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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 ...
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  • 4,773
16 votes
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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)....
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  • 7,421
15 votes
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Book recommendation for cobordism theory

Perhaps the Notes on cobordism by Haynes Miller could be of some help too. Another possibility (but geared primarily towards applications in symplectic geometry) is the book V. Guillemin, V. ...
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15 votes
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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}...
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  • 6,496
15 votes
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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 ...
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15 votes
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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 ...
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  • 2,937
14 votes
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Does $\mathfrak{N}_4$ contain at least four distinct elements?

Thom proved that the unoriented bordism is a polynomial ring over $\mathbb{F}_2$ generated by elements $x_i$ with $i$ running over all numbers not of the form $2^k-1$. Thus, $\mathfrak{N}_4 = \mathbb{...
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14 votes

How to write the Thom spectrum representing cobordism as an $\Omega$-spectrum?

A concrete model for Ω^∞ applied to Thom spectra (which is all what we need because Thom spectra are connective) was given by Quinn in his thesis. Very roughly, Ω^∞MG is a simplicial set whose n-...
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14 votes
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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 ...
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  • 33.3k
14 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 ...
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  • 12.7k
13 votes

Book recommendation for cobordism theory

you ask specifically for a book; one (expensive) option is On Thom Spectra, Orientability, and Cobordism, by Rudyak, announced as ... the first guide on the subject of cobordism since Stong's ...
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13 votes
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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{...
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13 votes
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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 ...
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  • 1,212
13 votes
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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 ...
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12 votes
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How do we handle the symmetry condition in nCob and TQFTs?

As Oscar has explained in comments, with the most common definitions it's just not true that $M \sqcup N$ is exactly the same as $N \sqcup M$. But even if you were working with some version of the ...
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  • 26.6k
12 votes
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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 ...
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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 ...
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  • 6,496
12 votes
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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" ...
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  • 26.6k
11 votes
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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 = (\...
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  • 6,496
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 ...
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  • 15.6k
11 votes
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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). ...
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