1,720,971 research outputs found
Generators of π»*(πππ;πβ) as a module over the Steenrod algebra, and the oriented cobordism ring
No full textIn this paper we will describe a minimal set of
A
A
-generators of
H
β
(
M
S
O
;
Z
2
)
{H^* }(MSO;{Z_2})
(where
A
A
is the
mod
β
2
\bmod {\mathbf { - }}2
Steenrod Algebra). The description is very much analogous to
R
{\text {R}}
. Thomβs description of generators for
H
β
(
M
O
;
Z
2
)
{H^*}(MO;{Z_2})
(see [7]). As a corollary, we give simple cohomological criteria for a manifold to be indecomposable in the oriented cobordism. Our proof relies on work of D. J. Pengelley (see [5]).
1
^{1}
</p
Imbeddings, immersions, and characteristic classes of differentiable manifolds
No full textLet
I
n
i
I_n^i
be the set of
mod
Β -Β
2
\bmod {\text { - }}2
characteristic classes which are of dimension i, and they are zero for all n-dimensional smooth manifolds. Let
I
n
,
k
i
I_{n,k}^i
be the set of i-dimensional
mod
Β -Β
2
\bmod {\text { - }}2
characteristic classes which are zero for all n-dimensional smooth manifolds which immerse in codimension k, (we are talking about normal characteristic classes). Let K be the (graded) ideal in
H
β
(
B
O
,
Z
2
)
{H^ \ast }(BO,{Z_2})
generated by
w
k
+
1
,
w
k
+
2
,
β¦
{w_{k + 1}},{w_{k + 2}}, \ldots
. Then if
i
β©½
(
n
+
k
)
/
2
i \leqslant (n + k)/2
, we have
I
n
,
k
i
=
I
n
i
+
K
i
I_{n,k}^i = I_n^i + {K^i}
. We have some related results for imbedded manifolds, and also for manifolds which immerse or imbed with an SO, U, SU, Spin, etc. structure on the normal bundle.</p
A note on Killing torsion of manifolds by surgery
No full textIn this note we prove that every manifold of dimension different than 3, oriented in a bundle theory (for most bundle theories), is cobordant to a manifold which contains not more torsion than the classifying space of the bundle theory.</p
The image of π»_{β}(π΅ππ;πβ) in π»_{β}(π΅π;πβ)
No full textWe construct explicit polynomial generators of the image of
H
β
(
B
S
O
;
Z
2
)
{H_ * }(BSO;{Z_2})
in
H
β
(
B
O
;
Z
2
)
{H_ * }(BO;{Z_2})
.</p
Killing characteristic classes by surgery
No full textLet M be an n-dimensional
C
β
{C^\infty }
manifold and let
be a characteristic class. Suppose that c as a factor annihilates all the characteristic numbers of M. We prove that if
dim
β‘
c
β©Ύ
(
n
+
1
)
/
2
\dim c \geqslant (n + 1)/2
then M is cobordant to a manifold which has the class c zero, in that way answering in the affirmative a question raised by C. T. C. Wall. We examine the same question for more general cobordism theories, and for
Z
p
{Z_p}
characteristic classes.</p
An identity on algebras over a Hopf algebra
No full textLet A be a connected Hopf algebra which has an associative comultiplication
Ο
:
A
β
A
β
A
\psi :A \to A \otimes A
. Let
Ο
:
A
β
A
\chi :A \to A
be the canonical conjugation on A. Let M be a graded algebra over the Hopf algebra A. If
x
,
y
β
M
,
Ο
(
a
)
=
Ξ£
a
β²
β
a
x,y \in M,\psi (a) = \Sigma aβ \otimes a
, then we have the identity
</p
The image of π»_{β}(π΅ππ;π_{π}) in π»_{β}(π΅π;π_{π})
No full textIn this note we construct explicit polynomial generators for the image of
H
β
(
B
S
U
;
Z
p
)
{H_ * }\left ( {BSU;{Z_p}} \right )
inside
H
β
(
B
U
;
Z
p
)
{H_ * }\left ( {BU;{Z_p}} \right )
.</p
Relations among characteristic classes of π-manifolds imbedded in π ^{π+π}
No full textLet
I
n
β
H
β
(
B
O
;
Z
2
)
{I_n} \subseteq {H^ \ast }(BO;{Z_2})
be the (graded) set of those normal characteristic classes which are zero on all compact, closed
C
β
{C^\infty }
manifolds. Let
I
n
,
k
β
H
β
(
B
O
;
Z
2
)
{I_{n,k}} \subseteq {H^ \ast }(BO;{Z_2})
be the set of those characteristic classes which are zero on all n-manifolds which imbed in
R
n
+
k
{R^{n + k}}
. Let K be the (graded) ideal in
H
β
(
B
O
;
Z
2
)
{H^ \ast }(BO;{Z_2})
generated by the Stiefel-Whitney classes
w
k
,
w
k
+
1
,
w
k
+
2
,
w
k
+
3
,
β¦
{w_k},{w_{k + 1}},{w_{k + 2}},{w_{k + 3}}, \ldots
. We will prove the following result: If
1
β©½
i
β©½
min
{
(
2
k
β
2
)
,
(
n
+
k
β
1
)
/
2
}
1 \leqslant i \leqslant \min \{ (2k - 2),(n + k - 1)/2\}
, then
I
n
,
k
i
=
I
n
i
+
K
i
I_{n,k}^i = I_n^i + {K^i}
. Also, we will prove an analogous result for manifolds with an extra structure.</p
- β¦
