The loop cohomology of a space with the polynomial cohomology algebra
Given a simply connected space X with polynomial cohomology H ∗ ( X ; Z 2 ) , we calculate the loop cohomology algebra H ∗ ( Ω X ; Z 2 ) by means of the action of the Steenrod cohomology operation S q 1 on H ∗ ( X ; Z 2 ) . This calculation uses an explicit construction of the minimal Hirsch filtere...
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Elsevier
2017-12-01
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| Series: | Transactions of A. Razmadze Mathematical Institute |
| Online Access: | http://www.sciencedirect.com/science/article/pii/S2346809217300090 |
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doaj-0fc15dcde76f41188d73efa9e0b2da612020-11-24T21:59:01ZengElsevierTransactions of A. Razmadze Mathematical Institute2346-80922017-12-011713389395The loop cohomology of a space with the polynomial cohomology algebraSamson Saneblidze0A. Razmadze Mathematical Institute, I. Javakhishvili Tbilisi State University 6, Tamarashvili st., Tbilisi 0177, GA, United StatesGiven a simply connected space X with polynomial cohomology H ∗ ( X ; Z 2 ) , we calculate the loop cohomology algebra H ∗ ( Ω X ; Z 2 ) by means of the action of the Steenrod cohomology operation S q 1 on H ∗ ( X ; Z 2 ) . This calculation uses an explicit construction of the minimal Hirsch filtered model of the cochain algebra C ∗ ( X ; Z 2 ) . As a consequence we obtain that H ∗ ( Ω X ; Z 2 ) is the exterior algebra if and only if S q 1 is multiplicatively decomposable on H ∗ ( X ; Z 2 ) . The last statement in fact contains a converse of a theorem of A. Borel (Switzer, 1975, Theorem 15.60). Keywords: Loop space, Polynomial cohomology, Hirsch algebra, Multiplicative resolution, Steenrod operationhttp://www.sciencedirect.com/science/article/pii/S2346809217300090 |
| collection |
DOAJ |
| language |
English |
| format |
Article |
| sources |
DOAJ |
| author |
Samson Saneblidze |
| spellingShingle |
Samson Saneblidze The loop cohomology of a space with the polynomial cohomology algebra Transactions of A. Razmadze Mathematical Institute |
| author_facet |
Samson Saneblidze |
| author_sort |
Samson Saneblidze |
| title |
The loop cohomology of a space with the polynomial cohomology algebra |
| title_short |
The loop cohomology of a space with the polynomial cohomology algebra |
| title_full |
The loop cohomology of a space with the polynomial cohomology algebra |
| title_fullStr |
The loop cohomology of a space with the polynomial cohomology algebra |
| title_full_unstemmed |
The loop cohomology of a space with the polynomial cohomology algebra |
| title_sort |
loop cohomology of a space with the polynomial cohomology algebra |
| publisher |
Elsevier |
| series |
Transactions of A. Razmadze Mathematical Institute |
| issn |
2346-8092 |
| publishDate |
2017-12-01 |
| description |
Given a simply connected space X with polynomial cohomology H ∗ ( X ; Z 2 ) , we calculate the loop cohomology algebra H ∗ ( Ω X ; Z 2 ) by means of the action of the Steenrod cohomology operation S q 1 on H ∗ ( X ; Z 2 ) . This calculation uses an explicit construction of the minimal Hirsch filtered model of the cochain algebra C ∗ ( X ; Z 2 ) . As a consequence we obtain that H ∗ ( Ω X ; Z 2 ) is the exterior algebra if and only if S q 1 is multiplicatively decomposable on H ∗ ( X ; Z 2 ) . The last statement in fact contains a converse of a theorem of A. Borel (Switzer, 1975, Theorem 15.60). Keywords: Loop space, Polynomial cohomology, Hirsch algebra, Multiplicative resolution, Steenrod operation |
| url |
http://www.sciencedirect.com/science/article/pii/S2346809217300090 |
| work_keys_str_mv |
AT samsonsaneblidze theloopcohomologyofaspacewiththepolynomialcohomologyalgebra AT samsonsaneblidze loopcohomologyofaspacewiththepolynomialcohomologyalgebra |
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